Geometry · Foundation · 7 min
Polygon angle sums
A polygon’s angle sum can be counted one triangle at a time.
WHAT HAPPENS IF…How many triangles fill a pentagon fan?
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Draw diagonals from one vertex of a convex polygon to every nonadjacent vertex. This partitions it into n−2 triangles without overlap. Adding their angle sums yields (n−2)π. A simple concave polygon can also be triangulated, but an arbitrary fan need not remain inside it.
A detail that matters
Dividing the total by n gives each angle only for an equiangular polygon.
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