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" diameter at the rim and 4" tall. r = 1.25 in. V = (1/3) × π × 1.5625 × 4 ≈ 6.55 in³ ≈ 0.107 liters ≈ about half a US cup.

Most actual ice-cream cones get a heaped scoop on top, so the served volume is roughly 1.5× the cone volume.

Worked example — traffic cone in cubic feet: An 18-inch orange traffic cone has a base diameter of about 10" and 18" tall. 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 — not that the inside is hollow space anyone uses; it’s just the displaced volume.

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 — important when sizing the bin’s true capacity (cylinder + cone tip).
• Heaped-pile volumes. Sand, gravel, salt — a free pile assumes the shape of a cone with an angle of repose for the material (typically 30-40°).
• Witch hats and party hats for SEO-friendly party-supply searches.
• Volcanic cones in geology — calculation for 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 — and the cone/cylinder having height 2r:

• 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. ✓