What is the diameter of a circle inscribed in a regular polygon of 12 sides, each of length 1 cm ?

• A

  $1+\sqrt{2} \mathrm{~cm}$
• B

  $2+\sqrt{2} \mathrm{~cm}$
• Correct answer
  C

  $2+\sqrt{3} \mathrm{~cm}$
• D

  $3+\sqrt{3} \mathrm{~cm}$

Solution

Central angle subtend by side $=\frac{360}{12}=30^{\circ}$
$\begin{aligned} & \tan 15^{\circ}=\frac{1 / 2}{r} \\ & \Rightarrow \frac{\sqrt{3}-1}{\sqrt{3}+1}=\frac{1}{2 r} \\ & \Rightarrow 2 r=\frac{\sqrt{3}+1}{\sqrt{3}-1}=\frac{4+2 \sqrt{3}}{2} \\ & \Rightarrow d=2+\sqrt{3} \mathrm{~cm}\end{aligned}$

Hence, the correct answer is option: 'C'.