✕
Angle Sum of Polygons
For any simple polygon (one whose sides don't cross), the interior angles always add up to:
Sum = (n − 2) × 180°
where n is the number of sides. A triangle (n=3) gives 180°, a quadrilateral (n=4) gives 360°, and so on.
Why it works: triangulation
Pick one vertex and draw a diagonal to every other vertex that isn't already next to it. That splits the polygon into exactly n − 2 triangles, and every triangle's angles sum to 180°. Add them all up and you get (n − 2) × 180° for the whole polygon. Toggle "Triangulation" above to see this fan of triangles.
Quick checks
Triangle: 1 × 180° = 180°. Quadrilateral: 2 × 180° = 360°. Hexagon: 4 × 180° = 720°.
Convex vs. concave
The formula holds for concave polygons too — a "caved-in" vertex just has an interior angle greater than 180°, and it still counts normally. Drag a vertex inward until the badge switches to "Concave."
A related fact: exterior angles
Walk all the way around the polygon and add up the exterior (turning) angles instead — those always sum to 360°, no matter how many sides there are. It's the companion theorem to this one.
One thing to watch for
This all assumes the sides never cross. Drag a vertex far enough to cross an edge and the measured sum will stop matching the formula — that's expected, and the tool will flag it.
A bit of history
Euclid's Elements (Book I, Proposition 32, around 300 BCE) proved the triangle case and extended it to general polygons by triangulation — essentially the same argument used here.