# What is the name of a polygon if the ratio of the interior to exterior angle is 7:2?

Jan 10, 2017

It is a nonagon.

#### Explanation:

It is assumed that it is a regular polygon and ratio of $7 : 2$ is between each pair of interior to exterior angle. As sum of the two angles is ${180}^{\circ}$,

and each interior angle is $\frac{7}{9} \times {180}^{\circ} = {140}^{\circ}$ and each exterior angle is $\frac{2}{9} \times {180}^{\circ} = {40}^{\circ}$

As sum of all exterior angles is ${360}^{\circ}$, total number of sides must be ${360}^{\circ} / {40}^{\circ} = 9$ and polygon is nonagon.

If it is not a regular polygon, $7 : 2$ ratio must be between sum of all interior angles and sum of all interior angles.

Hence as sum of exterior angle is always ${360}^{\circ}$, sum of interior angles is ${360}^{\circ} \times \frac{7}{2} = {1260}^{\circ}$ and as sum of interior angles of a polygon with $n$ sides is $\left(n - 2\right) \times {180}^{\circ}$, we have

$\left(n - 2\right) \times {180}^{\circ} = {1260}^{\circ}$ and $n - 2 = {1260}^{\circ} / {180}^{\circ} = 7$

and $n = 7 + 2 = 9$ and polygon is a nonagon.