Cylinder Volume vs Cone
A cylinder with base radius r and height h holds exactly three times the volume of a cone with the same dimensions. Cylinder: V = πr²h. Cone: V = (1/3)πr²h. This page explains why, with interactive comparisons and worked examples.
Cylinder vs Cone
The 3:1 Volume Ratio
The cylinder volume formula is V = πr²h. The cone formula is V = (1/3)πr²h. The cone holds exactly one-third of the cylinder.
This means it takes exactly 3 cones of water to fill 1 cylinder of the same base and height. You can test this physically — fill a cone-shaped cup and pour it into a matching cylinder. It takes three pours.
This ratio holds for any base radius and any height. It also holds for oblique versions — an oblique cone is still 1/3 of the matching oblique cylinder.
Why the Factor of 1/3?
A cone tapers from a full circular base (area πr²) to a point (area 0). The cross-sectional area at height y from the base is:
A(y) = π(r(1 − y/h))² = πr²(1 − y/h)²
Integrating from 0 to h: V = ∫₀ʰ πr²(1 − y/h)² dy = πr² × h/3 = (1/3)πr²h
The (1 − y/h)² factor — the squared linear taper — is what produces the 1/3. A cylinder has A(y) = πr² (constant), so its integral gives πr²h. The ratio is 1:3.
Practical Comparisons
Ice cream cones: A cone-shaped wafer holds about 1/3 the volume of a cylindrical cup of the same width and height. That's why cone-shaped cups look tall but hold less.
Funnels: A conical funnel's volume helps determine how much liquid it holds while draining. For a funnel with r = 5 cm, h = 10 cm: V = (1/3)π × 25 × 10 = 261.8 cm³.
Traffic cones, party hats, volcanic cinder cones — they all follow the same 1/3 relationship with their equivalent cylinders.