mathrokus

When a triangle rotates around one of it sides, it creates a cone or a bicone.

The formula for the volume of a cone is V = 1/3 h(pi*r^2).

In this situation, the Height is 5, since the cone was rotated around BC.

We know that the triangle is equilateral, so the altitude would be 5/2 * sqrt(3).

This means that the radius is 5/2 * sqrt(3).

V = 1/3 5(pi * (5/2 * sqrt(3))^2)

V = 5/3 (pi * (25/4 * 3))

V = 5/3pi * 75/4

V = 5pi * 25/4

V = \(\frac{125\pi}{4}\)

The radius of a circle can be found with the area of the circle.  The area of a sector is equal to n/360 * a where n is the angle of the sector and a is the area of the circle.

Using this formula,

60pi = 83/360 * a

a = 360/83 * 60pi

a ~ 260.24pi

The formula for the area of a circle is A = r^2 pi.

260.24pi = r^2 pi

260.24 = r^2

r ~ 16.13

So the radius of the circle is about 16.13.