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Geometry & measures · Angles & polygons

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Interior & exterior angles of polygons

Why the exterior angles of any polygon add up to 360°, where (n − 2) × 180° comes from, how regular polygons follow instantly, and the reliable route from a given interior angle back to the number of sides.

Geometry & measures · Angles & polygons

Interior & exterior angles of polygons

Why the exterior angles of any polygon add up to 360°, where (n − 2) × 180° comes from, how regular polygons follow instantly, and the reliable route from a given interior angle back to the number of sides.

Why it works

Walk round the polygon and count the turning. Start on one side, walk along it, and at each corner turn through the exterior angle to face along the next side. By the time you arrive back where you started, facing the way you first faced, you have made exactly one full turn. So the exterior angles of any polygon add up to 360° — triangle, hexagon, 100-gon, regular or wildly irregular. The total never depends on nn.

Interior and exterior angles are partners on a straight line. At each vertex the interior angle and the exterior angle sit together on the extended side, so interior + exterior = 180° (not 360° — a very common slip). That gives the interior-angle sum for free:

sum of interiors=180°n−360°=(n−2)×180°.\text{sum of interiors} = 180°n - 360° = (n - 2) \times 180°.

You can also see (n−2)×180°(n-2) \times 180° directly: from one vertex, diagonals fan the polygon into n−2n - 2 triangles.

Regular polygons split everything equally. All nn exterior angles are equal, so each is 360°n\dfrac{360°}{n}, and each interior angle is 180°−360°n180° - \dfrac{360°}{n}. For a regular octagon: exterior =360°÷8=45°= 360° \div 8 = 45°, interior =135°= 135°.

Finding nn from an interior angle — go via the exterior. Given "each interior angle of a regular polygon is 168°168°", don't divide anything into 360°360° yet: first find the exterior angle, 180°−168°=12°180° - 168° = 12°, then n=360°÷12°=30n = 360° \div 12° = 30. Dividing 360°360° by the interior angle is the classic wrong turn — the interior angles don't add up to 360°360°.

Mixed figures: angles at a point. When polygons tile around a shared point, the angles meeting there add up to 360°360° — work out each polygon's interior angle, then subtract from 360°360° to find the gap.