Year 6 Angles, Shapes and Circles: check-up and SATs-style questions

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.

1. Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).
   Total = 180°
2. Add the angles you know. Put them together to see how much of the total is used.
   75° = 75°
3. Take the angles you know away from the total. What is left over is the missing angle.
   180 - 75 = 105
4. 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.

1. Remember the total for angles at a point. Angles around a point add up to 360 degrees (a full turn).
   Total = 360°
2. Add the angles you know. Put them together to see how much of the total is used.
   90° + 110° + 70° = 270°
3. Take the angles you know away from the total. What is left over is the missing angle.
   360 - 270 = 90
4. 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.

1. Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).
   Total = 180°
2. Add the angles you know. Put them together to see how much of the total is used.
   48° = 48°
3. Take the angles you know away from the total. What is left over is the missing angle.
   180 - 48 = 132
4. 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.

1. Remember the total for a triangle. The three angles in any triangle add up to 180 degrees.
   Total = 180°
2. Add the angles you know. Put them together to see how much of the total is used.
   55° + 65° = 120°
3. Take the angles you know away from the total. What is left over is the missing angle.
   180 - 120 = 60
4. 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.

1. 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°
2. Add the angles you know. Put them together to see how much of the total is used.
   90° + 75° + 120° = 285°
3. Take the angles you know away from the total. What is left over is the missing angle.
   360 - 285 = 75
4. 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.

1. 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
2. Share the sum equally between the angles. All the angles of a regular polygon are the same size.
   1080 ÷ 8 = 135
3. 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.

1. Find the diameter first. The diameter is twice the radius.
   2 × 4 = 8
2. Write the formula. The circumference is pi times the diameter.
   C = π × d
3. Put in the numbers. The question says to use π = 3.14.
   C = 3.14 × 8
4. Work out the product. Multiply the diameter by pi.
   8 × 3.14 = 25.12
5. 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.

1. Write the formula. The circumference is pi times the diameter.
   C = π × d
2. Put in the numbers. The question says to use π = 22/7.
   C = 22/7 × 21
3. Work out the product. Multiply the diameter by pi.
   21 ÷ 7 = 3, then 3 × 22 = 66
4. 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.

1. Count the triangular faces. A square base has 4 sides, so 4 triangles meet at the top point.
   4 triangles
2. Add the base. The square base is one more face.
   4 + 1 = 5
3. 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.

1. Write the three angles and add them. The angles of a triangle add up to 180 degrees.
   2x + 3x + 4x = 180
2. Collect the x terms. 2x + 3x + 4x is 9 lots of x.
   9x = 180
3. Divide by 9 to find x. This gives the size of one x.
   180 ÷ 9 = 20, so x = 20
4. Find the largest angle. The largest angle is 4x.
   4 × 20 = 80
5. 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.

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.

1. Split the shape into triangles. Draw every diagonal from one corner. This makes 6 - 2 triangles.
   6 - 2 = 4 triangles
2. Add up the angles of the triangles. Each triangle has angles that add up to 180 degrees.
   4 × 180 = 720
3. 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.

1. Remember the total for a straight line. Angles on a straight line add up to 180 degrees (a half turn).
   Total = 180°
2. Add the angles you know. Put them together to see how much of the total is used.
   118° = 118°
3. Take the angles you know away from the total. What is left over is the missing angle.
   180 - 118 = 62
4. 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.

1. Write the formula. The circumference is pi times the diameter.
   C = π × d
2. Put in the numbers. The question says to use π = 3.14.
   C = 3.14 × 9
3. Work out the product. Multiply the diameter by pi.
   9 × 3.14 = 28.26
4. 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.

1. Write the angles as an equation. The three angles meet at a point, so they make a full turn.
   x + 2x + 3x = 360
2. Collect the x terms. x + 2x + 3x is 6 lots of x.
   6x = 360
3. Divide by 6. This gives the size of one x.
   360 ÷ 6 = 60
4. 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.

1. Know the rule for a parallelogram. The two angles next to each other on the same side add up to 180 degrees.
   72° + ? = 180°
2. Take the given angle away from 180. The missing angle is what is left over.
   180 - 72 = 108
3. 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.

1. Write the rule. The exterior angles of a regular polygon add up to 360 degrees.
   360 degrees
2. Divide 360 by the exterior angle. Each exterior angle is 24 degrees, so count how many 24s make 360.
   360 ÷ 24 = 15
3. 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.

1. Write the formula. The circumference is pi times the diameter.
   C = 3.14 × d
2. Undo the multiplication by 3.14. Divide the circumference by 3.14 to find the diameter.
   47.1 ÷ 3.14 = 15
3. 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.

1. Write the angles as an equation. The four angles of a quadrilateral add up to 360 degrees.
   x + 2x + 100 + 80 = 360
2. Collect the numbers and the x terms. The two known angles add up to 180, and x + 2x is 3x.
   3x + 180 = 360
3. Take 180 from 360. What is left is 3 lots of x.
   360 - 180 = 180, so 3x = 180
4. Divide by 3. This gives the size of x.
   180 ÷ 3 = 60
5. 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.

1. Find the distance in one turn. One turn of the wheel covers its circumference.
   C = 3.14 × 60 = 188.4 cm
2. Multiply by the number of turns. Ten turns cover ten circumferences.
   188.4 × 10 = 1884 cm
3. 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.