Question

The area of a circle inscribed in a regular hexagon is $100\pi$. The area of the hexagon is:

• A. $600$
• B. $300$
• C. $200\sqrt{2}$
• D. $200\sqrt{3}$
• E. $200\sqrt{5}$

Answer 1

The correct option is D $200\sqrt{3}$

Area of circle $=100\pi$
$\pi r^2=100\pi$
$r^2=100$
$r=10$
Now, a regular hexagon is made up of 6 equilateral $△s$ of equal areas. Now, height of equilateral $△$ is equal to radius of circle.Therefore, ar. of 1 equilateral $△=\frac{1}{2}$ x base x height
$\Rightarrow \frac{\sqrt{3}}{4}{a}^2=\frac{1}{2}\ast a\ast 10\Rightarrow a=\frac{4\ast 10}{2\sqrt{3}}=\frac{20\sqrt{3}}{3}$
Area of hexagon $6\ast \left(\frac{\sqrt{3}}{4}{a}^2\right)=6\ast \frac{\sqrt{3}}{4}\ast \frac{20\sqrt{3}}{3}\ast \frac{20\sqrt{3}}{3}=200\sqrt{3}$