Interior sum is (n−2)×180°
Split the polygon into triangles from one vertex; each triangle contributes . So the interior-angle sum is — e.g. a pentagon () gives , a hexagon () gives .
Exterior angles always sum to 360°
Walking once around the outside of a convex polygon turns you through one full turn, so the exterior angles always add to — no matter how many sides. For a regular polygon each exterior angle is therefore .
Interior + exterior = 180° at a vertex
At each vertex the interior and exterior angles lie on a straight line, so they add to . For a regular polygon: each interior angle .
Drawn from real examiner reports.
Multiplying by n instead of (n−2)
Using for the interior-angle sum (e.g. for a pentagon) instead of . Always subtract 2 from the number of sides first.
Dividing 360° by the interior angle
When given a regular polygon's interior angle, dividing by it directly. You must first find the exterior angle () and then do .
Treating an irregular polygon as regular
Dividing the total interior-angle sum by to get "each angle" only works for a regular polygon. For an irregular polygon you can only use the sum, then subtract the known angles.
Stopping at the exterior angle
After computing you have the EXTERIOR angle, not the interior. For a regular -gon each exterior angle is , so the interior angle is . Do not hand in the exterior value.
Thinking the exterior sum grows with sides
The exterior-angle SUM is a fixed for every polygon — it does not grow with the number of sides; only the interior sum grows. So never write the exterior sum as something: it is always .
Go through the exterior angle
To find the number of sides, work with the exterior angle: . Given an interior angle, first do to get the exterior angle. This avoids the heavier algebra.
Self-check: interior + exterior = 180°
Check any regular-polygon answer with interior + exterior . If your interior is , the exterior must be ; if the two do not add to , one of them is wrong. A fast, free check.
Irregular polygon: use the sum, then subtract
For a missing angle in an irregular polygon, find the interior SUM with , then subtract the angles you are given. Do not divide by — that only works when the polygon is regular.
Confirm the interior angle two ways
Confirm a regular interior angle both ways: and must agree. Use the second as a check if the exterior-angle route feels risky.
For a polygon with sides: The interior sum comes from cutting the polygon into triangles from a single vertex (each ). The exterior sum is always — one full turn as you walk around the outside — regardless of the number of sides.
| Polygon | Sides | Interior sum |
|---|---|---|
| Triangle | 3 | |
| Quadrilateral | 4 | |
| Pentagon | 5 | |
| Hexagon | 6 | |
| Heptagon | 7 | |
| Octagon | 8 | |
| Decagon | 10 |
What is the sum of the interior angles of an n-sided polygon?
Work out the sum of the interior angles of a polygon with sides (a nonagon).
