Question

An equilateral triangle that has an area of \( 9\sqrt{3} \) is inscribed in a circle. What is the area of the circle? [Official GMAT-2018]

Hint:

For an equilateral triangle inscribed in a circle, use the relationship between the side length and the radius to find the area of the circle.

Answer 1

Step 1: Use the formula for the area of an equilateral triangle.
The area \( A \) of an equilateral triangle with side length \( s \) is given by: \[ A = \frac{s^2 \sqrt{3}}{4} \] We are given that the area of the equilateral triangle is \( 9\sqrt{3} \), so: \[ \frac{s^2 \sqrt{3}}{4} = 9\sqrt{3} \] Simplify by dividing both sides by \( \sqrt{3} \): \[ \frac{s^2}{4} = 9 \] Multiply both sides by 4: \[ s^2 = 36 \] Thus, the side length of the triangle is: \[ s = 6 \] Step 2: Relate the side length of the equilateral triangle to the radius of the circle.
For an equilateral triangle inscribed in a circle, the radius \( r \) of the circle is related to the side length \( s \) by the formula: \[ r = \frac{s \sqrt{3}}{3} \] Substitute \( s = 6 \) into this formula: \[ r = \frac{6 \sqrt{3}}{3} = 2\sqrt{3} \] Step 3: Calculate the area of the circle.
The area \( A \) of the circle is given by: \[ A = \pi r^2 \] Substitute \( r = 2\sqrt{3} \): \[ A = \pi (2\sqrt{3})^2 = \pi \times 4 \times 3 = 12\pi \] Step 4: Conclusion.
Thus, the area of the circle is \( 12\pi \).