Which expression gives the volume of a right circular cone with radius r and height h?

• A. π r^2 h
• B. 1/3 π r^2 h
• C. 2π r h
• D. π r h^2

Answer: (B)

Volume of a cone is one third of the volume of a cylinder with the same base and height. The cylinder with base radius r and height h has volume π r^2 h. A cone with the same base and height fits inside that cylinder and takes exactly one third of its space, so its volume is (1/3) π r^2 h. You can see this by imagining slicing the cone into thin disks; as you go from the tip to the base, the disk radii grow linearly, and integrating the disk areas from 0 to h gives the factor 1/3. Therefore the correct expression is (1/3) π r^2 h. The other expressions either miss the 1/3 factor or have the wrong dependence on r and h.