Interior and Exterior Angles of Polygons

Geometry · Axiom Academy

LESSON Interior and Exterior Angles of Polygons Once you know the number of sides, every angle in a polygon is yours to compute — interior, exterior, and their stubborn 360° sum. The interior angles of any polygon add up to a fixed total that depends only on the number of sides. The key trick: cut the polygon into triangles from a single vertex. Watch the diagonals fan out below — each new triangle adds another 180°. Pick one vertex and draw every diagonal from it. A triangle is already one piece; a quadrilateral splits into 2; a pentagon into 3. In general a polygon with n sides breaks into (n − 2) triangles — always two fewer than the number of sides. Each triangle contributes 180°, so the interior angles of the whole polygon must total (n − 2) × 180° . That single idea is the engine behind every interior-angle calculation. Apply (n − 2) × 180° to the polygons you meet most often. Notice the sum climbs by exactly 180° each time you add a side. In a regular polygon all sides and all angles are equal, so each interior angle is just the sum divided by n. The exterior angles are even tidier — walk once around the boundary and the turns you make always come to a full circle, 360°. Find each interior and exterior angle of a regular octagon (8 sides). Divide the sum by the 8 equal angles: or equivalently (interior + exterior = 180°). Interior angle sum: — it works because any polygon splits into (n − 2) triangles. Each interior angle (regular polygon):

This is the written version of the interactive lesson above. See the full Geometry course.