Cone Volume Calculator

The formula

Volume = π × radius² × height ÷ 3

A cone holds exactly one third of a cylinder with the same base and height — a result already known to Greek mathematicians, and easy to check by filling a cone three times to empty a matching cylinder.

Worked examples

• A cone with a 3 cm radius and 12 cm height: π × 9 × 12 ÷ 3 = 113.1 cm³.
• An ice-cream cone 5 cm across (radius 2.5 cm) and 10 cm deep: 65.4 cm³, about 65 ml.
• A pile of sand 4 m across (radius 2 m) and 1.2 m high: π × 4 × 1.2 ÷ 3 ≈ 5.03 m³. With sand weighing about 1.6 tonnes per m³, that is around 8 tonnes.

Height vs slant height

The formula uses the vertical height, from the tip straight down to the centre of the base. The slant height s is measured along the side. They are linked by Pythagoras:

• h = √(s² − r²)
• A cone with radius 3 and slant height 5 has a height of √(25 − 9) = 4.

Stockpiles and the angle of repose

Loose materials form cones whose slope depends on the material — about 30–35° for dry sand and gravel. If you can only measure the circumference C at the base of a pile, find the radius with r = C ÷ (2π), estimate the height, and apply the formula. This is a common way to estimate quantities of sand, gravel, salt or grain.

Truncated cones

Many real objects — buckets, flower pots, lampshades — are truncated cones (frustums), with a radius R at the bottom and r at the top:

V = π × h × (R² + R × r + r²) ÷ 3

A bucket 30 cm high with radii of 15 cm at the top and 12 cm at the bottom holds π × 30 × (225 + 180 + 144) ÷ 3 ≈ 17,250 cm³, about 17 litres.

Surface area

Lateral surface = π × r × s; add π × r² for the base. Useful for making a paper or sheet-metal cone. The cylinder volume calculator handles straight-sided containers.

Formulas and content last reviewed: October 2026 · About