The area of a circle around a regular hexagon is 36 pi. Find the area of a regular hexagon.

The area of a regular hexagon in this case is:

S = (3 √3 * r²) / 2, where r is the radius of the circumscribed circle.

It remains to find the radius. It is known that the area of a circle is equal to 36π, and it is calculated by the formula

S = πr²,

from here

r = √S / π.

Now r can be found:

r = √36π / π = √36 = 6.

We found the radius, then we look for the area of the hexagon:

S = (3√3 * 6²) / 2 = (3√3 * 36) / 2 = 3√3 * 18 = 54√3.

Answer: the area of a regular hexagon is 54√3.