Interior sum = (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 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 .
Sum rule needs a regular polygon
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.
Thinking the exterior sum grows
The exterior-angle sum is a FIXED for every polygon — it does not increase with the number of sides. Only the interior sum grows. Do not scale up with .
n must be a whole number
The number of sides must be a positive integer. If comes out fractional, the angle cannot belong to a regular polygon, or an arithmetic slip has been made — recheck.
Go via the exterior angle
To find the number of sides, work with the exterior angle: . Given an interior angle, first do . This avoids the heavier algebra.
Self-check interior + exterior = 180°
After finding an interior and exterior angle, confirm they add to at a vertex. If they do not, one is wrong — a quick guard against interior/exterior mix-ups.
Know the small polygon sums
Memorise pentagon , hexagon , octagon . Recognising these instantly saves time and flags arithmetic slips in .
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 do the exterior angles of any polygon add up to?
Work out the sum of the interior angles of a polygon with sides (a nonagon).
