OE

Area of the circle = \(𝜋𝑟^2\) (r is the radius of the circle)

Given that area of circle = 36π

Hence, r = 6.

We know, when a circle is inscribed in an equilateral triangle, centroid of triangle is the
center of the circle, and centroid divided the median in the ratio 2:1.
i.e. AO: OD = 2:1



Now, we know OD = 6, therefore AO = 12
Hence, median = height (of the equilateral triangle) = 6 + 12 = 18
Also, the height of an equilateral triangle \(= \dfrac{\sqrt{3}}{2} \times side=18\)

\(side = \dfrac{36}{\sqrt{3}}\)

Therefore, area of the triangle \(= \dfrac{3}{4} \times side^2= \dfrac{\sqrt{3}}{4} \times \dfrac{36}{\sqrt{3}} \times \dfrac{36}{\sqrt{3}}= 108 \sqrt{3}\)

C is the answer