152k views
9 votes
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.

An equilateral triangle is inscribed in a circle, as shown below. The circle has a-example-1
User Trancot
by
6.0k points

1 Answer

6 votes

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!

User Wazy
by
5.8k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.