Polygons & Tilings

Interior and exterior angles

For any convex $n$-gon, the interior angles always sum to the same value:

For a regular polygon (all sides and angles equal), divide by $n$ to get each angle:

The exterior angles, by contrast, always sum to exactly $360°$ for any convex polygon — a fact that doesn't depend on $n$ at all, since walking all the way around any convex shape means turning through one full circle. For a regular polygon, each exterior angle is just $360°/n$.

Quick reference for common polygons

• Triangle ($n=3$): interior sum $180°$, each angle $60°$ if regular.
• Square ($n=4$): interior sum $360°$, each angle $90°$.
• Regular pentagon ($n=5$): each angle $108°$.
• Regular hexagon ($n=6$): each angle $120°$.

Practice problems