Year 6 Maths reasoning (SATs) Geometry: angles and shapes: check-up and practice test questions

Method. Take the sum of the known angles from 360.

1. Angles at a point add up to 360 degrees. A full turn is 360 degrees.
   360
2. Add the known angles. Add the angles you know.
   100 + 120 = 220
3. Take the total from 360. The missing angle is what is left.
   360 - 220 = 140
4. Write the answer.
   140

Check. 100 + 120 + 140 = 360. ✓

Watch out. Do not use 180 for angles at a point. A full turn is 360.

Method. Take the known angle from 180.

1. Angles on a straight line add up to 180 degrees. A straight line is half a turn.
   180
2. Subtract the known angle. Take 58 from 180.
   180 - 58 = 122
3. Write the answer.
   122

Check. 58 + 122 = 180. ✓

Watch out. Straight line: 180. Around a point: 360.

Method. Take the sum of the two known angles from 180.

1. Angles in a triangle add up to 180 degrees. Every triangle has three angles that total 180.
   180
2. Add the two known angles. Add the angles you know.
   47 + 73 = 120
3. Take the total from 180. Angle w is what is left.
   180 - 120 = 60
4. Write the answer.
   60

Check. 47 + 73 + 60 = 180. ✓

Watch out. Check that the three angles add up to 180.

Method. Take the apex from 180, then share the rest equally between the two base angles.

1. The two base angles are equal. In an isosceles triangle the base angles are the same.
   base angles are equal
2. Find the total of the two base angles. Take the apex angle from 180.
   180 - 70 = 110
3. Share it between the two base angles. Divide by 2.
   110 / 2 = 55
4. Write the answer.
   55

Check. 70 + 2 x 55 = 180. ✓

Watch out. The two base angles are equal, not the apex and a base angle.

Method. Take the total of the three known angles from 360.

1. Angles in a quadrilateral add up to 360 degrees. Split the shape into two triangles to see why.
   360
2. Add the three known angles. Add them together.
   75 + 110 + 95 = 280
3. Take the total from 360. The fourth angle is what is left.
   360 - 280 = 80
4. Write the answer.
   80

Check. 75 + 110 + 95 + 80 = 360. ✓

Watch out. Quadrilaterals have 360 degrees; triangles have 180.

Method. Find the sum of the interior angles, then divide by the number of sides.

1. Work out the total angle inside the shape. A shape with 10 sides has 8 triangles inside it, so the total is (10 - 2) x 180.
   (10 - 2) x 180 = 1440
2. Share the total equally between the angles. All 10 angles of a regular shape are the same size.
   1440 / 10 = 144
3. Write the answer.
   144

Check. 144 x 10 = 1440. ✓

Watch out. Check the total with the sum formula: (n - 2) x 180.

Method. Divide 360 by the number of sides.

1. Exterior angles of any polygon add up to 360 degrees. Walk round the shape and turn at each corner.
   360
2. Share equally for a regular shape. Each of the 10 turns is the same size.
   360 / 10 = 36
3. Write the answer.
   36

Check. 36 x 10 = 360. ✓

Watch out. Exterior angle plus interior angle = 180.

Method. Count the edges: the lines where two faces meet.

1. Picture the shape. A cube has 6 faces, 12 edges and 8 vertices.
   faces 6, edges 12, vertices 8
2. Count the edges. An edge is where two faces meet.
   12 edges
3. Write the answer.
   12

Check. 6 faces + 8 vertices - 12 edges = 2. ✓

Watch out. Edges are not faces. A cube has 12 edges, not 6.

Method. Check the angles of a rhombus.

1. Compare. A square is a rhombus with four right angles, but a rhombus can have other angles.
   not always
2. Answer. So Ben is wrong.
   no
3. Write the answer.
   no

Check. A rhombus with angles 60 and 120 is not right-angled. ✓

Method. Compare the size with 90 and 180 degrees.

1. Compare with 90 and 180. 160 degrees is compared with the rules for each type.
   160 is between 90 and 180 degrees
2. Name the angle. An angle that is between 90 and 180 degrees is obtuse.
   obtuse
3. Write the answer.
   obtuse

Check. 160 is between 90 and 180 degrees. ✓

Watch out. An angle of exactly 180 degrees is a straight line; more than 180 is reflex.

Method. Take the known angle from 180.

1. Angles on a straight line add up to 180 degrees. A straight line is half a turn.
   180
2. Subtract the known angle. Take 48 from 180.
   180 - 48 = 132
3. Write the answer.
   132

Check. 48 + 132 = 180. ✓

Watch out. Straight line: 180. Around a point: 360.

Method. Take the sum of the two known angles from 180.

1. Angles in a triangle add up to 180 degrees. Every triangle has three angles that total 180.
   180
2. Add the two known angles. Add the angles you know.
   65 + 55 = 120
3. Take the total from 180. Angle q is what is left.
   180 - 120 = 60
4. Write the answer.
   60

Check. 65 + 55 + 60 = 180. ✓

Watch out. Check that the three angles add up to 180.

Method. Take the apex from 180, then share the rest equally between the two base angles.

1. The two base angles are equal. In an isosceles triangle the base angles are the same.
   base angles are equal
2. Find the total of the two base angles. Take the apex angle from 180.
   180 - 56 = 124
3. Share it between the two base angles. Divide by 2.
   124 / 2 = 62
4. Write the answer.
   62

Check. 56 + 2 x 62 = 180. ✓

Watch out. The two base angles are equal, not the apex and a base angle.

Method. Find the sum of the interior angles, then divide by the number of sides.

1. Work out the total angle inside the shape. A shape with 12 sides has 10 triangles inside it, so the total is (12 - 2) x 180.
   (12 - 2) x 180 = 1800
2. Share the total equally between the angles. All 12 angles of a regular shape are the same size.
   1800 / 12 = 150
3. Write the answer.
   150

Check. 150 x 12 = 1800. ✓

Watch out. Check the total with the sum formula: (n - 2) x 180.

Method. Split the polygon into triangles from one corner, then multiply by 180.

1. Split the shape into triangles from one corner. A polygon with 9 sides splits into 7 triangles.
   9 - 2 = 7
2. Multiply by 180. Each triangle has angles of 180 degrees.
   7 x 180 = 1260
3. Write the answer.
   1260

Check. 7 triangles x 180 = 1260. ✓

Method. Count the edges: the lines where two faces meet.

1. Picture the shape. A triangular prism has 5 faces, 9 edges and 6 vertices.
   faces 5, edges 9, vertices 6
2. Count the edges. An edge is where two faces meet.
   9 edges
3. Write the answer.
   9

Check. 5 faces + 6 vertices - 9 edges = 2. ✓

Watch out. Edges are not faces. A cube has 12 edges, not 6.

Method. Use diameter = 2 x radius.

1. Write the rule. Radius = diameter / 2.
   6 / 2
2. Divide. The radius is half the diameter.
   3
3. Write the answer.
   3

Check. diameter = 2 x radius, so 6 = 2 x 3 or 6. ✓

Watch out. The radius is half the diameter. Do not double it when you need the radius.

Method. Compare a square and a rectangle.

1. Compare. A square has four right angles, like a rectangle.
   four right angles
2. Conclude. So a square is a special rectangle, and Sam is wrong.
   Sam is wrong
3. Write the answer.
   a square has four right angles and opposite sides equal

Check. A square is a special rectangle. ✓

Method. Find an obtuse angle that is less than 120 degrees.

1. Pick an obtuse angle. An obtuse angle is between 90 and 180 degrees.
   between 90 and 180
2. Find a counter-example. 100 degrees is obtuse and is less than 120.
   100 degrees
3. Write the answer.
   An angle of 100 degrees is obtuse but is not more than 120 degrees.

Check. 90 < 100 < 120. ✓

Method. Use the exterior angle, 180 - 108 = 72, and 360 / 72.

1. Exterior angle. 180 - 108 = 72 degrees.
   72
2. Number of sides. 360 / 72 = 5 sides.
   5
3. Write the answer.
   5

Check. 5 x 108 = 540 = (5 - 2) x 180. ✓