Inscribed in a circle

[alex.gellatly]

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

[eagleeye]

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

[Anurag@Gurome]

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

Refer to the following figure,

If length of each side of the triangle is a, then area of the triangle is given by (√3/4)a²

Hence, (√3/4)a² = 9√3 ---> a = √(9*4) = 6

Therefore, AB = BC = AC = 6
Hence, AD = Height of the triangle = (2*Area)/(Base) = 2*9√3/6 = 3√3

Now, O is the centroid of the triangle, i.e. where the medians of the triangles intersect each other at 2:1 ratio.

Hence, radius of the circle = OA = 2/3 of AD = (2/3)*3√3 = 2√3

Hence, area of the circle = π(2√3)² = 12π

The correct answer is C.

[Jeevanantham]

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??

[Anurag@Gurome]

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??

ABC is a equilateral triangle with all angles equal to 60°.
Whereas ABD and ACD are right angle triangle with angles 30°, 60° and 90°

Hope that helps.