Cone Volume Calculator

V = (1/3) × π × r² × h

A cone is exactly one-third the volume of a cylinder with the same radius and height. Same base, same height, but the cone tapers to a point, so it holds less.

Worked example: ice cream cone capacity A standard sugar cone is about 2.5" across at the rim and 4" tall.
r = 1.25 in. V = (1/3) × π × 1.5625 × 4 ≈ 6.55 in³ ≈ 0.107 liters, or a little under half a US cup.

Most real cones get a heaped scoop on top, so the served volume runs roughly 1.5× the cone itself.

Worked example: traffic cone in cubic feet An 18-inch orange traffic cone has a base diameter of about 10".
r = 5 in = 0.417 ft, h = 1.5 ft. V = (1/3) × π × 0.174 × 1.5 ≈ 0.273 ft³, about 7.7 liters.

Nobody stores anything in a traffic cone, so read that as displaced volume: the space the shape sweeps out. Note that stacking beats it easily, since cones nest, and a column of 40 takes nothing like 40 times this.

Where cone volumes show up:

• Ice cream cones. Most cones hold 1/3 to 1/2 cup of melted product if filled to the rim.
• Funnel volumes. Kitchen and lab funnels are cones in their lower half. Useful for sizing transfer-loss tolerances.
• Conical silo bottoms. The hopper bottom of a grain silo is a cone, and it matters when sizing the bin’s true capacity (cylinder plus cone tip).
• Heaped-pile volumes. Sand, gravel and road salt settle into a cone at the angle of repose for that material, typically 30 to 40°.
• Party hats and witch hats. Card stock volume is irrelevant here, but the cone dimensions drive the flat pattern you cut.
• Volcanic cones in geology, for estimating tephra deposit volume.

The “angle of repose” trick:

For a freely-poured pile (sand, salt, gravel), the cone has a fixed slope α (the angle of repose, usually 30-40° depending on grain size and moisture). Given a base radius r, the height is h = r × tan(α). So a pile of dry sand with r = 1 m has h ≈ 0.58 m (for tan 30°), and volume ≈ 0.61 m³.

Construction supervisors estimate “yardage” of a pile this way without ever climbing it. Measure the base diameter, look up the typical angle for the material, plug into V = (π/3) r² × r × tan(α) = (π/3) × tan(α) × r³.

Cone vs. cylinder vs. sphere (Archimedes’ ratio):

For a cone, sphere and cylinder all sharing the same radius r, with the cone and cylinder standing 2r tall:

• Cone: (1/3) × π × r² × 2r = (2/3)πr³
• Sphere: (4/3)πr³
• Cylinder: 2πr³

Ratio cone : sphere : cylinder = 1 : 2 : 3. Archimedes asked that this relationship be carved on his tomb. Smart move.

Sanity check:

• h = 0: V = 0 (zero-height cone is a flat circle). ✓
• r = 0: V = 0 (zero-radius is a line segment). ✓
• Volume of cone = (1/3) × volume of cylinder with same r, h. Always. ✓