What is the sum of the interior angle measures in a 20-gon?

May 16, 2018

Sum of all interior angles of $20$-gon is ${3240}^{\circ}$

Explanation:

Three things first.

1. A polygon with $n$ sides ($n$-gon) has $n$ interior angles and $n$ exterior angles.
2. As sum of each pair of exterior and interior angles is ${180}^{\circ}$, sum of all interior and exterior angles is $n \times {180}^{\circ}$
3. Sum of all the exterior angles of any polygon (i.e. irrespective of its number of sides) is always ${360}^{\circ}$.

Therefore $20$-gon has $20$ exterior angles whose sum is ${360}^{\circ}$.

As such, sum of its all interior and exterior angles is $20 \times {180}^{\circ} = {3600}^{\circ}$

Hence, sum of all interior angles is ${3600}^{\circ} - {360}^{\circ} = {3240}^{\circ}$