Geometry Lab formula guide
Polygon angles & diagonals
The interior angles of any simple n-sided polygon add up to (n − 2) × 180°. Each interior angle of a regular polygon is (n − 2) × 180° / n, each exterior angle is 360° / n, and the number of diagonals is n(n − 3) / 2.
Interior-angle sum = (n − 2) × 180°
| Use it to | Formula |
|---|---|
| Interior-angle sum | (n − 2) × 180° |
| Each interior angle (regular) | (n − 2) × 180° / n |
| Each exterior angle (regular) | 360° / n |
| Diagonals | n(n − 3) / 2 |
Try it with your own numbers
Polygon angles & diagonals calculator
A triangle-based explanation for many-sided shapes. Change a value and the diagram, result, and working update together.
Regular polygon explorerLIVE EXPLORATION
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Your measurements
A regular polygon has equal sides and equal interior angles. The diagram uses a normalized scale.
Why it works
Interior angle = (n − 2) × 180° / n
The idea
Draw diagonals from one vertex of a convex n-sided polygon. You obtain n − 2 triangles, so their angle sums add up to (n − 2) × 180°. In a regular polygon, every interior angle is equal: divide this sum by n. Each exterior turning angle is 360°/n.
Work through an example
1Work through an example
- A regular hexagon has interior-angle sum 720°, so each of its six angles measures 120°. It has n(n − 3)/2 = 6 × 3/2 = 9 diagonals. Each vertex connects diagonally to three non-neighboring vertices; divide by two because each diagonal was counted twice.
Common mistakes
- The sum formula applies to simple polygons, but equal individual angles require a regular or equiangular polygon. Do not divide the sum equally for an arbitrary irregular polygon.
Make the formula move.
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