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

Given the relationship between the number of sides, the radius, and the length of-example-1

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EXPLANATION

If I=r√2, replacing terms:


r\sqrt[]{2}=2r\sin (180)/(n)

Dividing both sides by 2r:


\frac{r\cdot\sqrt[]{2}}{2r}=\sin (180)/(n)

Simplifying:


\frac{\sqrt[]{2}}{2}=\sin (180)/(n)

Applying sin-1 to both sides:


\sin ^(-1)\frac{\sqrt[]{2}}{2}=(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=(180)/(45)

Simplifying:


n=4

The answer is n=4

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