Answer to Question #110115 in Geometry for jen

Question

The sum of the interior angles of an n-sided regular polygon is 720 degree. Find the size of each exterior angle of the polygon.

Answer 1

for a regular polygon, interior angle is given by,

$$interior\ angle=180^\circ-\frac{360^\circ}{n}\\$$

Since there are n edges,

The sum of the interior angles =$interior\ angle *n=(180^\circ-\frac{360^\circ}{n})n\\$

In this question sum of interior angles is $720^\circ$

Therefore number of sided(n),

$720^\circ=(180^\circ-\frac{360^\circ}{n})n\\ 4=(1-\frac{2}{n})n\\ \frac{4}{n}=1-\frac{2}{n}\\ \frac{6}{n}=1\\ n=6$

Therefore size of interior angle =$\frac{720^\circ}{6}=120^\circ$

$\therefore Each\ exterior\ angle=180^\circ-120^\circ\\ \therefore Each\ exterior\ angle=\bold{60^\circ }$

Also in another way,

$\therefore Each\ exterior\ angle=\frac{360^\circ}{6}=\bold{60^\circ}$