Answer to Question #154092 in Geometry for Janine Leodones

Question

How many sides has a regular polygon if the measure of one of its angles is 128 4/7º? 144º? 160º?

Answer 1

For a regular polygon, let $n$ be the number of sides and $\alpha$ be the interior angle.

The sum of interior angles for a regular polygon is $n\alpha$ .

Also we can use another formula for calculating the sum of interior angles: $180^\circ (n-2)$ .

$180^\circ (n-2)=n\alpha \\ 180^\circ n-360^\circ =n\alpha \\ n(180^\circ -\alpha )=360^\circ \\ n=\frac{360^\circ}{180^\circ -\alpha }$

1) $\alpha =128 \ 4/7 ^\circ$

$n=\frac{360^\circ}{180^\circ -128\ 4/7^\circ }=\frac{360^\circ}{51\ 3/7^\circ}=7$

2) $\alpha =144 ^\circ$

$n=\frac{360^\circ}{180^\circ -144^\circ }=\frac{360^\circ}{36^\circ}=10$

3) $\alpha =160^\circ$

$n=\frac{360^\circ}{180^\circ -160^\circ }=\frac{360^\circ}{20^\circ}=18$

Answer: 7 sides; 10 sides; 18 sides.