Question

The ratio of the area of a regular polygon of $n$ sides inscribed in a circle to that of the polygon of
same number of sides circumscribing the same is 3:4. Then the value of $n$ is?

• A. $6$
• B. $4$
• C. $8$
• D. $12$

Answer 1

The correct option is A $6$

Given ,ratio of Area of polygan $=\left(\frac{3}{4}\right)$

Since two polygon are having same number of sides and centre,then they are same number of sides and centre,then they are similar

ratio of Area = (ratio of sides)${}^2$

ratio of sides $=\sqrt{\frac{3}{4}}=\sqrt{\frac{3}{2}}$

ratio of altitude = radius of sides $=\sqrt{\frac{3}{2}}$

$\frac{R\left(\sin U\right)}{R}=\left(\frac{\sqrt{3}}{2}\right)\sin U=\left(\frac{\sqrt{3}}{2}\right)\sin U=\left(\frac{\sqrt{3}}{2}\right)U={60}^o,{120}^0$

number of sides = $3,6$

hence = $6$