# How do you find the sum of the interior angle measures of a convex 16 gon?

Feb 20, 2017

Sum of interior angles of a convex 16 sided-polygon
( hexakaidecagon ) would be ${2520}^{\circ}$.

#### Explanation:

Whatever the number of sides of a polygon,

the sum of its exterior angles is always ${360}^{\circ}$

Further, each pair of exterior angle and interior angle adds up to ${180}^{\circ}$

Hence in a polygon with $n$ sides (or angles),

the sum of all the interior and exterior angles would be ${180}^{\circ} \times n$

and sum of interior angles would be ${180}^{\circ} \times n - {360}^{\circ} = {180}^{\circ} \left(n - 2\right)$

Hence sum of interior angles of a convex 16-sided polygon would be ${180}^{\circ} \times \left(16 - 2\right) = {180}^{\circ} \times 14 = {2520}^{\circ}$.

Feb 20, 2017

Any polygon can be broken up into triangles by drawing in all the possible diagonals from one vertex.

You will find that the number of triangles is always 2 less than the number of sides.

The sum of the angles in any triangle is 180°

Consider the following:

$\text{shape"color(white)(......)"sides"color(white)(......)"triangles"color(white)(......)"sum int angles}$

"triangle"color(white)(......)"3"color(white)(...............)"1"color(white)(............)180xx1 = 180°

"quad"color(white)(..........)"4"color(white)(...............)"2"color(white)(............)180xx2 = 360°

"pentagon"color(white)(....)"5"color(white)(...............)"3"color(white)(..........)180xx3 = 540°

$\text{n-sides"color(white)(.........)"n} \textcolor{w h i t e}{\ldots \ldots \ldots} \left(n - 2\right) \textcolor{w h i t e}{\ldots \ldots} 180 \left(n - 2\right)$

"16-sides"color(white)(......)"16"color(white)(...........)14color(white)(..........)180xx14 = 2520°