Question

A cone is placed inside a cylinder as shown. The radius of the cone is half the radius of the cylinder. The height of the cone is equal to the radius of the cylinder. What is the volume of the cone in terms of the radius, r?

Answer 1

The answer is π r³/12.

The radius of the cylinder is r.
The radius of the cone is half of the radius of the cylinder: r/2.
The height of the cone is equal to the radius of the cylinder: r.

If the volume of the cone is π r²h/3, and the radius of the cone is r/2, and the height of the cone is r, then:
V = π × r² × h / 3
V = π × (r/2)² × r / 3
V = π × r²/4 × r / 3
V = π r³/12

Answer 2

Given:
radius of the cone = half the radius of the cylinder
height of the cone = radius of the cylinder

Volume of the cone = π r² h/3

let x be the radius of the cylinder

V = 3.14 * (x/2)² * x/3
V = 3.14 * (x/2 * x/2) * x/3
V = 3.14 * x²/4 * x/3
V = 3.14x³ / 12