How can I solve this problem

Question

A sphere is inscribed in a right circular cone whose slant
height has a length equal to the diameter of its base. What
is the length of the radius of the sphere if the slant height
and the diameter of the cone both measure 12 cm? 

Answer 1

Think of this as a circle inscribed in an equilateral triangle.

The center of the circle is at the "center" of the triangle.

Draw and label a diagram.

Bisect each angle.

The lines intersect at the "center."

The lines create 6 congruent 30-60-90 triangles.

The long side of each triangle is 6 (half the base/side).

The hypotenuse (h) of each triangle (and the radius of the circle) satisfies:

6 = h√3√ / 2

12 = h√3

12√3 / 3 = h

4√3 = h

 

The radius of the sphere is 4√3 cm