Question

Given the relationship between the number of sides, the radius, and the length of each side in a regular polygon, find n.

Answer 1

EXPLANATION

If I=r√2, replacing terms:

$r\sqrt[]{2}=2r\sin \frac{180}{n}$

Dividing both sides by 2r:

$\frac{r\cdot\sqrt[]{2}}{2r}=\sin \frac{180}{n}$

Simplifying:

$\frac{\sqrt[]{2}}{2}=\sin \frac{180}{n}$

Applying sin-1 to both sides:

$\sin ^{-1}\frac{\sqrt[]{2}}{2}=\frac{180}{n}$

Multiplying both sides by n:

$n\cdot\sin ^{-1}\frac{\sqrt[]{2}}{2}=180$

Dividing both sides by sin-1 (sqrt(2)/2):

$n=\frac{180}{\sin ^{-1}\frac{\sqrt[]{2}}{2}}$

Solving the argument:

$n=\frac{180}{45}$

Simplifying:

$n=4$

The answer is n=4