Year 6 Maths reasoning (SATs): Quadrilaterals and polygons
Part of the Year 6 Maths reasoning (SATs) geometry: angles and shapes chapter. Chapter overview · whole chapter on one page
Quadrilaterals and polygons
Rule
The angles in a quadrilateral add up to 360 degrees. A polygon with n sides has (n - 2) x 180 degrees of angles inside it.
A regular polygon has all sides equal and all angles equal. Each interior angle = (n - 2) x 180 / n.
Key ideas
- Quadrilateral: 360 degrees.
- Sum of interior angles: (n - 2) x 180.
- Regular polygon: divide the total by the number of sides.
Method. Take the total of the three known angles from 360.
- Angles in a quadrilateral add up to 360 degrees. Split the shape into two triangles to see why.360
- Add the three known angles. Add them together.80 + 95 + 110 = 285
- Take the total from 360. The fourth angle is what is left.360 - 285 = 75
- Write the answer. 75
Check. 80 + 95 + 110 + 75 = 360. ✓
Watch out. Quadrilaterals have 360 degrees; triangles have 180.
Method. Split the polygon into triangles from one corner, then multiply by 180.
- Split the shape into triangles from one corner. A polygon with 8 sides splits into 6 triangles.8 - 2 = 6
- Multiply by 180. Each triangle has angles of 180 degrees.6 x 180 = 1080
- Write the answer. 1080
Check. 6 triangles x 180 = 1080. ✓
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 5 sides has 3 triangles inside it, so the total is (5 - 2) x 180.(5 - 2) x 180 = 540
- Share the total equally between the angles. All 5 angles of a regular shape are the same size.540 / 5 = 108
- Write the answer. 108
Check. 108 x 5 = 540. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Watch out
Watch out. A regular pentagon has interior angles of 108 degrees, not 72. The exterior angle is 72.
Exam tip
Practice guide. Split the shape into triangles from one corner, then add up the triangles.
Exercise 5C Practice: quadrilaterals and polygons
Non-calculatorMethod. Take the total of the three known angles from 360.
- Angles in a quadrilateral add up to 360 degrees. Split the shape into two triangles to see why.360
- Add the three known angles. Add them together.90 + 90 + 60 = 240
- Take the total from 360. The fourth angle is what is left.360 - 240 = 120
- Write the answer. 120
Check. 90 + 90 + 60 + 120 = 360. ✓
Watch out. Quadrilaterals have 360 degrees; triangles have 180.
Method. Take the total of the three known angles from 360.
- Angles in a quadrilateral add up to 360 degrees. Split the shape into two triangles to see why.360
- Add the three known angles. Add them together.100 + 70 + 130 = 300
- Take the total from 360. The fourth angle is what is left.360 - 300 = 60
- Write the answer. 60
Check. 100 + 70 + 130 + 60 = 360. ✓
Watch out. Quadrilaterals have 360 degrees; triangles have 180.
Method. Split the polygon into triangles from one corner, then multiply by 180.
- Split the shape into triangles from one corner. A polygon with 6 sides splits into 4 triangles.6 - 2 = 4
- Multiply by 180. Each triangle has angles of 180 degrees.4 x 180 = 720
- Write the answer. 720
Check. 4 triangles x 180 = 720. ✓
Method. Split the polygon into triangles from one corner, then multiply by 180.
- Split the shape into triangles from one corner. A polygon with 10 sides splits into 8 triangles.10 - 2 = 8
- Multiply by 180. Each triangle has angles of 180 degrees.8 x 180 = 1440
- Write the answer. 1440
Check. 8 triangles x 180 = 1440. ✓
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 6 sides has 4 triangles inside it, so the total is (6 - 2) x 180.(6 - 2) x 180 = 720
- Share the total equally between the angles. All 6 angles of a regular shape are the same size.720 / 6 = 120
- Write the answer. 120
Check. 120 x 6 = 720. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 8 sides has 6 triangles inside it, so the total is (8 - 2) x 180.(8 - 2) x 180 = 1080
- Share the total equally between the angles. All 8 angles of a regular shape are the same size.1080 / 8 = 135
- Write the answer. 135
Check. 135 x 8 = 1080. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Method. Find the sum of the interior angles, then divide by the number of sides.
- Work out the total angle inside the shape. A shape with 3 sides has 1 triangles inside it, so the total is (3 - 2) x 180.(3 - 2) x 180 = 180
- Share the total equally between the angles. All 3 angles of a regular shape are the same size.180 / 3 = 60
- Write the answer. 60
Check. 60 x 3 = 180. ✓
Watch out. Check the total with the sum formula: (n - 2) x 180.
Method. Divide 360 by the number of sides.
- Exterior angles of any polygon add up to 360 degrees. Walk round the shape and turn at each corner.360
- Share equally for a regular shape. Each of the 5 turns is the same size.360 / 5 = 72
- Write the answer. 72
Check. 72 x 5 = 360. ✓
Watch out. Exterior angle plus interior angle = 180.
Method. Divide 360 by the number of sides.
- Exterior angles of any polygon add up to 360 degrees. Walk round the shape and turn at each corner.360
- Share equally for a regular shape. Each of the 12 turns is the same size.360 / 12 = 30
- Write the answer. 30
Check. 30 x 12 = 360. ✓
Watch out. Exterior angle plus interior angle = 180.
Method. The exterior angle is 180 - 150 = 30 degrees, and exterior angles add up to 360.
- Exterior angle. 180 - 150 = 30 degrees.30
- Number of sides. 360 / 30 = 12 sides.12
- Write the answer. 12
Check. 12 x 30 = 360. ✓
Method. The angles of a pentagon add to (5 - 2) x 180 = 540.
- Total of a pentagon. (5 - 2) x 180 = 540.540
- Add the four known angles. 100 + 110 + 95 + 120 = 425.425
- Find x. 540 - 425 = 115 degrees.115
- Write the answer. 115
Check. 425 + 115 = 540. ✓
Method. Test each choice against the question.
- Read what the question asks. Decide what the answer must do or show.
- Test the choice. A heptagon has 7 sides, so (7 - 2) x 180 = 900 degrees.heptagon fits
- Reject the other choices. A hexagon has 720 degrees and an octagon has 1080 degrees.hexagon and octagon do not fit
- Write the answer. heptagon
Check. heptagon is the one that fits. ✓
Method. Test each option.
- Check each option. Test each one against the rule or the calculation.
- Tick the true ones. In a regular polygon all sides and angles are equal, and a regular pentagon has 5 lines of symmetry.All sides and all angles are equal in a regular polygon, A regular pentagon has 5 lines of symmetry
- Leave the others unticked. These do not match.A regular hexagon has interior angles of 100 degrees, Every regular polygon has an even number of sides are not ticked
- Write the answer. All sides and all angles are equal in a regular polygon, A regular pentagon has 5 lines of symmetry
Check. Each ticked option is true. ✓
Watch out. A regular hexagon has interior angles of 120 degrees, and a triangle is a regular polygon with 3 sides.
Method. Draw a diagonal.
- Split the shape. Draw a diagonal to make two triangles.two triangles
- Add the triangles. Each triangle is 180 degrees, so 2 x 180 = 360.360 degrees
- Write the answer. Because a quadrilateral can be split into two triangles, and two triangles have 2 x 180 = 360 degrees.
Check. 2 x 180 = 360. ✓
Method. Use (n - 2) x 180 with n = 8.
- Use the formula. (8 - 2) x 180 = 6 x 180 = 1080.1080
- Write the answer. 1080
Check. 1080 / 8 = 135. ✓
Method. Work out the interior angle for 9 sides.
- Total. (9 - 2) x 180 = 1260 degrees.1260
- Each angle. 1260 / 9 = 140 degrees, not 120.140
- Write the answer. no
Check. 140 x 9 = 1260. ✓