Cone Volume Calculator

The volume of a cone formula

A cone narrows from a circular base to a single point, the apex. Its volume depends on just two measurements: the radius of the base and the height, measured straight from the centre of the base up to the apex.

V = ⅓ × π × r² × h

The expression π × r² × h is the volume of a cylinder with the same base and height. A cone takes up exactly one third of that cylinder, which is where the ⅓ comes from: three cones, filled and emptied one after another, fill the cylinder once. This holds for any cone, even a tilted one, as long as the height is measured perpendicular to the base.

How to calculate it with this tool

1. Enter the Base radius: half the width of the circular base.
2. Enter the Height: the vertical distance from the base to the tip.
3. Choose the Metric that both numbers use and press Calculate.

The result carries the cubed unit, for example 47.123890 cm³, and shows up to six decimals. If you only know the diameter, halve it before typing it in. If you know the slanted side (slant height) s instead of the vertical height, get the height first with h = √(s² − r²).

Example: find the volume of a cone

A waffle cone has an opening radius of 3 cm and is 12 cm tall. V = ⅓ × π × 3² × 12 = ⅓ × π × 108 = 36π ≈ 113.0973 cm³, close to 113 mL of ice cream before it is piled above the rim.

A heap of sand forms a rough cone 4 m across and 1.5 m high, so r = 2 m. V = ⅓ × π × 4 × 1.5 = 2π ≈ 6.2832 m³, which is about 8.22 cubic yards. The table lists a few more combinations, with the volume of the same-sized cylinder for comparison.

How the volume changes with size

The radius is squared in the formula and the height is not, so the two do not affect the volume equally. Doubling the height doubles the volume. Doubling the radius makes it four times as large. Doubling both gives eight times the volume, which is why a cone twice as big in every direction holds eight times as much.

Cutting the cone in half in height and radius does the reverse: with r = 1.5 and h = 2.5 instead of 3 and 5, the volume falls from 47.1239 to 5.8905, exactly one eighth.

Practical applications

• Funnels and hoppers: the volume of a conical hopper tells you how much grain or powder it holds before it must be refilled.
• Heaps of material: gravel, soil and grain tend to pile into cone shapes, so the cone formula gives a quick estimate of a stockpile, though real piles are rarely perfect cones. The volume calculator covers other simple solids in one place.
• Food and packaging: ice-cream cones, paper cups and party hats are all cones, and their capacity follows the same formula.
• Engineering and design: nose cones, speaker cones and road cones need volume figures for material and weight estimates.

For a cylinder of the same base and height, see the cylinder volume calculator; for a ball, the sphere volume calculator. A cone, a sphere and a cylinder of the same radius and height 2r have volumes in the ratio 1 : 2 : 3, an old result from Archimedes.

Common mistakes

• Using the diameter in place of the radius. The volume would be four times too large.
• Using the slant height as the height. The formula needs the perpendicular height; the slant is longer and overstates the volume.
• Leaving out the one third. Without it you have the volume of a cylinder.