Question

For a regular polygon, let r and R be the radii of the inscribed and the circumscribed circles. An incorrect statement among the following is

• A. there is a regular polygon with $\frac{r}{R}=\frac{1}{2}$
• B. there is a regular polygon with $\frac{r}{R}=\frac{1}{\sqrt{2}}$
• C. there is a regular polygon with $\frac{r}{R}=\frac{2}{3}$
• D. there is a regular polygon with $\frac{r}{R}=\frac{\sqrt{3}}{2}$

Answer 1

The correct option is C

there is a regular polygon with $\frac{r}{R}=\frac{2}{3}$

Explanation for correct options:

Step 1: Applying theorem

We know, for a regular polygon with $r$ and $R$ as the radii of the inscribed and the circumscribed circles and $n$ as the number of angles or sides of the polygon,

$r=\frac{a}{2}cot(\frac{\mathrm{\pi }}{n})$

Where, $a$ is the side of the polygon

$R=\frac{a}{2}\cos ec(\frac{\mathrm{\pi }}{n})$

Step 2: Finding the value of the ratio.

$\Rightarrow \frac{r}{R}=\frac{{\displaystyle \frac{a}{2}}cot({\displaystyle \frac{\mathrm{\pi }}{n}})}{{\displaystyle \frac{a}{2}}\cos ec({\displaystyle \frac{\mathrm{\pi }}{n}})}$

$\Rightarrow \frac{r}{R}=\cos (\frac{\mathrm{\pi }}{n})$

Step: 3 Considering values for n.

When n = $3$,

$\Rightarrow$ $\frac{r}{R}=\cos (\frac{\mathrm{\pi }}{3})$

$\Rightarrow$$\frac{r}{R}=\frac{1}{2}$

When n = $4$

$\Rightarrow \frac{r}{R}=\cos (\frac{\mathrm{\pi }}{4})$

$\Rightarrow \frac{r}{R}=\frac{1}{\sqrt{2}}$

When n = $6$

$\Rightarrow \frac{r}{R}=\cos (\frac{\mathrm{\pi }}{6})$

$\Rightarrow \frac{r}{R}=\frac{1}{2}$

Since the value of $\frac{r}{R}$ is not equal to $\frac{2}{3}$

Therefore, the statement that there is a regular polygon with $\frac{r}{R}=\frac{2}{3}$ is incorrect.

Hence, Option (C) is correct.