Question

The area of a regular polygon of n sides is (where r is inradius, R is circumradius, and a is side of the triangle)

• A. $\frac{n{R}^2}{2}\sin \left(\frac{2\pi }{n}\right)$
• B. $n{r}^2\tan \left(\frac{\pi }{n}\right)$
• C. $\frac{n{a}^2}{4}\cot \frac{\pi }{n}$
• D. $n{R}^2\tan \left(\frac{\pi }{n}\right)$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $\frac{n{R}^2}{2}\sin \left(\frac{2\pi }{n}\right)$
B $n{r}^2\tan \left(\frac{\pi }{n}\right)$
C $\frac{n{a}^2}{4}\cot \frac{\pi }{n}$

Area of the regular polygon will be
$=\frac{n{R}^2}{2}sin\left(\frac{2\pi }{n}\right)$.
Now
$R=\frac{s}{2sin\left(\frac{\pi }{n}\right)}$
Hence
$A=\frac{n{s}^2}{8si{n}^2\frac{\pi }{n}}.2sin\left(\frac{\pi }{n}\right).cos\left(\frac{\pi }{n}\right)$
$=\frac{n{s}^2}{4}.cot\left(\frac{\pi }{n}\right)$. where s is the side of the polygon.
$=n{r}^2.tan\left(\frac{\pi }{n}\right)$ where r is the incentre.