Year 6 Angles, Shapes and Circles: worked examples and questions with answers
By the end of this chapter you will be able to
- Find a missing angle on a straight line (180 degrees), at a point (360 degrees) and opposite a crossing
- Find the missing angles in triangles and quadrilaterals, and the angles of an isosceles triangle
- Work out the interior angles of a polygon, and each angle of a regular polygon
- Name the radius, diameter and circumference of a circle, and use the diameter is twice the radius
- Work out the circumference of a circle using pi = 3.14 or 22/7
- Draw a shape from its sides and angles, and count the faces, edges and vertices of a 3-D shape
- Check whether a net folds up into a cube
You should already know
- Angles and shapes: acute, obtuse and reflex angles, and angles on a line and at a point (Year 5, Chapter 9)
- Perimeter of rectangles and other shapes (Year 5, Chapter 8; Year 6, Chapter 9)
- Multiplying and dividing decimals (Year 6, Chapter 5)
- Simple equations with one unknown, used for missing angles (Year 6, Chapter 8)
Where this leads
- Coordinates, reflection and translation of shapes (Year 6, Chapter 11)
- Circle area, and angles in parallel lines and circles (KS3)
- Using angle facts to prove geometric results (KS3)
National Curriculum for England, Key Stage 2 mathematics, Year 6. Content: Draw 2-D shapes using given dimensions and angles, Recognise, describe and build simple 3-D shapes, including making nets, Compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals and regular polygons, Illustrate and name parts of circles, including radius, diameter and circumference, and know that the diameter is twice the radius, Recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles.
Exercise P Do you remember?
Non-calculatorKey words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Compare the angle with a right angle, which is 90°.
- Look at the size of the angle. A right angle is 90 degrees. This angle is bigger than that.110 > 90
- Compare it with a straight line. An obtuse angle is bigger than 90 and smaller than 180.90 < 110 < 180
- Write the answer. obtuse
Check. An acute angle is less than 90 degrees, and a reflex angle is more than 180 degrees. ✓
Watch out. Obtuse angles are between 90 and 180 degrees.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Take the smaller number away from the bigger one.
- Subtract the smaller number from the bigger one. The answer is the difference between the two numbers.180 - 67 = 113
- Write the answer. 113
Check. 113 + 67 = 180. ✓
Watch out. Check by adding the answer back on.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Add the lengths of all the sides.
- Add the lengths of all the sides. The perimeter is the distance all the way round the outside.9 + 4 + 9 + 4 = 26
- Write the answer. The perimeter is in cm.26
Check. Go round the shape once more and check that each side is counted once. ✓
Watch out. Count every side once. Do not forget the sides you cannot see on a drawing.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Think about the shape with square faces.
- Count the faces. Six square faces, all the same size, close up to make a box with equal sides.six square faces
- Name the shape. A 3-D shape with six equal square faces is a cube.cube
Check. A cube has 6 faces, 12 edges and 8 vertices.
Watch out. A cuboid has rectangular faces. A cube has square faces.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Multiply the digits first, then put the decimal point back in the answer.
- Multiply as if there were no decimal point. Ignore the point for now.42 × 3 = 126
- Put the decimal point back. 4.2 has 1 decimal place, so the answer has 1 decimal place.12.6
Check. 12.6 ÷ 3 = 4.2. ✓
Watch out. Count the decimal places in the question and keep the same number in the answer.
Angles on a line, at a point and opposite
An angle is measured in degrees. Three facts help you find a missing angle.
On a straight line the angles add up to 180°, which is half a turn.
At a point the angles add up to 360°, which is a full turn.
When two straight lines cross, the angles that face each other are called vertically opposite angles. They are equal. The angles next to each other around a crossing add up to 180°.
Always name the fact you used, and then show the sum you did.
Key ideas
- Angles on a straight line add up to 180°.
- Angles at a point add up to 360°.
- Vertically opposite angles are equal.
- Angles next to each other on a straight line make 180° together.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a straight line.
- Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.37° = 37°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 37 = 143
- Write the answer. The missing angle is 143 degrees.143
Check. 143 + 37 = 180, so the angles add up correctly. ✓
Watch out. Use 180 for a straight line, not 360. The angles at a point add up to 360.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for angles at a point.
- Remember the total for angles at a point. Angles around a point add up to 360 degrees (a full turn).Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.90° + 105° + 60° = 255°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 255 = 105
- Write the answer. The missing angle is 105 degrees.105
Check. 105 + 255 = 360, so the angles add up correctly. ✓
Watch out. Use 360 for angles at a point, not 180. A straight line is only 180.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Use the two facts for crossing lines.
- Find c, the vertically opposite angle. Vertically opposite angles are equal.c = 124°
- Find b, the angle next to 124°. The two angles on a straight line add up to 180 degrees.180 - 124 = 56, so b = 56°
- Write the answers. c is the same as the given angle, and b is the rest of the line.124 and 56
Check. 124 + 56 = 180, the straight line. ✓
Watch out. Check your answers with the other fact. The two angles give a straight line.
Exercise 10A Practice: straight lines, points and crossing lines
Non-calculatorKey words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a straight line.
- Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.63° = 63°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 63 = 117
- Write the answer. The missing angle is 117 degrees.117
Check. 117 + 63 = 180, so the angles add up correctly. ✓
Watch out. Use 180 for a straight line, not 360. The angles at a point add up to 360.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a straight line.
- Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.36° + 90° = 126°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 126 = 54
- Write the answer. The missing angle is 54 degrees.54
Check. 54 + 126 = 180, so the angles add up correctly. ✓
Watch out. Use 180 for a straight line, not 360. The angles at a point add up to 360.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for angles at a point.
- Remember the total for angles at a point. Angles around a point add up to 360 degrees (a full turn).Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.120° + 95° = 215°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 215 = 145
- Write the answer. The missing angle is 145 degrees.145
Check. 145 + 215 = 360, so the angles add up correctly. ✓
Watch out. Use 360 for angles at a point, not 180. A straight line is only 180.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for angles at a point.
- Remember the total for angles at a point. Angles around a point add up to 360 degrees (a full turn).Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.80° + 75° + 110° = 265°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 265 = 95
- Write the answer. The missing angle is 95 degrees.95
Check. 95 + 265 = 360, so the angles add up correctly. ✓
Watch out. Use 360 for angles at a point, not 180. A straight line is only 180.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Vertically opposite angles are equal, so copy the size of the angle.
- Look at the two straight lines that cross. Angles that face each other at a crossing are called vertically opposite angles.The angle given is 47°
- Use the rule: vertically opposite angles are equal. So the angle opposite is the same size.Opposite angle = 47°
- Write the answer. The angle opposite is the same size as the one given.47
Check. The angle next to it on a straight line is 180 - 47 = 133, and 47 + 133 = 180. ✓
Watch out. Vertically opposite means the angles face each other, not the ones next to each other.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Vertically opposite angles are equal, so copy the size of the angle.
- Look at the two straight lines that cross. Angles that face each other at a crossing are called vertically opposite angles.The angle given is 35°
- Use the rule: vertically opposite angles are equal. So the angle opposite is the same size.Opposite angle = 35°
- Write the answer. The angle opposite is the same size as the one given.35
Check. The angle next to it on a straight line is 180 - 35 = 145, and 35 + 145 = 180. ✓
Watch out. Vertically opposite means the angles face each other, not the ones next to each other.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a straight line.
- Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.35° = 35°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 35 = 145
- Write the answer. The missing angle is 145 degrees.145
Check. 145 + 35 = 180, so the angles add up correctly. ✓
Watch out. Use 180 for a straight line, not 360. The angles at a point add up to 360.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a straight line.
- Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.40° + 65° = 105°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 105 = 75
- Write the answer. The missing angle is 75 degrees.75
Check. 75 + 105 = 180, so the angles add up correctly. ✓
Watch out. Use 180 for a straight line, not 360. The angles at a point add up to 360.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Share the full turn equally between the angles.
- Remember: the angles at a point add up to 360 degrees. That is a full turn.360 degrees
- Share 360 equally between the angles. All the angles are the same size, so divide by how many there are.360 ÷ 5 = 72
- Write the answer. Each angle is this many degrees.72
Check. 72 × 5 = 360. ✓
Watch out. Use 360 for a full turn around a point. Do not use 180.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Share the total into equal parts using the ratio, then work out each angle.
- Add the parts of the ratio. Each part of the ratio is one share of the total.2 : 3: 5 parts in all
- Find the size of one part. Share the total of 180 degrees into 5 equal parts.180 ÷ 5 = 36
- Work out each angle. Multiply each ratio number by the size of one part.2 × 36 = 72, 3 × 36 = 108
- Write the answer. The largest angle is 108 degrees.108
Check. The angles add up to 72 + 108 = 180. ✓
Watch out. Share the total by the number of parts, not by the number of angles.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Use the fact that angles at a point add up to 360, not 180.
- Check the rule for a straight line. A straight line makes 180 degrees. That is half a turn.straight line = 180
- Check the rule for a full turn. Angles at a point go all the way round, so they make a full turn.angles at a point = 360
- Compare the two rules. Ali has used the rule for a straight line, but the angles are at a point.180 is not 360, so Ali is wrong
- Write the answer. Ali is not right.No
Check. Four angles at a point could be 90, 90, 90 and 90, which add up to 360. ✓
Watch out. Angles at a point add up to 360. A straight line is only 180.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. The vertically opposite angle is the same size, so it is not the angle next to it.
- Find the vertically opposite angle. Opposite angles at a crossing are equal.opposite angle = 70°
- Find the angle next to 70°. That angle is on a straight line with 70, so it is 180 - 70.180 - 70 = 110°
- Write the answer. Sam has used the angle next to it, not the opposite one.No
Check. 70 + 110 = 180. ✓
Watch out. Vertically opposite angles are equal. Angles next to each other are not.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Set up an equation with the four angles, then solve it.
- Write the four angles as an equation. Angles at a point add up to 360 degrees.x + 2x + 3x + 90 = 360
- Collect the x terms. x + 2x + 3x is 6 lots of x.6x + 90 = 360
- Take 90 from 360. What is left is the 6 lots of x.360 - 90 = 270, so 6x = 270
- Divide by 6 to find x. This gives the size of one x.270 ÷ 6 = 45
- Write the answer. x is 45 degrees.45
Check. 45 + 90 + 135 + 90 = 360, the full turn. ✓
Watch out. Keep the 90 degree angle separate. It is not an x angle.
Watch out
Common mistake: using 180 for angles at a point, or 360 for a straight line. Look at the picture first and decide which total you need.
Exam tip
In the test, write down the fact you used, such as "angles on a straight line add up to 180", before you do the sum.
Angles in triangles and quadrilaterals
The angles inside a triangle always add up to 180°. The angles inside a quadrilateral (a four-sided shape) always add up to 360°. A diagonal splits a quadrilateral into two triangles, so 360 = 2 × 180.
| Shape | The angles inside add up to |
|---|---|
| Triangle | 180° |
| Quadrilateral | 360° |
We can classify a shape by its angles and sides. An isosceles triangle has two equal sides, and its two base angles are equal. An equilateral triangle has three equal sides and three angles of 60°. In a parallelogram the opposite angles are equal, and the two angles next to each other add up to 180°.
Key ideas
- A triangle has angles that add up to 180°.
- A quadrilateral has angles that add up to 360°.
- In an isosceles triangle, the two base angles are equal.
- In a parallelogram, opposite angles are equal and adjacent angles add up to 180°.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.47° + 68° = 115°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 115 = 65
- Write the answer. The missing angle is 65 degrees.65
Check. 65 + 115 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a quadrilateral.
- Remember the total for a quadrilateral. The four angles in any quadrilateral add up to 360 degrees. A diagonal splits it into two triangles, and 2 × 180 = 360.Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.90° + 75° + 110° = 275°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 275 = 85
- Write the answer. The missing angle is 85 degrees.85
Check. 85 + 275 = 360, so the angles add up correctly. ✓
Watch out. A four-sided shape has 360 degrees in total. Do not use 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Take the top angle from 180, then share what is left equally between the two base angles.
- Remember: the three angles of a triangle add up to 180 degrees. So the top angle and the two base angles share 180.180 total
- Take the top angle away. This leaves the two base angles together.180 - 36 = 144
- Share the rest equally between the two base angles. The two base angles are equal, so divide by 2.144 ÷ 2 = 72
- Write the answer. Each base angle is this size.72
Check. 72 + 72 + 36 = 180. ✓
Watch out. Only divide the rest by 2. Do not divide the top angle.
Exercise 10B Practice: triangles, quadrilaterals and special shapes
Non-calculatorKey words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.50° + 70° = 120°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 120 = 60
- Write the answer. The missing angle is 60 degrees.60
Check. 60 + 120 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.38° + 92° = 130°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 130 = 50
- Write the answer. The missing angle is 50 degrees.50
Check. 50 + 130 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.90° + 27° = 117°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 117 = 63
- Write the answer. The missing angle is 63 degrees.63
Check. 63 + 117 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.115° + 28° = 143°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 143 = 37
- Write the answer. The missing angle is 37 degrees.37
Check. 37 + 143 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Take the top angle from 180, then share what is left equally between the two base angles.
- Remember: the three angles of a triangle add up to 180 degrees. So the top angle and the two base angles share 180.180 total
- Take the top angle away. This leaves the two base angles together.180 - 40 = 140
- Share the rest equally between the two base angles. The two base angles are equal, so divide by 2.140 ÷ 2 = 70
- Write the answer. Each base angle is this size.70
Check. 70 + 70 + 40 = 180. ✓
Watch out. Only divide the rest by 2. Do not divide the top angle.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a quadrilateral.
- Remember the total for a quadrilateral. The four angles in any quadrilateral add up to 360 degrees. A diagonal splits it into two triangles, and 2 × 180 = 360.Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.85° + 110° + 95° = 290°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 290 = 70
- Write the answer. The missing angle is 70 degrees.70
Check. 70 + 290 = 360, so the angles add up correctly. ✓
Watch out. A four-sided shape has 360 degrees in total. Do not use 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a quadrilateral.
- Remember the total for a quadrilateral. The four angles in any quadrilateral add up to 360 degrees. A diagonal splits it into two triangles, and 2 × 180 = 360.Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.90° + 90° + 70° = 250°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 250 = 110
- Write the answer. The missing angle is 110 degrees.110
Check. 110 + 250 = 360, so the angles add up correctly. ✓
Watch out. A four-sided shape has 360 degrees in total. Do not use 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Adjacent angles in a parallelogram add up to 180 degrees, so subtract the given angle from 180.
- Know the rule for a parallelogram. The two angles next to each other on the same side add up to 180 degrees.64° + ? = 180°
- Take the given angle away from 180. The missing angle is what is left over.180 - 64 = 116
- Write the answer. The angle is 116 degrees.116
Check. 64 + 116 = 180. ✓
Watch out. Next to each other means on the same side. Do not copy the angle.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Opposite angles in a parallelogram are equal, so copy the angle.
- Know the rule for a parallelogram. Opposite angles in a parallelogram are equal.Opposite angle = 64°
- Write the answer. The opposite angle is the same size.64
Check. The four angles are 64, 116, 64 and 116, and 64 + 116 + 64 + 116 = 360. ✓
Watch out. Opposite angles are equal. Angles next to each other add up to 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Subtract the known angles from the total, then share the rest equally.
- Add the angles you know. Put them together first.100° + 100° = 200°
- Take the total away. What is left is shared by the 2 equal angles.360 - 200 = 160
- Share the rest equally. Each of the equal angles is the same size.160 ÷ 2 = 80
- Write the answer. Each equal angle is this size.80
Check. 80 × 2 = 160, and 160 + 200 = 360. ✓
Watch out. Share only the angles that are left over. Do not share the total.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Share the total into equal parts using the ratio, then work out each angle.
- Add the parts of the ratio. Each part of the ratio is one share of the total.1 : 2 : 3: 6 parts in all
- Find the size of one part. Share the total of 180 degrees into 6 equal parts.180 ÷ 6 = 30
- Work out each angle. Multiply each ratio number by the size of one part.1 × 30 = 30, 2 × 30 = 60, 3 × 30 = 90
- Write the answer. The largest angle is 90 degrees.90
Check. The angles add up to 30 + 60 + 90 = 180. ✓
Watch out. Share the total by the number of parts, not by the number of angles.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Use the rule that the angles of a triangle add up to 180 degrees.
- Add the two right angles. Each right angle is 90 degrees.90 + 90 = 180
- Look at what is left for the third angle. The three angles must add up to 180, so nothing is left.180 - 180 = 0
- Write the answer. A triangle cannot have an angle of 0 degrees, so Sam is not right.No
Check. Two right angles would already make the full 180 degrees of a triangle. ✓
Watch out. A triangle can have at most one right angle.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Write the three angles in terms of x, add them, and solve.
- Write the three angles and add them up. The angles of a triangle add up to 180 degrees.x + (x + 20) + (x + 40) = 180
- Collect the x terms and the numbers. x + x + x is 3x, and 20 + 40 is 60.3x + 60 = 180
- Take 60 from 180. What is left is 3 lots of x.180 - 60 = 120, so 3x = 120
- Divide by 3 to find x. This gives the size of x.120 ÷ 3 = 40, so x = 40
- Find the largest angle. The three angles are 40, 60 and 80, so the largest is x + 40.x + 40 = 40 + 40 = 80
- Write the answer. The largest angle is 80 degrees.80
Check. The three angles are 40, 60 and 80, and 40 + 60 + 80 = 180. ✓
Watch out. Find x first. Then work out each angle before you choose the largest.
Watch out
Common mistake: using 180 for a quadrilateral. A four-sided shape has 360 degrees inside it. Also check that a missing angle is positive.
Exam tip
If a question gives a diagram that is not drawn to scale, use the numbers on it, not the size of the drawn angles.
Polygons and their angles
A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An irregular polygon does not have both.
To find the total of the interior angles of any polygon with n sides, split it into triangles from one corner. There are n − 2 triangles, and each has 180°, so the total is (n − 2) × 180°.
| Sides | Name | Interior angles add up to |
|---|---|---|
| 3 | triangle | 180° |
| 4 | quadrilateral | 360° |
| 5 | pentagon | 540° |
| 6 | hexagon | 720° |
| 8 | octagon | 1080° |
| 10 | decagon | 1440° |
In a regular polygon every interior angle is the same, so divide the total by the number of sides. Each exterior angle is found from the turn, and the exterior angles of any polygon add up to 360°.
Key ideas
- Interior angles of an n-sided polygon add up to (n − 2) × 180°.
- In a regular polygon, each interior angle = total ÷ number of sides.
- Each exterior angle of a regular polygon = 360° ÷ number of sides.
- Regular means equal sides and equal angles.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Split the shape into triangles, then multiply the number of triangles by 180.
- Split the shape into triangles. Draw every diagonal from one corner. This makes 6 - 2 triangles.6 - 2 = 4 triangles
- Add up the angles of the triangles. Each triangle has angles that add up to 180 degrees.4 × 180 = 720
- Write the answer. The interior angles of this shape add up to this total.720
Check. A shape with 6 sides makes 4 triangles inside it. ✓
Watch out. Use (n - 2) × 180. Do not multiply the number of sides by 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Find the total of the interior angles, then share it equally between the sides.
- Find the sum of the interior angles. The shape has 10 sides, so its angles add up to (10 - 2) × 180.(10 - 2) × 180 = 1440
- Share the sum equally between the angles. All the angles of a regular polygon are the same size.1440 ÷ 10 = 144
- Write the answer. Each interior angle is this many degrees.144
Check. The exterior angle is 360 ÷ 10 = 36, and 144 + 36 = 180. ✓
Watch out. In a regular polygon every interior angle is the same, so divide the total by the number of sides.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a shape with 5 sides.
- Remember the total for a shape with 5 sides. A shape with 5 sides has interior angles that add up to (5 - 2) × 180 = 540 degrees.Total = 540°
- Add the angles you know. Put them together to see how much of the total is used.100° + 120° + 90° + 110° = 420°
- Take the angles you know away from the total. What is left over is the missing angle.540 - 420 = 120
- Write the answer. The missing angle is 120 degrees.120
Check. 120 + 420 = 540, so the angles add up correctly. ✓
Watch out. Use (n - 2) × 180 for the total. Each extra side adds 180 degrees.
Exercise 10C Practice: polygons, regular shapes and exterior angles
Non-calculatorKey words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Count the sides, then use the name for that number.
- Count the sides. The shape has 8 straight sides.8 sides
- Use the name. A polygon with 8 sides is called an octagon.octagon
Check. An octagon has 8 sides and 8 corners. ✓
Watch out. Octa means 8, as in an octopus with 8 arms.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Split the shape into triangles, then multiply the number of triangles by 180.
- Split the shape into triangles. Draw every diagonal from one corner. This makes 5 - 2 triangles.5 - 2 = 3 triangles
- Add up the angles of the triangles. Each triangle has angles that add up to 180 degrees.3 × 180 = 540
- Write the answer. The interior angles of this shape add up to this total.540
Check. A shape with 5 sides makes 3 triangles inside it. ✓
Watch out. Use (n - 2) × 180. Do not multiply the number of sides by 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Split the shape into triangles, then multiply the number of triangles by 180.
- Split the shape into triangles. Draw every diagonal from one corner. This makes 8 - 2 triangles.8 - 2 = 6 triangles
- Add up the angles of the triangles. Each triangle has angles that add up to 180 degrees.6 × 180 = 1080
- Write the answer. The interior angles of this shape add up to this total.1080
Check. A shape with 8 sides makes 6 triangles inside it. ✓
Watch out. Use (n - 2) × 180. Do not multiply the number of sides by 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Find the total of the interior angles, then share it equally between the sides.
- Find the sum of the interior angles. The shape has 6 sides, so its angles add up to (6 - 2) × 180.(6 - 2) × 180 = 720
- Share the sum equally between the angles. All the angles of a regular polygon are the same size.720 ÷ 6 = 120
- Write the answer. Each interior angle is this many degrees.120
Check. The exterior angle is 360 ÷ 6 = 60, and 120 + 60 = 180. ✓
Watch out. In a regular polygon every interior angle is the same, so divide the total by the number of sides.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Share 360 degrees equally between the sides of the regular polygon.
- Remember: the exterior angles of any shape add up to 360 degrees. Going all the way round, the turns make one full turn.360 degrees
- Share 360 equally between the sides. In a regular polygon every exterior angle is the same size.360 ÷ 8 = 45
- Write the answer. Each exterior angle is this many degrees.45
Check. 45 × 8 = 360. ✓
Watch out. The exterior angle is a turn, not an inside angle. Use 360, not 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Find the total of the interior angles, then share it equally between the sides.
- Find the sum of the interior angles. The shape has 5 sides, so its angles add up to (5 - 2) × 180.(5 - 2) × 180 = 540
- Share the sum equally between the angles. All the angles of a regular polygon are the same size.540 ÷ 5 = 108
- Write the answer. Each interior angle is this many degrees.108
Check. The exterior angle is 360 ÷ 5 = 72, and 108 + 72 = 180. ✓
Watch out. In a regular polygon every interior angle is the same, so divide the total by the number of sides.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a shape with 5 sides.
- Remember the total for a shape with 5 sides. A shape with 5 sides has interior angles that add up to (5 - 2) × 180 = 540 degrees.Total = 540°
- Add the angles you know. Put them together to see how much of the total is used.105° + 130° + 95° + 85° = 415°
- Take the angles you know away from the total. What is left over is the missing angle.540 - 415 = 125
- Write the answer. The missing angle is 125 degrees.125
Check. 125 + 415 = 540, so the angles add up correctly. ✓
Watch out. Use (n - 2) × 180 for the total. Each extra side adds 180 degrees.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Find the exterior angle, then divide 360 by it to find the number of sides.
- Find the exterior angle. An interior angle and its exterior angle make a straight line, so they add up to 180.180 - 140 = 40
- Divide 360 by the exterior angle. The exterior angles of a regular polygon add up to 360 and are all equal.360 ÷ 40 = 9
- Write the answer. The polygon has this many sides.9
Check. A regular 9-sided shape has interior angles of 140°, which is 180 - 40. ✓
Watch out. Find the exterior angle first. Do not divide the interior angle into 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Find the total of the interior angles, then share it equally between the sides.
- Find the sum of the interior angles. The shape has 12 sides, so its angles add up to (12 - 2) × 180.(12 - 2) × 180 = 1800
- Share the sum equally between the angles. All the angles of a regular polygon are the same size.1800 ÷ 12 = 150
- Write the answer. Each interior angle is this many degrees.150
Check. The exterior angle is 360 ÷ 12 = 30, and 150 + 30 = 180. ✓
Watch out. In a regular polygon every interior angle is the same, so divide the total by the number of sides.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a shape with 6 sides.
- Remember the total for a shape with 6 sides. A shape with 6 sides has interior angles that add up to (6 - 2) × 180 = 720 degrees.Total = 720°
- Add the angles you know. Put them together to see how much of the total is used.110° + 110° + 110° + 110° + 110° = 550°
- Take the angles you know away from the total. What is left over is the missing angle.720 - 550 = 170
- Write the answer. The missing angle is 170 degrees.170
Check. 170 + 550 = 720, so the angles add up correctly. ✓
Watch out. Use (n - 2) × 180 for the total. Each extra side adds 180 degrees.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Remember the two conditions for a regular polygon.
- Write down what a regular polygon needs. A regular polygon needs equal sides and equal angles.equal sides + equal angles
- Check what Ali said. Ali only mentions the sides, so he has left out the angles.sides only is not enough
- Write the answer. Ali is not right, because the angles must be equal as well.No
Check. A square is regular, because its sides and its angles are all equal. ✓
Watch out. Both the sides and the angles must be equal for a regular polygon.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Find the exterior angle, then divide 360 by it to find the number of sides.
- Find the exterior angle. An interior angle and its exterior angle make a straight line, so they add up to 180.180 - 162 = 18
- Divide 360 by the exterior angle. The exterior angles of a regular polygon add up to 360 and are all equal.360 ÷ 18 = 20
- Write the answer. The polygon has this many sides.20
Check. A regular 20-sided shape has interior angles of 162°, which is 180 - 18. ✓
Watch out. Find the exterior angle first. Do not divide the interior angle into 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Split the shape into triangles, then multiply the number of triangles by 180.
- Split the shape into triangles. Draw every diagonal from one corner. This makes 20 - 2 triangles.20 - 2 = 18 triangles
- Add up the angles of the triangles. Each triangle has angles that add up to 180 degrees.18 × 180 = 3240
- Write the answer. The interior angles of this shape add up to this total.3240
Check. A shape with 20 sides makes 18 triangles inside it. ✓
Watch out. Use (n - 2) × 180. Do not multiply the number of sides by 180.
Watch out
Common mistake: using n × 180 for the sum of the interior angles. The right total is (n − 2) × 180. Also, a regular polygon is not the same as any polygon with equal sides.
Exam tip
Each interior angle and its exterior angle make a straight line, so they add up to 180 degrees.
Parts of a circle and the circumference
A circle has a centre. The radius goes from the centre to the edge. The diameter goes from one side of the circle to the other, through the centre. The diameter is twice the radius, so the radius is half the diameter.
The circumference is the distance all the way round the edge. For every circle, the circumference is about 3 times the diameter. The exact number is called pi (π). In this book we use two values, and the question always tells you which to use:
- π = 3.14
- π = 22/7 (good when the diameter is a multiple of 7)
The formula is C = π × d, where d is the diameter.
Key ideas
- The radius goes from the centre to the edge.
- The diameter goes across the circle through the centre.
- Diameter = 2 × radius, and radius = diameter ÷ 2.
- Circumference = π × diameter.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Double the radius to find the diameter.
- Remember: the diameter is twice the radius. The diameter goes across the whole circle, through the centre, so it is two radii.diameter = 2 × radius
- Multiply the radius by 2. This gives the diameter.2 × 5 = 10
- Write the answer. The diameter is in the same unit as the radius.10
Check. 10 ÷ 2 = 5, the radius. ✓
Watch out. The diameter is double the radius, not half of it.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Multiply the diameter by 3.14.
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 3.14.C = 3.14 × 12
- Work out the product. Multiply the diameter by pi.12 × 3.14 = 37.68
- Write the answer. The circumference is in the same unit as the diameter.37.68
Check. 37.68 ÷ 3.14 = 12. ✓
Watch out. The circumference is about three times the diameter, but use the exact value of pi given in the question.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Divide the diameter by 7, then multiply by 22.
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 22/7.C = 22/7 × 14
- Work out the product. Multiply the diameter by pi.14 ÷ 7 = 2, then 2 × 22 = 44
- Write the answer. The circumference is in the same unit as the diameter.44
Check. 44 ÷ 22 × 7 = 14. ✓
Watch out. The circumference is about three times the diameter, but use the exact value of pi given in the question.
Exercise 10D Practice: radius, diameter and circumference
Non-calculatorKey words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Think about where the line starts and ends.
- Find the line. It starts at the centre and ends on the edge.centre to edge
- Name it. A line from the centre to the edge is called the radius.radius
Check. The diameter is twice this line, and goes through the centre. ✓
Watch out. The radius is half of the diameter. It starts at the centre.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Double the radius to find the diameter.
- Remember: the diameter is twice the radius. The diameter goes across the whole circle, through the centre, so it is two radii.diameter = 2 × radius
- Multiply the radius by 2. This gives the diameter.2 × 6 = 12
- Write the answer. The diameter is in the same unit as the radius.12
Check. 12 ÷ 2 = 6, the radius. ✓
Watch out. The diameter is double the radius, not half of it.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Halve the diameter to find the radius.
- Remember: the radius is half the diameter. The radius goes from the centre to the edge, which is half of the way across.radius = diameter ÷ 2
- Divide the diameter by 2. This gives the radius.14 ÷ 2 = 7
- Write the answer. The radius is in the same unit as the diameter.7
Check. 7 + 7 = 14, the diameter. ✓
Watch out. Halve the diameter to get the radius. Do not double it.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Multiply the diameter by 3.14.
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 3.14.C = 3.14 × 10
- Work out the product. Multiply the diameter by pi.10 × 3.14 = 31.4
- Write the answer. The circumference is in the same unit as the diameter.31.4
Check. 31.4 ÷ 3.14 = 10. ✓
Watch out. The circumference is about three times the diameter, but use the exact value of pi given in the question.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Divide the diameter by 7, then multiply by 22.
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 22/7.C = 22/7 × 7
- Work out the product. Multiply the diameter by pi.7 ÷ 7 = 1, then 1 × 22 = 22
- Write the answer. The circumference is in the same unit as the diameter.22
Check. 22 ÷ 22 × 7 = 7. ✓
Watch out. The circumference is about three times the diameter, but use the exact value of pi given in the question.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Double the radius to get the diameter, then multiply the diameter by 3.14.
- Find the diameter first. The diameter is twice the radius.2 × 3 = 6
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 3.14.C = 3.14 × 6
- Work out the product. Multiply the diameter by pi.6 × 3.14 = 18.84
- Write the answer. The circumference is in the same unit as the radius.18.84
Check. 18.84 ÷ 3.14 = 6, the diameter. ✓
Watch out. Use the diameter in the formula, not the radius.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Divide the circumference by 3.14 to find the diameter.
- Write the formula. The circumference is pi times the diameter.C = 3.14 × d
- Undo the multiplication by 3.14. Divide the circumference by 3.14 to find the diameter.31.4 ÷ 3.14 = 10
- Write the answer. The diameter is in the same unit as the circumference.10
Check. 10 × 3.14 = 31.4. ✓
Watch out. Divide the circumference by pi. Do not divide it by 2.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Multiply the diameter by 3.14.
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 3.14.C = 3.14 × 2.5
- Work out the product. Multiply the diameter by pi.2.5 × 3.14 = 7.85
- Write the answer. The circumference is in the same unit as the diameter.7.85
Check. 7.85 ÷ 3.14 = 2.5. ✓
Watch out. The circumference is about three times the diameter, but use the exact value of pi given in the question.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Multiply the diameter by 3.14.
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 3.14.C = 3.14 × 70
- Work out the product. Multiply the diameter by pi.70 × 3.14 = 219.8
- Write the answer. The circumference is in the same unit as the diameter.219.8
Check. 219.8 ÷ 3.14 = 70. ✓
Watch out. The circumference is about three times the diameter, but use the exact value of pi given in the question.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Double the radius to get the diameter, then multiply the diameter by 3.14.
- Find the diameter first. The diameter is twice the radius.2 × 4 = 8
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 3.14.C = 3.14 × 8
- Work out the product. Multiply the diameter by pi.8 × 3.14 = 25.12
- Write the answer. The circumference is in the same unit as the radius.25.12
Check. 25.12 ÷ 3.14 = 8, the diameter. ✓
Watch out. Use the diameter in the formula, not the radius.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Compare the claim with the value of pi.
- Write the formula. Circumference = π × diameter.C = π × d
- Compare pi with 3. Pi is 3.14, which is a little more than 3.3.14 is not 3
- Write the answer. Ruby is not exactly right, because the circumference is about 3.14 times the diameter.No
Check. For a diameter of 10 cm, 3 times 10 is 30 cm, but the circumference is 31.4 cm. ✓
Watch out. Three is only an estimate. The exact multiplier is pi, which is 3.14 to two decimal places.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Find the diameter from the circumference using 22/7, then halve it.
- Write the formula. The circumference is 22/7 times the diameter.C = 22/7 × d
- Find the diameter. Divide the circumference by 22, then multiply by 7.22 ÷ 22 × 7 = 7
- Halve the diameter to get the radius. The diameter is twice the radius.7 ÷ 2 = 3.5
- Write the answer. The radius is in the same unit as the circumference.3.5
Check. 3.5 + 3.5 = 7, and 22/7 × 7 = 22. ✓
Watch out. Halve the diameter to get the radius at the end.
Watch out
Common mistake: using the radius in the formula C = π × d. Double the radius first to get the diameter, then multiply by pi. Also, the diameter is double the radius, not half of it.
Exam tip
Write the number of pi that the question gives, such as 3.14, in your working. This shows the examiner which value you used.
Drawing 2-D shapes from their dimensions and angles
When you draw a shape, the question gives you its dimensions: some lengths and some angles. Use a ruler for the lengths and a protractor for the angles. Mark each corner with a letter, so you can name the shape.
Once a shape is drawn, you can still use the angle rules. A triangle still has angles that add up to 180°, and a quadrilateral has angles that add up to 360°. You can also add up the sides to find the perimeter, and check that the sides can make a shape at all.
A drawing must be accurate, which means it is drawn to the lengths and angles that are given. A picture that is not to scale does not use the true sizes, so do not measure it.
Key ideas
- Use a ruler for the lengths and a protractor for the angles.
- Name each corner with a capital letter.
- A triangle drawn with two angles has the third angle left over: 180 minus the two.
- The perimeter is the sum of all the sides.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.40° + 75° = 115°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 115 = 65
- Write the answer. The missing angle is 65 degrees.65
Check. 65 + 115 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Find the length of each of the four sides, then add them up.
- Write down the lengths of the sides. A parallelogram has two sides of each length: the base, and the slanted side.base = 8 cm, slanted side = 5 cm
- Add the four sides. The sides are 8, 5, 8 and 5. The angles do not change the perimeter.8 + 5 + 8 + 5 = 26
- Write the answer. The perimeter is in centimetres.26
Check. 2 × (8 + 5) = 26, the same answer. ✓
Watch out. The angles are not needed for the perimeter. Use only the lengths of the sides.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Take the top angle from 180, then share what is left equally between the two base angles.
- Remember: the three angles of a triangle add up to 180 degrees. So the top angle and the two base angles share 180.180 total
- Take the top angle away. This leaves the two base angles together.180 - 120 = 60
- Share the rest equally between the two base angles. The two base angles are equal, so divide by 2.60 ÷ 2 = 30
- Write the answer. Each base angle is this size.30
Check. 30 + 30 + 120 = 180. ✓
Watch out. Only divide the rest by 2. Do not divide the top angle.
Exercise 10E Practice: drawing and checking shapes
Non-calculatorKey words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Add the lengths of all the sides.
- Add the lengths of all the sides. The perimeter is the distance all the way round the outside.6 + 4 + 6 + 4 = 20
- Write the answer. The perimeter is in cm.20
Check. Go round the shape once more and check that each side is counted once. ✓
Watch out. Count every side once. Do not forget the sides you cannot see on a drawing.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Add the lengths of all the sides.
- Add the lengths of all the sides. The perimeter is the distance all the way round the outside.4.5 + 4.5 + 4.5 + 4.5 = 18
- Write the answer. The perimeter is in cm.18
Check. Go round the shape once more and check that each side is counted once. ✓
Watch out. Count every side once. Do not forget the sides you cannot see on a drawing.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Add the lengths of all the sides.
- Add the lengths of all the sides. The perimeter is the distance all the way round the outside.6 + 6 + 4 = 16
- Write the answer. The perimeter is in cm.16
Check. Go round the shape once more and check that each side is counted once. ✓
Watch out. Count every side once. Do not forget the sides you cannot see on a drawing.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.52° + 48° = 100°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 100 = 80
- Write the answer. The missing angle is 80 degrees.80
Check. 80 + 100 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Take the top angle from 180, then share what is left equally between the two base angles.
- Remember: the three angles of a triangle add up to 180 degrees. So the top angle and the two base angles share 180.180 total
- Take the top angle away. This leaves the two base angles together.180 - 40 = 140
- Share the rest equally between the two base angles. The two base angles are equal, so divide by 2.140 ÷ 2 = 70
- Write the answer. Each base angle is this size.70
Check. 70 + 70 + 40 = 180. ✓
Watch out. Only divide the rest by 2. Do not divide the top angle.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a quadrilateral.
- Remember the total for a quadrilateral. The four angles in any quadrilateral add up to 360 degrees. A diagonal splits it into two triangles, and 2 × 180 = 360.Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.90° + 90° + 60° = 240°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 240 = 120
- Write the answer. The missing angle is 120 degrees.120
Check. 120 + 240 = 360, so the angles add up correctly. ✓
Watch out. A four-sided shape has 360 degrees in total. Do not use 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Add the lengths of all the sides.
- Add the lengths of all the sides. The perimeter is the distance all the way round the outside.4 + 4 + 4 + 4 + 4 = 20
- Write the answer. The perimeter is in cm.20
Check. Go round the shape once more and check that each side is counted once. ✓
Watch out. Count every side once. Do not forget the sides you cannot see on a drawing.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Add the lengths of all the sides.
- Add the lengths of all the sides. The perimeter is the distance all the way round the outside.5.5 + 5.5 + 5.5 + 5.5 = 22
- Write the answer. The perimeter is in cm.22
Check. Go round the shape once more and check that each side is counted once. ✓
Watch out. Count every side once. Do not forget the sides you cannot see on a drawing.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.35° + 90° = 125°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 125 = 55
- Write the answer. The missing angle is 55 degrees.55
Check. 55 + 125 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Add the lengths of all the sides.
- Add the lengths of all the sides. The perimeter is the distance all the way round the outside.7 + 7 + 2 + 2 = 18
- Write the answer. The perimeter is in cm.18
Check. Go round the shape once more and check that each side is counted once. ✓
Watch out. Count every side once. Do not forget the sides you cannot see on a drawing.
Key words. To draw a shape from its dimensions, use a ruler for the lengths and a protractor for the angles. A dimension is a length or an angle that gives the size of a shape.
Method. Check whether the two shorter sides can reach across to the longest side.
- Add the two shorter sides. The two shorter sides must be able to reach the longest side.3 + 4 = 7
- Compare with the longest side. The longest side is 8 cm, which is longer than 7 cm.7 is less than 8
- Write the answer. The sides cannot meet, so the triangle cannot be drawn.No
Check. A triangle with sides 3 cm, 4 cm and 5 cm can be drawn, because 3 + 4 is more than 5. ✓
Watch out. Two short sides must add up to more than the longest side.
Key words. To draw a shape from its dimensions, use a ruler for the lengths and a protractor for the angles. A dimension is a length or an angle that gives the size of a shape.
Method. Split the rectangle into two triangles with the diagonal, and add the angles of each triangle.
- Find the angles in one triangle. Each triangle has angles that add up to 180 degrees.180 degrees
- Find the angles in the other triangle. The second triangle also has angles that add up to 180 degrees.180 degrees
- Add the two totals. This gives all the angles in both triangles.180 + 180 = 360
- Write the answer. The total of all the angles in both triangles is 360 degrees.360
Check. This is the same as the four angles of a quadrilateral, which also add up to 360. ✓
Watch out. The total of two triangles is 360, not 180. Each triangle gives 180.
Watch out
Common mistake: measuring a diagram that is not drawn to scale. Use the numbers that the question gives. Also, when a triangle is drawn with two angles, the third is 180 minus the other two, not 360.
Exam tip
Before you draw, check the dimensions. Two short sides must be long enough to reach the longest side.
3-D shapes and nets
A 3-D shape is a solid. Its flat sides are called faces. Two faces meet at an edge, and edges meet at a vertex.
A prism has two identical ends, and the same shape all the way along. A triangular prism has two triangular faces and three rectangular faces. A pyramid has a base, and triangular faces that all meet at one point called the apex.
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube | 6 | 12 | 8 |
| Triangular prism | 5 | 9 | 6 |
| Square-based pyramid | 5 | 8 | 5 |
A net is a flat pattern that folds up into a 3-D shape. Each square of the net becomes one face, so a cube needs six squares. A net of a cube cannot have four squares that form a 2 by 2 block, because four faces cannot meet at one corner of a cube.
Key ideas
- A face is a flat side. An edge is where two faces meet. A vertex is a corner.
- A cube has 6 faces, 12 edges and 8 vertices.
- A net folds up to make a 3-D shape.
- A net of a cube has six squares, and each square covers a different face.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Count the faces, then the edges, then the vertices, one shape part at a time.
- Count the faces. Two triangles and three rectangles.2 + 3 = 5 faces
- Count the edges. The two triangles have 3 edges each, and the three rectangles share the other 3 edges.3 + 3 + 3 = 9 edges
- Count the vertices. Each triangle has 3 corners, and the two triangles give 6 corners in total.3 + 3 = 6 vertices
Check. Check: faces + vertices = edges + 2 for a prism, so 5 + 6 = 9 + 2. ✓
Watch out. A triangular prism has 5 faces, 9 edges and 6 vertices. Do not count a hidden edge twice.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Check that the net has six squares, and that it has no 2 by 2 block.
- Count the squares. A cube has 6 faces, so the net needs 6 squares.6 squares
- Look for a 2 by 2 block. Four squares in a block would all meet at one corner, which cannot happen on a cube.no 2 by 2 block
- Check the shape. The row of four wraps round the sides. One square goes on top and one goes underneath.row of 4, 1 above, 1 below
- Write the answer. This net folds up into a cube.Yes
Check. Each of the six squares folds onto a different face. ✓
Watch out. The cross shape with a row of four is a cube net. Count the squares first.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Find the area of one square face, then multiply by the six faces of the net.
- Find the area of one square. Area of a square = side × side.3 × 3 = 9 cm²
- Count the squares in the net. A cube has 6 faces, so the net has 6 squares.6 squares
- Multiply the area of one square by 6. Each square is the same size, so the total area is 6 lots of 9.6 × 9 = 54
- Write the answer. The total area of the net is in square centimetres.54
Check. 54 ÷ 6 = 9, which is the area of one face. ✓
Watch out. Find the area of one face first. Do not use the length of the edge as the area.
Exercise 10F Practice: faces, edges, vertices and nets
Non-calculatorKey words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Count the flat faces of the prism.
- Count the triangles. There are two triangular ends.2 triangles
- Count the rectangles. The three sides between the ends are rectangles.3 rectangles
- Write the answer. Add the two kinds of face.2 + 3 = 5
Check. A prism with a triangular end has 3 + 2 = 5 faces. ✓
Watch out. Do not forget the two ends.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Each end has 3 edges, and the three sides join the ends.
- Count the edges on one end. A triangle has 3 edges.3 edges
- Count the edges on the other end. The other end also has 3 edges.3 + 3 = 6
- Add the edges that join the ends. There are 3 of these, one at each corner.6 + 3 = 9
Check. Each of the 6 vertices has 3 edges, so the edges are 6 × 3 ÷ 2 = 9. ✓
Watch out. The edges that join the two ends are easy to miss. There are three of them.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. A vertex is a corner. Count the corners on each end.
- Count the corners on one end. A triangle has 3 corners.3 vertices
- Count the corners on both ends. The other end has 3 more.3 + 3 = 6
Check. Each end has 3 corners, and 3 + 3 = 6. ✓
Watch out. A vertex is a corner. Do not count the edges.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Look at the faces that meet at the top point.
- Find the triangles. Each side of the square base has one triangle that meets the top point.4 sides = 4 triangles
- Write the answer. The base is a square, not a triangle, so it is not counted.4
Check. A square-based pyramid has 4 triangular faces and 1 square base, 5 faces altogether. ✓
Watch out. The base is a square, so it is not a triangular face.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Count the edges: 4 around the base and 4 going up to the top point.
- Count the edges around the base. A square base has 4 edges.4 edges
- Count the edges up to the top point. Four edges join the four corners of the base to the apex.4 edges
- Add them up. This gives the number of edges.4 + 4 = 8
Check. A square-based pyramid has 8 edges. ✓
Watch out. Count the sloping edges as well as the base edges.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. A vertex is a corner. Count the corners of the base and the apex.
- Count the corners of the base. A square has 4 corners.4 corners
- Add the apex. The top point is also a corner.4 + 1 = 5
Check. A square-based pyramid has 5 vertices, which is 4 + 1. ✓
Watch out. The top point counts as a vertex.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Fold the net in your mind. Each square becomes one face of the cube.
- Count the squares. The net has a row of four squares, one above and one below.4 + 1 + 1 = 6 squares
- Think about the shape the net makes. A row of four wraps round four sides, and the two single squares make the top and bottom.four sides, top and bottom
- Write the answer. The shape is a cube, because all six faces are squares.cube
Check. A cube has 6 square faces, so this net folds into a cube. ✓
Watch out. Look at the number of squares and their shape to name the 3-D shape.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Use the rule for cube nets: six squares, and no 2 by 2 block of squares.
- Check net A. Two squares sit on the same side of the row, so two faces fold onto the same face of the cube. One face is missing.A: not a cube
- Check net B. Squares in a 2 by 2 block would meet at one corner, which cannot happen on a cube.B: 2 by 2 block
- Check net C. The row of four wraps round the sides, with one square above and one below, at different places. Each square folds onto a different face.C: folds into a cube
- Check net D. Net D has a 2 by 3 block of squares, so it cannot fold into a cube.D: 2 by 3 block
- Write the answer. Only net C folds into a cube.C
Check. The nets are checked by rolling a cube over each one: only C gives six different faces. ✓
Watch out. Check each square of the net against the cube. A correct net has no 2 by 2 block and no two squares on the same face.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Find the area of one square face, then multiply by the six faces of the cube net.
- Find the area of one square. Area of a square = side × side.3 × 3 = 9 cm²
- Multiply by the six squares. The net has six squares, one for each face.6 × 9 = 54
- Write the answer. The total area is in square centimetres.54
Check. 54 ÷ 9 = 6 squares. ✓
Watch out. Area is in square units, so write cm².
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Multiply the three lengths of the cuboid.
- Find the number of cubes along the length, width and height. The cuboid is 4 cubes long, 3 cubes wide and 2 cubes high.4, 3 and 2
- Multiply the three numbers. Each cube fits into one place in the layers.4 × 3 × 2 = 24
- Write the answer. It takes 24 cubes to build the cuboid.24
Check. Two layers of 4 × 3 = 12 cubes each, so 12 + 12 = 24. ✓
Watch out. Multiply the three lengths. Do not add them.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Subtract the vertices of the pyramid from the vertices of the cube.
- Find the vertices of the cube. A cube has 8 corners.8 vertices
- Find the vertices of the pyramid. A square-based pyramid has 5 corners.5 vertices
- Find the difference. The cube has this many more.8 - 5 = 3
Check. 5 + 3 = 8. ✓
Watch out. Subtract the smaller number from the bigger one.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Follow the squares round the cube, one face at a time.
- Fold the row of six. Four squares wrap round the sides of a cube, so the fifth square lands on the first side again.fifth square = first face
- Check the faces. Two squares cover the same face, so there are not six different faces.faces repeat
- Write the answer. Sam is not right. A row of six squares does not fold into a cube.No
Check. A row of four squares wraps round the cube, so a row of six must overlap itself. ✓
Watch out. A net of a cube is never a single row of six squares.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Think about the corners of a cube and how many faces meet at each one.
- Look at a corner of a cube. Three faces meet at each corner of a cube.3 faces at each corner
- Look at the four squares in a 2 by 2 block. In a block of four squares, all four would meet at the same point.4 squares at one point
- Write the answer. Four squares cannot all meet at one corner of a cube, so the block cannot fold up.4 > 3, so it cannot fold
Check. A cube has no corner where four faces meet, so a 2 by 2 block can never be part of a cube net. ✓
Watch out. Think about the corners of the cube, not the faces in the middle.
Watch out
Common mistake: counting the faces of a prism without the two ends, or counting edges and vertices as if they were faces. Also, a net must have one square for each face: a row of six does not fold into a cube.
Exam tip
To check a net, count the squares first. Then look for any block of four squares, because a cube has no 2 by 2 block in its net.
Chapter summary
| Skill | Method |
|---|---|
| Straight line | The angles on a straight line add up to 180 degrees. |
| Angles at a point | The angles at a point add up to 360 degrees. |
| Vertically opposite | Angles that face each other at a crossing are equal. Angles next to each other add up to 180 degrees. |
| Triangle | The three angles add up to 180 degrees. An isosceles triangle has two equal base angles. |
| Quadrilateral and parallelogram | The four angles add up to 360 degrees. In a parallelogram, opposite angles are equal. |
| Polygon | The interior angles add up to (n − 2) × 180 degrees. In a regular polygon, share the total equally. |
| Exterior angle | The exterior angles of a polygon add up to 360 degrees. In a regular polygon, each one is 360 ÷ n. |
| Circle | The diameter is 2 × the radius. Circumference = π × diameter, with π = 3.14 or 22/7. |
| 3-D shapes and nets | A cube has 6 faces, 12 edges and 8 vertices. A net needs one square for each face. |
Can you…?
- I can find a missing angle on a straight line, at a point and opposite a crossing.
- I can find missing angles in triangles and quadrilaterals.
- I can work out the interior angles of a polygon, and of a regular polygon.
- I can name the radius, diameter and circumference of a circle.
- I can find the circumference of a circle using 3.14 or 22/7.
- I can draw a shape from its dimensions and angles, and name its sides and angles.
- I can count the faces, edges and vertices of a 3-D shape, and check a net.
Key terms
- Radius
- The distance from the centre of a circle to its edge.
- Diameter
- The distance across a circle through its centre. It is twice the radius.
- Circumference
- The distance all the way round the edge of a circle.
- Pi (π)
- The number that links the circumference and the diameter. We use 3.14 or 22/7.
- Interior angle
- An angle inside a shape.
- Regular polygon
- A shape with all its sides equal and all its angles equal.
- Vertically opposite
- The equal angles that face each other where two straight lines cross.
- Net
- A flat pattern that folds up to make a 3-D shape.
- Prism
- A 3-D shape with the same cross-section all the way along, such as a triangular prism.
- Pyramid
- A 3-D shape with a flat base and triangular faces that meet at a point.
Exercise 10Z Check-up
Non-calculatorKey words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a straight line.
- Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.75° = 75°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 75 = 105
- Write the answer. The missing angle is 105 degrees.105
Check. 105 + 75 = 180, so the angles add up correctly. ✓
Watch out. Use 180 for a straight line, not 360. The angles at a point add up to 360.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for angles at a point.
- Remember the total for angles at a point. Angles around a point add up to 360 degrees (a full turn).Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.90° + 110° + 70° = 270°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 270 = 90
- Write the answer. The missing angle is 90 degrees.90
Check. 90 + 270 = 360, so the angles add up correctly. ✓
Watch out. Use 360 for angles at a point, not 180. A straight line is only 180.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a straight line.
- Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.48° = 48°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 48 = 132
- Write the answer. The missing angle is 132 degrees.132
Check. 132 + 48 = 180, so the angles add up correctly. ✓
Watch out. Use 180 for a straight line, not 360. The angles at a point add up to 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a triangle.
- Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.55° + 65° = 120°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 120 = 60
- Write the answer. The missing angle is 60 degrees.60
Check. 60 + 120 = 180, so the angles add up correctly. ✓
Watch out. A triangle has 180 degrees in total, not 360. Quadrilaterals have 360.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a quadrilateral.
- Remember the total for a quadrilateral. The four angles in any quadrilateral add up to 360 degrees. A diagonal splits it into two triangles, and 2 × 180 = 360.Total = 360°
- Add the angles you know. Put them together to see how much of the total is used.90° + 75° + 120° = 285°
- Take the angles you know away from the total. What is left over is the missing angle.360 - 285 = 75
- Write the answer. The missing angle is 75 degrees.75
Check. 75 + 285 = 360, so the angles add up correctly. ✓
Watch out. A four-sided shape has 360 degrees in total. Do not use 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Find the total of the interior angles, then share it equally between the sides.
- Find the sum of the interior angles. The shape has 8 sides, so its angles add up to (8 - 2) × 180.(8 - 2) × 180 = 1080
- Share the sum equally between the angles. All the angles of a regular polygon are the same size.1080 ÷ 8 = 135
- Write the answer. Each interior angle is this many degrees.135
Check. The exterior angle is 360 ÷ 8 = 45, and 135 + 45 = 180. ✓
Watch out. In a regular polygon every interior angle is the same, so divide the total by the number of sides.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Double the radius to get the diameter, then multiply the diameter by 3.14.
- Find the diameter first. The diameter is twice the radius.2 × 4 = 8
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 3.14.C = 3.14 × 8
- Work out the product. Multiply the diameter by pi.8 × 3.14 = 25.12
- Write the answer. The circumference is in the same unit as the radius.25.12
Check. 25.12 ÷ 3.14 = 8, the diameter. ✓
Watch out. Use the diameter in the formula, not the radius.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Divide the diameter by 7, then multiply by 22.
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 22/7.C = 22/7 × 21
- Work out the product. Multiply the diameter by pi.21 ÷ 7 = 3, then 3 × 22 = 66
- Write the answer. The circumference is in the same unit as the diameter.66
Check. 66 ÷ 22 × 7 = 21. ✓
Watch out. The circumference is about three times the diameter, but use the exact value of pi given in the question.
Key words. A face is a flat side. An edge is where two faces meet. A vertex (plural vertices) is a corner. A net is a flat pattern that folds up to make a 3-D shape.
Method. Count the faces of the pyramid: the square base and the triangles.
- Count the triangular faces. A square base has 4 sides, so 4 triangles meet at the top point.4 triangles
- Add the base. The square base is one more face.4 + 1 = 5
- Write the answer. The pyramid has this many faces altogether.5
Check. Four triangles and one square give 5 faces. ✓
Watch out. Count the square base as a face too.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Write the three angles in terms of x, then add them up.
- Write the three angles and add them. The angles of a triangle add up to 180 degrees.2x + 3x + 4x = 180
- Collect the x terms. 2x + 3x + 4x is 9 lots of x.9x = 180
- Divide by 9 to find x. This gives the size of one x.180 ÷ 9 = 20, so x = 20
- Find the largest angle. The largest angle is 4x.4 × 20 = 80
- Write the answer. The largest angle is 80 degrees.80
Check. The three angles are 40, 60 and 80, and 40 + 60 + 80 = 180. ✓
Watch out. Find x first, then multiply to find the angle you need.
Total 21 marks. Allow about 21 minutes.
Exercise 10X SATs-style questions
Non-calculatorKey words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Split the shape into triangles, then multiply the number of triangles by 180.
- Split the shape into triangles. Draw every diagonal from one corner. This makes 6 - 2 triangles.6 - 2 = 4 triangles
- Add up the angles of the triangles. Each triangle has angles that add up to 180 degrees.4 × 180 = 720
- Write the answer. The interior angles of this shape add up to this total.720
Check. A shape with 6 sides makes 4 triangles inside it. ✓
Watch out. Use (n - 2) × 180. Do not multiply the number of sides by 180.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Add the angles you know, then take them away from the total for a straight line.
- Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).Total = 180°
- Add the angles you know. Put them together to see how much of the total is used.118° = 118°
- Take the angles you know away from the total. What is left over is the missing angle.180 - 118 = 62
- Write the answer. The missing angle is 62 degrees.62
Check. 62 + 118 = 180, so the angles add up correctly. ✓
Watch out. Use 180 for a straight line, not 360. The angles at a point add up to 360.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Multiply the diameter by 3.14.
- Write the formula. The circumference is pi times the diameter.C = π × d
- Put in the numbers. The question says to use π = 3.14.C = 3.14 × 9
- Work out the product. Multiply the diameter by pi.9 × 3.14 = 28.26
- Write the answer. The circumference is in the same unit as the diameter.28.26
Check. 28.26 ÷ 3.14 = 9. ✓
Watch out. The circumference is about three times the diameter, but use the exact value of pi given in the question.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Angles at a point add up to 360 degrees, so write the three angles as one equation.
- Write the angles as an equation. The three angles meet at a point, so they make a full turn.x + 2x + 3x = 360
- Collect the x terms. x + 2x + 3x is 6 lots of x.6x = 360
- Divide by 6. This gives the size of one x.360 ÷ 6 = 60
- Write the answer. x is 60 degrees.60
Check. 60 + 120 + 180 = 360. ✓
Watch out. Use 360 for a full turn, not 180.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Adjacent angles in a parallelogram add up to 180 degrees, so subtract the given angle from 180.
- Know the rule for a parallelogram. The two angles next to each other on the same side add up to 180 degrees.72° + ? = 180°
- Take the given angle away from 180. The missing angle is what is left over.180 - 72 = 108
- Write the answer. The angle is 108 degrees.108
Check. 72 + 108 = 180. ✓
Watch out. Next to each other means on the same side. Do not copy the angle.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Use the exterior angles, which share a full turn of 360 degrees.
- Write the rule. The exterior angles of a regular polygon add up to 360 degrees.360 degrees
- Divide 360 by the exterior angle. Each exterior angle is 24 degrees, so count how many 24s make 360.360 ÷ 24 = 15
- Write the answer. The polygon has 15 sides.15
Check. 15 × 24 = 360. ✓
Watch out. Use the exterior angle, not the interior angle, to find the number of sides.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Divide the circumference by 3.14 to find the diameter.
- Write the formula. The circumference is pi times the diameter.C = 3.14 × d
- Undo the multiplication by 3.14. Divide the circumference by 3.14 to find the diameter.47.1 ÷ 3.14 = 15
- Write the answer. The diameter is in the same unit as the circumference.15
Check. 15 × 3.14 = 47.1. ✓
Watch out. Divide the circumference by pi. Do not divide it by 2.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Add the known angles, then solve for x.
- Write the angles as an equation. The four angles of a quadrilateral add up to 360 degrees.x + 2x + 100 + 80 = 360
- Collect the numbers and the x terms. The two known angles add up to 180, and x + 2x is 3x.3x + 180 = 360
- Take 180 from 360. What is left is 3 lots of x.360 - 180 = 180, so 3x = 180
- Divide by 3. This gives the size of x.180 ÷ 3 = 60
- Write the answer. x is 60 degrees.60
Check. 60 + 120 + 100 + 80 = 360. ✓
Watch out. Write the equation with all four angles, not just three.
Key words. The radius goes from the centre to the edge. The diameter goes from edge to edge through the centre, so it is two radii. The circumference is the distance all the way round the edge.
Method. Work out the distance round one turn, then multiply by 10 turns.
- Find the distance in one turn. One turn of the wheel covers its circumference.C = 3.14 × 60 = 188.4 cm
- Multiply by the number of turns. Ten turns cover ten circumferences.188.4 × 10 = 1884 cm
- Write the answer. The wheel travels 1884 cm in 10 turns.1884
Check. 1884 ÷ 10 = 188.4 cm, and 188.4 ÷ 3.14 = 60 cm. ✓
Watch out. Find the circumference first, then multiply by the number of turns.
Key words. An angle is the turn between two lines. Angles are measured in degrees (°). A straight line makes a half turn of 180°. A full turn around a point is 360°.
Method. Share the total into equal parts using the ratio, then work out each angle.
- Add the parts of the ratio. Each part of the ratio is one share of the total.1 : 2 : 6: 9 parts in all
- Find the size of one part. Share the total of 180 degrees into 9 equal parts.180 ÷ 9 = 20
- Work out each angle. Multiply each ratio number by the size of one part.1 × 20 = 20, 2 × 20 = 40, 6 × 20 = 120
- Write the answer. The largest angle is 120 degrees.120
Check. The angles add up to 20 + 40 + 120 = 180. ✓
Watch out. Share the total by the number of parts, not by the number of angles.
Key words. The interior angles are the angles inside a shape. A polygon is a flat shape with straight sides. A regular polygon has all its sides equal and all its angles equal.
Method. Find the three angles, then check whether any of them is 90 degrees.
- Find each angle. Share 180 into 9 parts, so one part is 20 degrees.20, 40 and 120 degrees
- Look for a right angle. A right-angled triangle has an angle of exactly 90 degrees.none of the angles is 90
- Write the answer. Sam is not right, because the triangle has no right angle.No
Check. 20 + 40 + 120 = 180. ✓
Watch out. The largest angle here is obtuse, because it is more than 90 degrees.