Right Circular Cone Calculator

Background.

Pour dry sand from a single point and it settles into a cone; so do gravel stockpiles, grain heaped on a floor and the tip of a sharpened pencil. Cones appear wherever material spreads out from a point, and three measurements describe every one of them: the radius of the circular base, the perpendicular height from base to tip, and the slant height along the sloping side.

The volume is one third of the cylinder with the same base and height, V = πr²h/3, and that third is the fact worth remembering. A conical paper cup 8 cm across and 10 cm deep holds about 168 mL, while a straight-sided cup of the same width and depth would hold about 503. Cones nearly always hold less than they look as if they should.

Because the radius is squared, it dominates. Doubling the base radius multiplies the volume by four, while doubling the height only doubles it, so when you estimate a stockpile, a small error in the measured base costs far more than one in the height.

The page works in any direction. Give it radius and height for the volume, a target volume and a height to find the radius needed, or a volume and a radius to find the height. Whatever you solve for, it then reports the whole set: diameter, slant height √(r² + h²), and the total surface area of base plus sloping side.

It assumes a right circular cone, with the tip directly above the centre of the base, and a solid shape with no wall thickness. A bucket or lampshade is a cone with its top cut off, a frustum, and needs its own formula; nothing is added for the overlap needed when the side is cut from flat sheet.

What is right circular cone calculator?

A right circular cone is the solid formed by joining every point of a circle to a single apex directly above the circle's centre. Its radius r is the radius of that base circle, its height h is the perpendicular distance from base to apex, and its slant height l is the straight distance from the rim of the base to the apex along the surface. Radius, height and slant height form a right triangle, so l² = r² + h².

How to use this calculator.

1. 01Choose what to Solve for. Volume needs Radius and Height; Radius needs Volume and Height; Height needs Volume and Radius. Whatever sits in the field being solved for is ignored.
2. 02Measure Height perpendicular to the base, from the centre of the base straight up to the tip, not along the sloping side. If you only know the slant height l and the radius, the height is √(l² − r²).
3. 03For a pile of material, wrap a tape around the base, then divide that circumference by 2π (about 6.283) to get the radius. That is usually easier and more accurate than trying to reach the centre.
4. 04Keep every length in one unit. Volume then comes out in that unit cubed and surface area in that unit squared; 1,000 cubic centimetres make a litre.

The formula.

V = πr²h ÷ 3, where πr² is the area of the circular base and h the perpendicular height. The factor of one third applies to every cone and pyramid: each fills exactly a third of the prism or cylinder sharing its base and height. Rearranged, the radius for a given volume is r = √(3V ÷ πh) and the height is h = 3V ÷ πr². The slant height comes from Pythagoras, l = √(r² + h²). Total surface area is πr(r + l): the base contributes πr², and the sloping side contributes πrl, which is the area of the circular sector you would get by slitting the side and unrolling it flat. The radius enters both terms of the area and is squared in the volume, so errors in it weigh more than errors in the height.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.