Cone Calculator

How it works

A right circular cone has a circular base and an apex directly above the centre of that base. Its radius r, its height h and its slant height l (measured along the surface from the rim to the apex) form a right triangle, so l = √(r² + h²); given the slant height instead, the height is √(l² − r²), which is why the slant height must be longer than the radius.

The volume of a cone is exactly one third of the cylinder that would enclose it, ⅓πr²h, a result that goes back to Euclid. The curved surface, cut along one line and flattened, is a sector of a circle of radius l whose arc is the base circumference 2πr, so its area is πrl; adding the base circle gives the total surface area πr(r + l).

The unit you choose labels the lengths, areas and volumes; the numbers are the same whatever the unit. It matters for the capacity line, where the volume in cubic inches, feet, yards, centimetres or metres is converted to US gallons and litres from the legal definitions of those units.

Formula

l = √(r² + h²)        h = √(l² − r²)
V       = ⅓ × π × r² × h
base    = π × r²
lateral = π × r × l
total   = π × r × (r + l)
capacity: 1 US gal = 231 in³ = 3.785411784 L;  1 ft³ = 7.48052 gal;  1 m³ = 1,000 L

Example

A cone with a 3 ft radius and a 4 ft height has a slant height of √(9 + 16) = 5 ft. Its base area is 9π ≈ 28.2743 ft², its volume is ⅓ × π × 9 × 4 = 12π ≈ 37.6991 ft³, its lateral area is π × 3 × 5 = 15π ≈ 47.1239 ft², and its total surface area is π × 3 × (3 + 5) = 24π ≈ 75.3982 ft². Filled to the rim it would hold 37.6991 × 7.48052 ≈ 282.01 US gallons, or 1,067.52 litres.

Entering the same cone as a 3 ft radius and a 5 ft slant height instead gives the height √(25 − 9) = 4 ft and identical results. A cone 1 ft in radius and 3 ft tall holds π ≈ 3.1416 ft³, a third of the 3π ft³ cylinder of the same size, which is 23.50 gallons.

Assumptions and limitations

• The cone is a right circular cone: the base is a circle and the apex is directly above its centre. An oblique cone has the same volume but a different surface area.
• The capacity assumes the whole volume is usable, apex included; a real hopper or funnel has an outlet and is rarely filled to the rim.
• The gallon is the US liquid gallon of 231 cubic inches, not the imperial gallon.
• Results are rounded for display only.

Frequently asked questions