Angles and shapes

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Count the faces of the shape, one at a time.

1. Picture the shape. A cube has six square faces, one on each side.
2. Count the faces. Count each one once and do not miss any.
   6 faces

Check. Counting in a different order gives 6 again. ✓

Watch out. Faces, edges and vertices are different things. Read the question carefully.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Count the edges of the shape, one at a time.

1. Picture the shape. An edge is a line where two faces meet. A cube has 4 edges on top, 4 on the bottom and 4 going up.
2. Count the edges. Count each one once and do not miss any.
   12 edges

Check. Counting in a different order gives 12 again. ✓

Watch out. Faces, edges and vertices are different things. Read the question carefully.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Count the vertices of the shape, one at a time.

1. Picture the shape. A vertex is a corner. A cuboid has 4 corners on the top and 4 on the bottom.
2. Count the vertices. Count each one once and do not miss any.
   8 vertices

Check. Counting in a different order gives 8 again. ✓

Watch out. Faces, edges and vertices are different things. Read the question carefully.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. List the faces, then find the one that matches the top face.

1. List the faces. The top and bottom are 4 cm by 2 cm. The front and back are 4 cm by 3 cm. The sides are 2 cm by 3 cm.
   top 4 by 2, bottom 4 by 2, front and back 4 by 3, sides 2 by 3
2. Find the match to the top. Only the bottom face is the same size and shape as the top face.
   bottom = top
3. Write the answer. Apart from the top, only one face matches it.
   1 other face

Check. The front and back are 4 by 3, so they do not match the top. ✓

Watch out. Check the size of each face, not just its shape.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Count the faces of the net and look at their shapes. Then name the 3-D shape that it folds into.

1. Count the squares. Each square becomes one face of the solid.
   6 squares
2. Check the faces. All six faces are squares of the same size.
   six equal squares
3. Name the solid. A 3-D shape with six equal square faces is a cube.
   cube

Check. The solid has 6 faces, the same as the net. ✓

Watch out. A cube has square faces. A cuboid has rectangular faces, and a cube is a special cuboid.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Count the faces of the net and look at their shapes. Then name the 3-D shape that it folds into.

1. Count the rectangles. Each rectangle becomes one face of the solid.
   6 rectangles
2. Look for matching faces. Opposite faces of a cuboid are the same size.
   the band has two pairs of equal faces, and the top and bottom match
3. Name the solid. A 3-D shape with six rectangular faces is a cuboid.
   cuboid

Check. The solid has 6 faces, the same as the net. ✓

Watch out. A cube has square faces. A cuboid has rectangular faces, and a cube is a special cuboid.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Count the faces of a cube, then compare the answer with what each child says.

1. Count the faces of a cube. A cube has six square faces, one on each side.
   6 faces
2. Compare with what each child says. Ruby says 8, and Tom says 6.
   6 ≠ 8, and 6 = 6
3. Decide. Tom is right, because a cube has 6 faces.
   Tom is right

Check. A cube has 6 faces, so the answer is 6. ✓

Watch out. Count the faces, do not guess the number.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Find the number of edges, then multiply by the length of one edge.

1. Write the number of edges. A cube has 12 edges.
   12 edges
2. Multiply by the length of one edge. All the edges of a cube are the same length.
   12 × 4 = 48
3. Write the answer.
   48 cm

Check. 48 ÷ 12 = 4. ✓

Watch out. Count the edges first. A cube has 12 of them.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Count the edges on the top, the bottom and the vertical edges.

1. Count the edges on the top. The top face is a rectangle with 4 edges.
   4 edges
2. Count the edges on the bottom. The bottom face is also a rectangle with 4 edges.
   4 edges
3. Count the vertical edges. Four edges join the top corners to the bottom corners.
   4 edges
4. Add them up. Add the three groups of 4.
   4 + 4 + 4 = 12 edges
5. Decide. A cuboid has 12 edges, so Sam is wrong.
   No

Check. A cuboid has the same 12 edges as a cube. ✓

Watch out. Count every edge, including the ones on the bottom.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Write the three numbers, then add them together.

1. Write the three numbers. A cube has 6 faces, 12 edges and 8 vertices.
   6 + 12 + 8
2. Add the numbers. Add the first two, then add the third.
   6 + 12 = 18, and 18 + 8 = 26
3. Write the answer.
   26

Check. 26 − 8 = 18, and 18 − 12 = 6. ✓

Watch out. Add the three numbers. Do not write them as a number of angles.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Think about what a cube and a cuboid have in common.

1. Is every cube a cuboid? A cube has 6 rectangular faces, so it is a cuboid.
   Yes
2. Is every cuboid a cube? A cuboid only becomes a cube if all its edges are the same length.
   No, not always
3. Write the answer.
   A cube is always a cuboid. A cuboid is only a cube if all its edges are equal.

Check. A cube is a special cuboid. ✓

Watch out. A cube is a special cuboid. Not every cuboid is a cube.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare the size of the angle with 90°, 180° and 360° to find its type.

1. Compare the angle with a right angle. A right angle is 90°.
   60° < 90°
2. Name the type of angle. An angle less than 90° is acute.
   acute
3. Write the answer. So the angle 60° is acute.
   acute

Check. 60° is less than 90°, so it is acute. ✓

Watch out. Look at the number of degrees, not at how long the arms look.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare the size of the angle with 90°, 180° and 360° to find its type.

1. Compare the angle with a right angle. A right angle is 90°.
   150° > 90°
2. Compare the angle with a straight line. A straight line is 180°.
   150° < 180°
3. Name the type of angle. An angle more than 90° and less than 180° is obtuse.
   obtuse
4. Write the answer. So the angle 150° is obtuse.
   obtuse

Check. 150° is between 90° and 180°, so it is obtuse. ✓

Watch out. Look at the number of degrees, not at how long the arms look.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare the size of the angle with 90°, 180° and 360° to find its type.

1. Compare the angle with a straight line. A straight line is 180°.
   250° > 180°
2. Compare the angle with a full turn. A full turn is 360°.
   250° < 360°
3. Name the type of angle. An angle more than 180° and less than 360° is reflex.
   reflex
4. Write the answer. So the angle 250° is reflex.
   reflex

Check. 250° is between 180° and 360°, so it is reflex. ✓

Watch out. Look at the number of degrees, not at how long the arms look.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare the angle with the three choices.

1. Compare with a right angle. The arms are wider than a right angle, so the angle is more than 90°.
   110° > 90°
2. Compare with a straight line. The arms are not as wide as a straight line, so the angle is less than 180°.
   110° < 180°
3. Choose the best estimate. Of the three choices, 110° is the only one between 90° and 180°.
   110°

Check. 110° is between 90° and 180°, which matches the picture. ✓

Watch out. 40° is too small and 250° is too big for this picture.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare the angle with a right angle, then look at the choices.

1. Compare with a right angle. The arms are much closer together than a right angle, so the angle is acute.
   less than 90°
2. Look at the choices. Only 30° is close to this size. The choice of 300° is reflex, and 60° is too big.
   30° is closest
3. Choose the best estimate. The arms are about one third of a right angle, so 30° is the best estimate.
   30°

Check. 30° is a third of a right angle, which matches. ✓

Watch out. Think of the right angle as 90° and compare with it.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Put the angles in order by their number of degrees, starting with the smallest.

1. Write the angles. These are the angles we need to put in order.
   150°, 45°, 260°, 90°
2. Compare the numbers. The smallest number is the smallest angle.
   45 < 90 < 150 < 260
3. Write the answer, smallest first.
   45°, 90°, 150°, 260°

Check. Each angle is smaller than the next one. ✓

Watch out. Do not put angles in order by how long their arms look.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Give the range for each type of angle, then compare the two ranges.

1. Give the range for a reflex angle. A reflex angle is more than 180° and less than 360°.
   180° < reflex < 360°
2. Give the range for an obtuse angle. An obtuse angle is more than 90° and less than 180°.
   90° < obtuse < 180°
3. Compare the two ranges. Every reflex angle is more than 180°, and every obtuse angle is less than 180°.
   Yes, Ali is right

Check. The two ranges do not overlap, and reflex is the bigger one. ✓

Watch out. Check the numbers at the ends of each range.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare each angle with 90° and 180°.

1. Compare each angle with 90°. We need an angle bigger than 90°.
   85 no, 95 yes, 180 yes, 270 yes
2. Compare with 180°. We need an angle smaller than 180°.
   95 < 180 yes, 180 is not less
3. Write the answer.
   95°

Check. 95° is more than 90° and less than 180°. ✓

Watch out. The angle must be bigger than a right angle and smaller than a straight line.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare 90° with the ranges for each type of angle.

1. Compare with a right angle. A right angle is exactly 90°.
   90° = 90°
2. Compare with the type names. An obtuse angle must be more than 90°, so 90° is not obtuse.
   90° is not more than 90°
3. Write the answer.
   No, 90° is a right angle

Check. Obtuse angles are more than 90°, so 90° is not obtuse. ✓

Watch out. A right angle is exactly 90°, and it is not obtuse.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Pick numbers in each range.

1. Acute angles. Acute angles are less than 90°, so pick three numbers from 1 to 89.
   20°, 45°, 80°
2. Obtuse angles. Obtuse angles are more than 90° and less than 180°.
   100°, 130°, 170°
3. Check the answer. Each angle must be in the right range.
   Any answers in these ranges are correct.

Check. Each acute angle is less than 90°, and each obtuse angle is between 90° and 180°. ✓

Watch out. An angle of exactly 90° is right, not acute or obtuse.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Put the centre on the vertex, line up the 0 line and read the scale where the other arm crosses it.

1. Put the centre of the protractor on the vertex. The centre must be at the corner of the angle.
   centre on the vertex
2. Line up one arm with the 0 line. Then the scale starts at 0 on that arm.
   start at 0 on the base line
3. Read the number where the other arm crosses the scale. Count the scale in 5s and 10s to find the number.
   the arm crosses the scale at 40
4. Write the answer with the degree sign. The angle is acute.
   40°

Check. 40° is acute, which matches the picture. ✓

Watch out. Read the scale that starts at 0 on the arm you lined up. Do not read the other scale by mistake.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Put the centre on the vertex, line up the 0 line and read the scale where the other arm crosses it.

1. Put the centre of the protractor on the vertex. The centre must be at the corner of the angle.
   centre on the vertex
2. Line up one arm with the 0 line. Then the scale starts at 0 on that arm.
   start at 0 on the base line
3. Read the number where the other arm crosses the scale. Count the scale in 5s and 10s to find the number.
   the arm crosses the scale at 115
4. Write the answer with the degree sign. The angle is obtuse.
   115°

Check. 115° is obtuse, which matches the picture. ✓

Watch out. Read the scale that starts at 0 on the arm you lined up. Do not read the other scale by mistake.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Put the centre on the vertex, line up the 0 line and read the scale where the other arm crosses it.

1. Put the centre of the protractor on the vertex. The centre must be at the corner of the angle.
   centre on the vertex
2. Line up one arm with the 0 line. Then the scale starts at 0 on that arm.
   start at 0 on the base line
3. Read the number where the other arm crosses the scale. Count the scale in 5s and 10s to find the number.
   the arm crosses the scale at 150
4. Write the answer with the degree sign. The angle is obtuse.
   150°

Check. 150° is obtuse, which matches the picture. ✓

Watch out. Read the scale that starts at 0 on the arm you lined up. Do not read the other scale by mistake.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Put the centre on the vertex, line up the 0 line and read the scale where the other arm crosses it.

1. Put the centre of the protractor on the vertex. The centre must be at the corner of the angle.
   centre on the vertex
2. Line up one arm with the 0 line. Then the scale starts at 0 on that arm.
   start at 0 on the base line
3. Read the number where the other arm crosses the scale. Count the scale in 5s and 10s to find the number.
   the arm crosses the scale at 25
4. Write the answer with the degree sign. The angle is acute.
   25°

Check. 25° is acute, which matches the picture. ✓

Watch out. Read the scale that starts at 0 on the arm you lined up. Do not read the other scale by mistake.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Put the centre on the vertex, line up the 0 line and read the scale where the other arm crosses it.

1. Put the centre of the protractor on the vertex. The centre must be at the corner of the angle.
   centre on the vertex
2. Line up one arm with the 0 line. Then the scale starts at 0 on that arm.
   start at 0 on the base line
3. Read the number where the other arm crosses the scale. Count the scale in 5s and 10s to find the number.
   the arm crosses the scale at 75
4. Write the answer with the degree sign. The angle is acute.
   75°

Check. 75° is acute, which matches the picture. ✓

Watch out. Read the scale that starts at 0 on the arm you lined up. Do not read the other scale by mistake.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Put the centre on the vertex, line up the 0 line and read the scale where the other arm crosses it.

1. Put the centre of the protractor on the vertex. The centre must be at the corner of the angle.
   centre on the vertex
2. Line up one arm with the 0 line. Then the scale starts at 0 on that arm.
   start at 0 on the base line
3. Read the number where the other arm crosses the scale. Count the scale in 5s and 10s to find the number.
   the arm crosses the scale at 135
4. Write the answer with the degree sign. The angle is obtuse.
   135°

Check. 135° is obtuse, which matches the picture. ✓

Watch out. Read the scale that starts at 0 on the arm you lined up. Do not read the other scale by mistake.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known angles at a point, then take them away from 360.

1. Add the parts we know. The angles at a point that we know are 130.
   130
2. Take this away from 360. The angles at a point add up to 360, so the missing one is what is left.
   360 − 130 = 230
3. Write the answer.
   230°

Check. 130 + 230 = 360. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Check each part of the drawing: the centre, the 0 line and the scale.

1. Check where the centre was. The centre must be on the vertex, the corner of the angle.
   centre not on the vertex
2. Check the 0 line. One arm must lie along the 0 line of the scale.
   the 0 line was not lined up
3. Measure again. Read the scale that starts at 0 on the arm.
   55° is the correct size

Check. Measure the drawing again: it should read 55°. ✓

Watch out. A small mistake in the centre or the 0 line gives the wrong answer.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Draw one arm, put the protractor on it, mark the angle, then draw the second arm.

1. Draw one straight arm. This arm is the 0 line.
   one arm drawn
2. Put the centre on the end of the arm. The end of the arm is the vertex, where the two arms meet.
   centre on the vertex
3. Find the number on the scale and make a dot. Count round the scale to the number in the question.
   dot at 70°
4. Join the dot to the vertex. The second arm makes the angle.
   70° angle drawn

Check. Measure your drawing: it should read 70°. ✓

Watch out. Use the scale that starts at 0 on the arm you drew.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Draw one arm, put the protractor on it, mark the angle, then draw the second arm.

1. Draw one straight arm. This arm is the 0 line.
   one arm drawn
2. Put the centre on the end of the arm. The end of the arm is the vertex, where the two arms meet.
   centre on the vertex
3. Find the number on the scale and make a dot. Count round the scale to the number in the question.
   dot at 125°
4. Join the dot to the vertex. The second arm makes the angle.
   125° angle drawn

Check. Measure your drawing: it should read 125°. ✓

Watch out. Use the scale that starts at 0 on the arm you drew.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Put the centre on the vertex, line up the 0 line and read the scale where the other arm crosses it.

1. Put the centre of the protractor on the vertex. The centre must be at the corner of the angle.
   centre on the vertex
2. Line up one arm with the 0 line. Then the scale starts at 0 on that arm.
   start at 0 on the base line
3. Read the number where the other arm crosses the scale. Count the scale in 5s and 10s to find the number.
   the arm crosses the scale at 170
4. Write the answer with the degree sign. The angle is obtuse.
   170°

Check. 170° is obtuse, which matches the picture. ✓

Watch out. Read the scale that starts at 0 on the arm you lined up. Do not read the other scale by mistake.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Estimate first, then measure.

1. Estimate first. The arm is past the right angle line, so the angle is more than 90°.
   more than 90°
2. Measure with the protractor. Put the centre on the vertex and line up the 0 line, then read the scale.
   the arm crosses the scale at 110
3. Write the answer. The estimate and the measurement agree.
   110°

Check. 110° is more than 90°, which matches the estimate. ✓

Watch out. An estimate tells you which scale and which end of the scale to read.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known angles on a straight line, then take them away from 180.

1. Add the parts we know. The angles on a straight line that we know are 125.
   125
2. Take this away from 180. The angles on a straight line add up to 180, so the missing one is what is left.
   180 − 125 = 55
3. Write the answer.
   55°

Check. 125 + 55 = 180. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known angles on a straight line, then take them away from 180.

1. Add the parts we know. The angles on a straight line that we know are 30.
   30
2. Take this away from 180. The angles on a straight line add up to 180, so the missing one is what is left.
   180 − 30 = 150
3. Write the answer.
   150°

Check. 30 + 150 = 180. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known angles at a point, then take them away from 360.

1. Add the parts we know. The angles at a point that we know are 100 + 85 + 60.
   100 + 85 + 60 = 245
2. Take this away from 360. The angles at a point add up to 360, so the missing one is what is left.
   360 − 245 = 115
3. Write the answer.
   115°

Check. 245 + 115 = 360. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known angles at a point, then take them away from 360.

1. Add the parts we know. The angles at a point that we know are 120 + 50 + 60.
   120 + 50 + 60 = 230
2. Take this away from 360. The angles at a point add up to 360, so the missing one is what is left.
   360 − 230 = 130
3. Write the answer.
   130°

Check. 230 + 130 = 360. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known turns, then take them away from 360.

1. Add the parts we know. The turns that we know are 270.
   270
2. Take this away from 360. The turns add up to 360, so the missing one is what is left.
   360 − 270 = 90
3. Write the answer.
   90°

Check. 270 + 90 = 360. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the two angles, then think of other sizes that also work.

1. Add the two angles. Angles on a straight line add up to 180°.
   90 + 90 = 180
2. Try other sizes. Two angles on a straight line can be 60° and 120°, and these also add up to 180°.
   60 + 120 = 180
3. Decide. Sam is right only in the special case when both angles are 90°.
   Sometimes

Check. 90 + 90 = 180, and 60 + 120 = 180 too. ✓

Watch out. One example that works does not prove that something is always true.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add up the four angles, then try different sizes.

1. Add four right angles. Four angles of 90° add up to 360°.
   4 × 90 = 360
2. Try four different angles. Angles at a point do not all have to be equal.
   100 + 80 + 120 + 60 = 360
3. Decide. So Tom is wrong. The four angles only need to add up to 360°.
   No

Check. Both sets add up to 360°. ✓

Watch out. The total must be 360°. The sizes of the angles can be different.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Pick four angles that add up to 360°, in two different ways.

1. First set. Four right angles.
   90 + 90 + 90 + 90
2. Second set. Four different angles, added together.
   100 + 80 + 120 + 60
3. Check the answer. Both sets add up to the total for a point.
   Both totals are 360

Check. 90 + 90 + 90 + 90 = 360 and 100 + 80 + 120 + 60 = 360. ✓

Watch out. Check the total of each set. A set that adds up to 350 is not right.

Key words. A rectangle has four right angles. Its opposite sides are equal in length. The perimeter is the distance all the way round the outside.

Method. The perimeter is the distance all the way round, so add all four sides.

1. Add the two long sides. A rectangle has two long sides of the same length.
   15 + 15 = 30
2. Add the two short sides. It also has two short sides of the same length.
   6 + 6 = 12
3. Add all four sides. The perimeter is the total of all the sides.
   30 + 12 = 42
4. Write the answer.
   42 cm

Check. 2 × (15 + 6) = 42. ✓

Watch out. Do not forget the second long side and the second short side.

Key words. A rectangle has four right angles. Its opposite sides are equal in length. The perimeter is the distance all the way round the outside.

Method. Half of the perimeter is one length and one width. Find the missing side from that.

1. Halve the perimeter. Half of the perimeter is one long side and one short side added together.
   30 ÷ 2 = 15
2. Take away the side we know. What is left is the other side.
   15 − 9 = 6
3. Write the answer.
   6 cm

Check. 2 × (6 + 9) = 30. ✓

Watch out. Halve the perimeter first. Do not take the known side away from the whole perimeter.

Key words. A rectangle has four right angles. Its opposite sides are equal in length. The perimeter is the distance all the way round the outside.

Method. Half of the perimeter is one length and one width. Find the missing side from that.

1. Halve the perimeter. Half of the perimeter is one long side and one short side added together.
   24 ÷ 2 = 12
2. Take away the side we know. What is left is the other side.
   12 − 5 = 7
3. Write the answer.
   7 cm

Check. 2 × (7 + 5) = 24. ✓

Watch out. Halve the perimeter first. Do not take the known side away from the whole perimeter.

Method. Add the known parts of the long side, then take them away from 15.

1. Add the parts we know. The parts of the long side that we know are 6.
   6
2. Take this away from 15. The parts of the long side add up to 15, so the missing one is what is left.
   15 − 6 = 9
3. Write the answer.
   9 cm

Check. 6 + 9 = 15. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Method. Add the known angles at the corner, then take them away from 90.

1. Add the parts we know. The angles at the corner that we know are 62.
   62
2. Take this away from 90. The angles at the corner add up to 90, so the missing one is what is left.
   90 − 62 = 28
3. Write the answer.
   28°

Check. 62 + 28 = 90. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Method. Add the known angles at the corner, then take them away from 90.

1. Add the parts we know. The angles at the corner that we know are 48.
   48
2. Take this away from 90. The angles at the corner add up to 90, so the missing one is what is left.
   90 − 48 = 42
3. Write the answer.
   42°

Check. 48 + 42 = 90. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. A rectangle has four right angles. Its opposite sides are equal in length. The perimeter is the distance all the way round the outside.

Method. Half of the perimeter is one length and one width. Find the missing side from that.

1. Halve the perimeter. Half of the perimeter is one long side and one short side added together.
   20 ÷ 2 = 10
2. Take away the side we know. What is left is the other side.
   10 − 6 = 4
3. Write the answer.
   4 cm

Check. 2 × (4 + 6) = 20. ✓

Watch out. Halve the perimeter first. Do not take the known side away from the whole perimeter.

Key words. A rectangle has four right angles. Its opposite sides are equal in length. The perimeter is the distance all the way round the outside.

Method. Find the length first, then add all four sides.

1. Find the length. The length is 4 cm more than the width.
   3 + 4 = 7 cm
2. Add the four sides. Two sides are 7 cm and two sides are 3 cm.
   7 + 3 + 7 + 3 = 20 cm
3. Write the answer.
   20 cm

Check. 2 × (7 + 3) = 20. ✓

Watch out. Find the length before you add the sides.

Key words. A rectangle has four right angles. Its opposite sides are equal in length. The perimeter is the distance all the way round the outside.

Method. Think about the rules for the sides of a rectangle.

1. Think about a square. A square has four equal sides and four right angles, so it is a rectangle.
   square = rectangle
2. Push the shape over. The sides stay equal when the shape leans, but the corners are no longer right angles.
   equal sides, not right angles
3. Decide. So a shape with four equal sides is not always a rectangle.
   No

Check. A rectangle needs four right angles as well as opposite sides that are equal. ✓

Watch out. Four equal sides are not enough. A rectangle also needs four right angles.

Key words. A rectangle has four right angles. Its opposite sides are equal in length. The perimeter is the distance all the way round the outside.

Method. Look at the pairs of opposite sides.

1. Look at the pairs of opposite sides. A rectangle has two pairs of equal opposite sides.
   pairs: 5 and 5, 6 and 7
2. Compare the second pair. The two sides of this pair must be equal.
   6 ≠ 7
3. Decide. So this shape cannot be a rectangle.
   No

Check. The other pair, 5 cm and 5 cm, is equal, but 6 cm and 7 cm are not. ✓

Watch out. Check both pairs of opposite sides, not just one.

Key words. 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.

Method. Count the sides, then use the name for that number of sides.

1. Count the sides. Each side is one straight edge.
   5 sides
2. Use the name for that number of sides. A shape with 5 sides is called a pentagon.
   pentagon

Check. A pentagon has 5 sides. ✓

Watch out. A quadrilateral has 4 sides, a pentagon 5, a hexagon 6 and an octagon 8.

Key words. 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.

Method. Count the sides, then use the name for that number of sides.

1. Count the sides. Each side is one straight edge.
   6 sides
2. Use the name for that number of sides. A shape with 6 sides is called a hexagon.
   hexagon

Check. A hexagon has 6 sides. ✓

Watch out. A quadrilateral has 4 sides, a pentagon 5, a hexagon 6 and an octagon 8.

Key words. 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.

Method. A regular shape has equal sides, so multiply the number of sides by the length of one side.

1. Count the sides. A regular shape has the same length of side all the way round.
   6 sides
2. Multiply by the length of one side. Every side is the same length, so multiply.
   6 × 5 = 30
3. Write the answer.
   30 cm

Check. 30 ÷ 6 = 5. ✓

Watch out. Count the sides carefully. A hexagon has 6 sides, not 5.

Key words. 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.

Method. Share the perimeter equally between the sides.

1. Divide the perimeter by the length of one side. Every side is 6 cm, so count how many 6 cm sides fit in 36 cm.
   36 ÷ 6 = 6
2. Write the number of sides. A shape with 6 sides is a hexagon.
   6 sides

Check. 6 × 6 = 36 cm. ✓

Watch out. Divide the perimeter by the length of one side to find the number of sides.

Key words. 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.

Method. A regular shape has equal sides, so multiply the number of sides by the length of one side.

1. Count the sides. A regular shape has the same length of side all the way round.
   8 sides
2. Multiply by the length of one side. Every side is the same length, so multiply.
   8 × 7 = 56
3. Write the answer.
   56 cm

Check. 56 ÷ 8 = 7. ✓

Watch out. Count the sides carefully. A hexagon has 6 sides, not 5.

Key words. 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.

Method. Look at the lengths of the sides.

1. Look at the picture. The sides are different lengths. Some are long and some are short.
   long and short sides
2. Check the rule. A regular shape must have all its sides equal.
   not all equal
3. Decide. So this hexagon is irregular.
   No

Check. The rule for regular is broken, so the answer is no. ✓

Watch out. One different side makes the shape irregular.

Key words. 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.

Method. Think about the rules: a regular shape has equal sides and equal angles.

1. Look at each shape. A square, a regular pentagon and a regular triangle all have equal sides.
   square, pentagon, triangle: regular
2. Check the rectangle. A rectangle has equal angles, but its sides are not all equal.
   rectangle: not regular
3. Decide. The rectangle is the odd one out, because it is the only shape that is not regular.
   The rectangle

Check. Only the rectangle has sides that are not all equal. ✓

Watch out. Check the sides as well as the angles.

Key words. 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.

Method. Think about an example of a regular shape.

1. Pick a regular shape. A regular triangle has 3 equal sides and 3 equal angles.
   regular triangle
2. Check its angles. The angles are equal, but they are not right angles.
   not 90°
3. Decide. So Sam is wrong: regular shapes do not have to have right angles.
   No

Check. A regular triangle has equal sides and angles, but no right angles. ✓

Watch out. A right angle is only in some regular shapes, such as a square.

Key words. 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.

Method. Share the perimeter equally between the five sides.

1. Count the sides. A regular pentagon has 5 equal sides.
   5 sides
2. Divide the perimeter by 5. All the sides are equal, so share the perimeter equally.
   45 ÷ 5 = 9
3. Write the answer.
   9 cm

Check. 5 × 9 = 45 cm. ✓

Watch out. The perimeter is the total of all the sides, so divide it to find one side.

Key words. 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.

Method. Find each perimeter, then take the smaller from the bigger.

1. Find the perimeter of the hexagon. A hexagon has 6 sides of 4 cm.
   6 × 4 = 24 cm
2. Find the perimeter of the triangle. A triangle has 3 sides of 4 cm.
   3 × 4 = 12 cm
3. Find the difference. Take the smaller perimeter from the bigger one.
   24 − 12 = 12 cm

Check. 12 + 12 = 24, the hexagon perimeter. ✓

Watch out. Find each perimeter first. Do not subtract the side lengths.

Key words. 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.

Method. Think about what a regular shape needs.

1. Write the rule. A regular shape has all its sides equal and all its angles equal.
   all sides equal
2. Compare with the shape. If the sides are not all the same length, the shape cannot be regular.
   unequal sides
3. Write the answer.
   No

Check. Unequal sides break the rule for a regular shape. ✓

Watch out. Equal sides are needed for a regular shape.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare the size of the angle with 90°, 180° and 360° to find its type.

1. Compare the angle with a right angle. A right angle is 90°.
   135° > 90°
2. Compare the angle with a straight line. A straight line is 180°.
   135° < 180°
3. Name the type of angle. An angle more than 90° and less than 180° is obtuse.
   obtuse
4. Write the answer. So the angle 135° is obtuse.
   obtuse

Check. 135° is between 90° and 180°, so it is obtuse. ✓

Watch out. Look at the number of degrees, not at how long the arms look.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare the size of the angle with 90°, 180° and 360° to find its type.

1. Compare the angle with a straight line. A straight line is 180°.
   250° > 180°
2. Compare the angle with a full turn. A full turn is 360°.
   250° < 360°
3. Name the type of angle. An angle more than 180° and less than 360° is reflex.
   reflex
4. Write the answer. So the angle 250° is reflex.
   reflex

Check. 250° is between 180° and 360°, so it is reflex. ✓

Watch out. Look at the number of degrees, not at how long the arms look.

Key words. A protractor measures angles. Its centre goes on the vertex, and one arm lies along the 0 line.

Method. Put the centre on the vertex, line up the 0 line and read the scale where the other arm crosses it.

1. Put the centre of the protractor on the vertex. The centre must be at the corner of the angle.
   centre on the vertex
2. Line up one arm with the 0 line. Then the scale starts at 0 on that arm.
   start at 0 on the base line
3. Read the number where the other arm crosses the scale. Count the scale in 5s and 10s to find the number.
   the arm crosses the scale at 120
4. Write the answer with the degree sign. The angle is obtuse.
   120°

Check. 120° is obtuse, which matches the picture. ✓

Watch out. Read the scale that starts at 0 on the arm you lined up. Do not read the other scale by mistake.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known angles on a straight line, then take them away from 180.

1. Add the parts we know. The angles on a straight line that we know are 40.
   40
2. Take this away from 180. The angles on a straight line add up to 180, so the missing one is what is left.
   180 − 40 = 140
3. Write the answer.
   140°

Check. 40 + 140 = 180. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known angles at a point, then take them away from 360.

1. Add the parts we know. The angles at a point that we know are 100 + 90 + 70.
   100 + 90 + 70 = 260
2. Take this away from 360. The angles at a point add up to 360, so the missing one is what is left.
   360 − 260 = 100
3. Write the answer.
   100°

Check. 260 + 100 = 360. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. An angle is the amount of turn between two lines (the arms) that meet at a point (the vertex). We measure angles in degrees, written with the symbol °.

Method. Each quarter turn is a right angle, so multiply the number of quarter turns by 90.

1. Count the quarter turns. A whole turn is four quarter turns.
   3 quarter turns
2. Multiply by 90°. Each quarter turn is a right angle, which is 90°.
   3 × 90 = 270
3. Write the answer.
   270°

Check. 270 ÷ 90 = 3. ✓

Watch out. A right angle is 90°, not 9° or 100°.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Find the number of edges, then multiply by the length of one edge.

1. Write the number of edges. A cube has 12 edges.
   12 edges
2. Multiply by the length of one edge. All the edges of a cube are the same length.
   12 × 5 = 60
3. Write the answer.
   60 cm

Check. 60 ÷ 12 = 5. ✓

Watch out. Count the edges first. A cube has 12 of them.

Key words. A rectangle has four right angles. Its opposite sides are equal in length. The perimeter is the distance all the way round the outside.

Method. Half of the perimeter is one length and one width. Find the missing side from that.

1. Halve the perimeter. Half of the perimeter is one long side and one short side added together.
   26 ÷ 2 = 13
2. Take away the side we know. What is left is the other side.
   13 − 8 = 5
3. Write the answer.
   5 cm

Check. 2 × (5 + 8) = 26. ✓

Watch out. Halve the perimeter first. Do not take the known side away from the whole perimeter.

Method. Add the known angles at the corner, then take them away from 90.

1. Add the parts we know. The angles at the corner that we know are 40.
   40
2. Take this away from 90. The angles at the corner add up to 90, so the missing one is what is left.
   90 − 40 = 50
3. Write the answer.
   50°

Check. 40 + 50 = 90. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. 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.

Method. Check the sides and the angles of a square.

1. Check the sides. A square has 4 equal sides.
   4 equal sides
2. Check the angles. A square has 4 equal right angles.
   4 angles of 90°
3. Decide. Both rules are true, so a square is regular.
   Yes

Check. A regular shape has equal sides and equal angles, and a square does. ✓

Watch out. Both rules must be true.

Key words. An acute angle is less than 90°. A right angle is exactly 90°. An obtuse angle is more than 90° and less than 180°. A reflex angle is more than 180° and less than 360°.

Method. Compare the size of the angle with 90°, 180° and 360° to find its type.

1. Compare the angle with a right angle. A right angle is 90°.
   98° > 90°
2. Compare the angle with a straight line. A straight line is 180°.
   98° < 180°
3. Name the type of angle. An angle more than 90° and less than 180° is obtuse.
   obtuse
4. Write the answer. So the angle 98° is obtuse.
   obtuse

Check. 98° is between 90° and 180°, so it is obtuse. ✓

Watch out. Look at the number of degrees, not at how long the arms look.

Key words. An angle is the amount of turn between two lines (the arms) that meet at a point (the vertex). We measure angles in degrees, written with the symbol °.

Method. Learn these facts about turns: a right angle is 90, a half turn is 180 and a whole turn is 360.

1. Recall the fact. A half turn is half of a full turn, and a full turn is 360°.
   a half turn = 180°
2. Write the answer.
   180°

Check. 180 is the right number for a half turn. ✓

Watch out. A whole turn is bigger than a half turn, and a half turn is bigger than a right angle.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known angles on a straight line, then take them away from 180.

1. Add the parts we know. The angles on a straight line that we know are 72.
   72
2. Take this away from 180. The angles on a straight line add up to 180, so the missing one is what is left.
   180 − 72 = 108
3. Write the answer.
   108°

Check. 72 + 108 = 180. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. Angles on a straight line add up to 180°, which is a half turn. Angles at a point add up to 360°, which is a whole turn.

Method. Add the known angles at a point, then take them away from 360.

1. Add the parts we know. The angles at a point that we know are 120 + 85.
   120 + 85 = 205
2. Take this away from 360. The angles at a point add up to 360, so the missing one is what is left.
   360 − 205 = 155
3. Write the answer.
   155°

Check. 205 + 155 = 360. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Count the edges of the shape, one at a time.

1. Picture the shape. A cube has 4 edges on the top, 4 on the bottom and 4 going up.
2. Count the edges. Count each one once and do not miss any.
   12 edges

Check. Counting in a different order gives 12 again. ✓

Watch out. Faces, edges and vertices are different things. Read the question carefully.

Key words. A face is a flat side of a 3-D shape. An edge is where two faces meet. A vertex is a corner (more than one is vertices). A net is a flat pattern that folds up into a 3-D shape.

Method. Count the faces of the net and look at their shapes. Then name the 3-D shape that it folds into.

1. Count the rectangles. Each rectangle becomes one face of the solid.
   6 rectangles
2. Look for matching faces. Opposite faces of a cuboid are the same size.
   the band has two pairs of equal faces, and the top and bottom match
3. Name the solid. A 3-D shape with six rectangular faces is a cuboid.
   cuboid

Check. The solid has 6 faces, the same as the net. ✓

Watch out. A cube has square faces. A cuboid has rectangular faces, and a cube is a special cuboid.

Method. Add the known parts of the long side, then take them away from 18.

1. Add the parts we know. The parts of the long side that we know are 7.
   7
2. Take this away from 18. The parts of the long side add up to 18, so the missing one is what is left.
   18 − 7 = 11
3. Write the answer.
   11 cm

Check. 7 + 11 = 18. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Method. Add the known angles at the corner, then take them away from 90.

1. Add the parts we know. The angles at the corner that we know are 28.
   28
2. Take this away from 90. The angles at the corner add up to 90, so the missing one is what is left.
   90 − 28 = 62
3. Write the answer.
   62°

Check. 28 + 62 = 90. ✓

Watch out. Do not add the missing part to the total. Take the known parts away from the total.

Key words. 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.

Method. A regular shape has equal sides, so multiply the number of sides by the length of one side.

1. Count the sides. A regular shape has the same length of side all the way round.
   6 sides
2. Multiply by the length of one side. Every side is the same length, so multiply.
   6 × 7 = 42
3. Write the answer.
   42 cm

Check. 42 ÷ 6 = 7. ✓

Watch out. Count the sides carefully. A hexagon has 6 sides, not 5.

Key words. A rectangle has four right angles. Its opposite sides are equal in length. The perimeter is the distance all the way round the outside.

Method. Halve the perimeter, then use the rule that the length is 4 m more than the width.

1. Halve the perimeter. Half of the perimeter is one length and one width.
   64 ÷ 2 = 32
2. Take away the extra 4 m. The length is 4 m more than the width, so take 4 away first.
   32 − 4 = 28
3. Find the width. Two widths make 28 m, so one width is half of 28.
   28 ÷ 2 = 14 m
4. Find the length. The length is 4 m more than the width.
   14 + 4 = 18 m
5. Write the answer.
   Length 18 m, width 14 m

Check. 2 × (18 + 14) = 64 m, which is the perimeter. ✓

Watch out. Halve the perimeter first. Do not take 4 m away from the whole perimeter.