Sum of Internal Angles of Polygon/Proof 1

Theorem

The sum $S$ of all internal angles of a polygon with $n$ sides is given by the formula $S = \paren {n - 2} 180 \degrees$.

Proof

The Polygon Triangulation Theorem shows that there exists a triangulation of the polygon that consists of $n - 2$ triangles.

The sides of these triangles are sides and chords of the polygon, where the chords lie completely in the interior of $P$.

Hence the vertices of the triangles are vertices of the polygon.

Sum of Angles of Triangle equals Two Right Angles shows that the sum of the internal angles of a triangle is $180 \degrees$.

As the triangulation covers the polygon, the sum of the internal angles of the vertices of the triangles in the triangulation is equal to $S$.

So:

$S = \paren {n - 2} 180 \degrees$

$\blacksquare$

Sources

• 1987: Joseph O'Rourke: Art Gallery Theorems and Algorithms: $\S 1.3.1$