# Sum of Internal Angles of Polygon

## Theorem

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

### Corollary

The size $A$ of each internal angle of a regular $n$-gon is given by:

$A = \dfrac {\left({n - 2}\right) 180^\circ} n$

## Proof

For convex polygons, name a vertex as $A_1$, go clockwise and name the vertices as $A_2, A_3, \ldots, A_n$.

By joining $A_1$ to every vertex except $A_2$ and $A_n$, one can form $\paren {n - 2}$ triangles.

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

Therefore, the sum of internal angles of a polygon with $n$ sides is $\paren {n - 2} 180 \degrees$.

$\blacksquare$