Interior angle of a regular polygon

Geometry Level 4

Let $x^\circ$ be the interior angle of a $n$ -sided regular polygon. How many different $n$ are there so that $x$ is an integer?

1 solution

Chan Lye Lee
Apr 4, 2017

The interior angle of a $n$ -sided regular polygon is $\frac{(n-2)180^\circ}{n} =180^\circ -\frac{360^\circ}{n} = x^\circ$ .

If $x$ is an integer, $\frac{360}{n}$ must be an integer and $n\ge 3$ .

Now $360 = 2^3 3^2 5$ and hence 360 has $4 \times 3 \times 2 =24$ factors including $n=1$ and $n=2$ .

Finally, there are 22 values of n such that $\frac{360}{n}$ is an integer and $n\ge 3$ .

Please change the wording to n-sided polygon. "Polygon of side n" would imply that the length of the side is equal n. Which makes no sense.

Marta Reece - 4 years, 2 months ago

Thanks for pointing out the mistake.

Chan Lye Lee - 4 years, 2 months ago