Question

If $r$ is the radius of the inscribed circle of a regular polygon of $n$ sides, then $r$ is equal to?

• A. $\frac{a}{2}cot\left(\frac{\pi }{2n}\right)$
• B. $\frac{a}{2}cot\left(\frac{\pi }{n}\right)$
• C. $\frac{a}{2}tan\left(\frac{\pi }{n}\right)$
• D. $\frac{a}{2}cos\left(\frac{\pi }{n}\right)$

Answer 1

The correct option is B $\frac{a}{2}cot\left(\frac{\pi }{n}\right)$

in $\Delta ABL$, $AL$ is the radius of the inscribed circle.

$BL=\frac{BC}{2}=\frac{a}{2}$

$\cot \left(\frac{\pi }{n}\right)=\frac{AL}{BL}=\frac{r}{\frac{a}{2}}$

Hence $r=\frac{a\left(\cot \right(\frac{\pi }{n}\left)\right)}{2}$