# How to find length of sides of an equilateral triangle inscribed in a circle with a radius of 36?

The title says it all :) How do you find the area and length of the sides of an equilateral triangle inscribed in a circle with a radius of 36?

Relevance

The way I solved this is with the following property:

Draw a circle and a diameter.

Draw a point on the circle, say P.

Draw two segments from P, one to each end of the diameter.

The resulting triangle is a right triangle, where the diameter is the hypotenuse.

(Basically, the theorem says that any triangle inscribed in a circle where one of the sides is a diameter is a right triangle.)

Now, to find the length of the sides of this triangle:

First draw a circle.

Then draw an equilateral triangle inscribed in the circle. Let the points of the triangle be A,B,C, for reference.

Bisect angle A with a diameter. That is, draw the diameter where A is one endpoint.

Let the other endpoint be called D.

Since angle A is 60° (because the triangle is equilateral), then after bisecting, angle BAD and CAD are each 30°.

But now, triangle BAD is right, because one of the sides, AD, is a diameter.

In triangle BAD, angle BAD is 30°, angle ABD is right, which means the other angle, angle ADB, must be 60°. Therefore, triangle ABD is a 30-60-90 triangle.

Which means the ratios of the sides are 1:√3:2.

Since the hypotenuse of triangle ABD is AD, which is a diameter, it's length is twice the radius. So it has length 72.

We're looking for the length of one of the sides of triangle ABC. Side AB is in that triangle, and also in ABD.

In triangle ABD, AB must have have a length (√3)/2 times that of AD.

AD is 72, so AB is 36√3

With that, you can find the area of the triangle ABC

• Ok, we know that an equilateral triangle is a triangle in which all sides are equal in length.

If the circle has a radius of 36, then the diameter is 72.

That being said, any segment of the triangle should have two sides touching the circle if it is inscribed. Therefore, the length of the sides should be 72.