The volumes of a cone, a cylinder, and a hemisphere with the same radius are compared.
Problem
Find and compare the volumes of the three shapes shown below:
- (a) A cone with height \( r \) and radius of the base \( r \)
- (b) A cylinder of height \( r \) and radius of the base \( r \)
- (c) A hemisphere of radius \( r \)
Solution
Step 1: Volume of the Cone
Volume (\(V_{\text{co}}\)) of the cone is given by:
\[ V_{\text{co}} = \frac{1}{3} \times \text{area of base} \times \text{height} = \frac{1}{3} \pi r^2 \times r = \frac{1}{3} \pi r^3 \]Step 2: Volume of the Cylinder
Volume (\(V_{\text{cy}}\)) of the cylinder is given by:
\[ V_{\text{cy}} = \text{area of base} \times \text{height} = \pi r^2 \times r = \pi r^3 \]Step 3: Volume of the Hemisphere
Volume (\(V_{\text{he}}\)) of the hemisphere is given by:
\[ V_{\text{he}} = \frac{1}{2} \times \text{volume of a sphere} = \frac{1}{2} \times \frac{4}{3} \pi r^3 = \frac{2}{3} \pi r^3 \]Step 4: Comparison of Volumes
The volume of the cylinder is the largest. The volume of the cone is one third of the volume of the cylinder and it is the smallest. The volume of the hemisphere is twice the volume of the cone or two thirds the volume of the cylinder.