Question

An equilateral triangle is inscribed in a circle, as shown below. The circle has a radius of 5 m and the equilateral triangle has side lengths of 8.66 m. Find the shaded area.

Answer 1

Easiest method.

Suppose the side of the equilateral triangle is a . Then radius,

R = a/root 3

Explanation: In an equilateral triangle, it’s centroid lies upon it’s altitude. An altitude is the median in an equilateral triangle. So by pythagoras theorem,

H^2 + (a/2)^2 = a^2, where H is the altitude.

Solving this, you’ll get H = [root(3)/2] x a

Now, we know that centroid divides a median (here, median = altitude cause it’s equilateral) in 2:1

So, 2x + x = H

You’ll get x = [root(3)/ 6 ] * a

But we need 2x (from centroid),

Therefore, 2x = [root(3)/3] * a = a / root(3)

But the centroid will be the center of the circle in this case, therefore

R = a / root(3)

This is the easiest possible way, use this shortcut especially in competitive exams!