If the radius of a cone is doubled while height remains fixed, by what factor does the volume increase?

If the radius of a cone is doubled while height remains fixed, by what factor does the volume increase?

• A. 2
• B. 3
• C. 4
• D. 8

Answer: (C)

Key idea: for a cone with fixed height, its volume scales with the base area, which depends on the radius squared. The volume is V = (1/3)π r^2 h. If the radius doubles, r becomes 2r, so r^2 becomes (2r)^2 = 4r^2. The height stays the same, so the whole volume is multiplied by 4: V' = (1/3)π (2r)^2 h = 4 × (1/3)π r^2 h = 4V. Therefore, the volume increases by a factor of 4.

Key idea: for a cone with fixed height, its volume scales with the base area, which depends on the radius squared.

The volume is V = (1/3)π r^2 h. If the radius doubles, r becomes 2r, so r^2 becomes (2r)^2 = 4r^2. The height stays the same, so the whole volume is multiplied by 4: V' = (1/3)π (2r)^2 h = 4 × (1/3)π r^2 h = 4V. Therefore, the volume increases by a factor of 4.