# Sum of Internal Angles of Polygon

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## Contents

## 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 $\paren {n - 2} 180 \degrees$.

$\blacksquare$

## Sources

- 1968: M.N. Aref and William Wernick:
*Problems & Solutions in Euclidean Geometry*... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.9$: Corollary - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**polygon**