Year 6: Polygons and their angles
Part of the Year 6 angles, shapes and circles chapter. Chapter overview · whole chapter on one page
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.