Interior and Exterior Angle Sums of Polygons

Pick any vertex of a polygon and draw diagonals from it to every other vertex that isn't next to it. That chops the polygon into triangles, and you always get exactly n-2 of them for an n-sided polygon. A quadrilateral splits into 2 triangles, a pentagon into 3, a hexagon into 4. Since each triangle contributes 180 degrees of interior angle, the whole polygon's interior angles sum to (n-2) times 180. That's why a quadrilateral is stuck at 360 regardless of whether it's a square or some stretched, lopsided kite — the triangle count never changes, only how the angles are distributed inside.

For a regular polygon, every interior angle is identical, so divide that total by n to get one angle: (n-2) times 180, all over n. A decagon has n=10, so its angles sum to 8 times 180 = 1440 degrees, and each angle is 1440/10 = 144 degrees. Exterior angles work differently: at each vertex, the exterior angle is just 180 minus the interior angle, and walking all the way around the polygon's boundary always turns you through one full rotation, 360 degrees, no matter how many vertices you pass through. More sides just means more, smaller turns adding up to the same lap.

This connects straight to the volume and surface area work nearby: regular polygons as cross-sections or faces need these angle rules before you can find side lengths or apportion area. As n grows large, each interior angle (n-2)x180/n creeps toward 180 degrees — the polygon flattens out and starts looking like a circle, which is exactly the limit that connects polygons to circular geometry.