Interior angles of a polygon

Topic guide

1. What an interior angle is

An interior angle is an angle inside a polygon at one of its vertices.

2. Why the sum formula works

Any simple n-sided polygon can be divided into n − 2 non-overlapping triangles. For a convex polygon, this can be seen by drawing diagonals from one vertex to all the non-adjacent vertices. Because each triangle has an angle sum of 180°, the polygon’s interior-angle sum is (n − 2) × 180°.

3. The sum of interior angles

To find the total sum of the interior angles of a polygon with n sides, use the formula:

Sum = (n − 2) × 180°

4. Finding one interior angle of a regular polygon

A regular polygon has all sides equal and all interior angles equal. Once you find the total sum, you can find the size of one interior angle by dividing the total by the number of sides, n.

5. Finding a missing angle in an irregular polygon

If you know all but one of the interior angles of a polygon, you can find the missing angle by first calculating the total sum using the formula, then subtracting all the known angles from that total.

6. Finding the number of sides from a known total

If you are given the total sum of the interior angles, you can find the number of sides by reversing the formula. First divide the total by 180, then add 2.

7. Finding the number of sides from one regular interior angle

If you know the size of one interior angle of a regular polygon, let's call it I, you can set up the equation:

I × n = 180(n − 2)

You can then solve this equation using ordinary algebra to find n.

8. Concave polygons

The same sum formula, (n − 2) × 180°, applies to concave polygons (polygons that "dent inwards"). One or more of the interior angles in a concave polygon will be a reflex angle, which means it is greater than 180°.

9. Joined regular polygons

When regular polygons meet at a point, you can use the rule that angles around a point total 360°. Once you have calculated the interior angles of the known shapes, subtract them from 360° to find the interior angle of the unknown shape.

10. Worked example: Interior-angle sum

Find the sum of the interior angles of a hexagon.

A hexagon has 6 sides, so n = 6.

Sum = (6 − 2) × 180°

Sum = 4 × 180°

Sum = 720°

11. Worked example: One angle of a regular polygon

Find the size of one interior angle of a regular octagon.

An octagon has 8 sides.

Sum = (8 − 2) × 180° = 6 × 180° = 1080°

One angle = 1080° ÷ 8 = 135°

12. Worked example: Number of sides from one regular angle

A regular polygon has an interior angle of 156°. How many sides does it have?

Set up the equation:

156n = 180(n − 2)

156n = 180n − 360

Rearrange to solve for n:

24n = 360

n = 15

13. Worked example: A joined-polygon problem

A regular pentagon and a square meet at a point alongside a regular polygon P. How many sides does polygon P have?

Interior angle of a square = 90°

Interior angle of a regular pentagon = 540° ÷ 5 = 108°

Angles around a point total 360°.

Interior angle of P = 360° − 90° − 108° = 162°

Now find the number of sides of P:

162n = 180(n − 2)

162n = 180n − 360

18n = 360

n = 20

14. Common mistakes

• miscounting the number of sides;
• using n rather than n − 2 in the sum formula;
• dividing by n − 2 rather than n when finding one regular angle;
• assuming an irregular polygon has equal angles;
• treating a reflex interior angle as though it were below 180°;
• forgetting that angles around a point total 360°.