# What is the sum of the measures of the interior angles of an octagon?

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Jan 1, 2017

In the figure above we have an irregular octagon $A B C D E F G H$.

5 diagonals each originating from A are drawn.These diagonals divide the octagon into 6 triangles.

It is obvious from the figure that the sum of the measures of the interior angles of the octagon is equal to the sum of the interior angles of 6 triangles.

As we know that sum of the interior angles of a triangle is ${180}^{\circ}$, the sum of 8 interior angles of octagon will be $6 \times {180}^{\circ} = {1080}^{\circ}$

Extension

Comparing with this situation we can say that a polygon of n-sides can be divided into $n - 2$ triangles and thereby the sum of the interior angles of polygon of n-sides will be

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

Exercise

How will you get the sum of 8 interior angles of a concave irregular octagon using following figure?

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