alex.gellatly wrote:Note: For this problem I will use ~ as a square root sign, sorry for the inconvenience.

An equilateral triangle that has an area of 9~3 is inscribe in a circle. What is the area of the circle?

A. 6pi
B. 9pi
C. 12pi
D. 9pi~3
E. 18pi~3

Thanks

Nice sign convention. I will just use sqrt for clarity

We know that if side of an equilateral triangle is a, then.

Height = sqrt(3)*a/2.
Area = sqrt(3)*a^2/4.

Also, When an equilateral triangle is inscribed in a circle, the median (which is the same as the height) is divided into 1:2 ratio. so for the circle r = 2/3*sqrt(3)/2 = a/sqrt(3)
So for the circle we have:
r= a/sqrt(3)
Area of circle = pi*r^2 = pi* a^2/3.

Now we are given that for the triangle area = 9*sqrt(3).
So we have sqrt(3)*a^2/4 = 9* sqrt(3) => a^2 = 4*9.

So finally, Area of circle = pi* a^2/3 = pi*4*9/3 = 12pi.

Hence C is the right answer.

Let me know if this helps

I have one doubt here. Here Equilateral triangle is considered as 30 : 60 :90 . And taken as x:xsqrt3 :2x .. Why is it so?? Equilateral triangle will be 60 : 60 : 60 . right??