How to use the Cone Calculator
- Enter the radius of the circular base.
- Enter the vertical height from base to apex — not the sloping edge.
- Read the volume, which is exactly one third of the enclosing cylinder.
- Read the surface area, which combines the base circle with the curved lateral surface.
- Use the litre row when working in centimetres.
How the calculation works
A cone holds exactly one third of the cylinder that encloses it, a result Archimedes proved and calculus later confirmed by integrating circular slices of shrinking radius. That gives volume as pi r squared h over three. The height in this formula is strictly the perpendicular height from base to apex; the sloping edge is longer and using it inflates the volume noticeably.
The slant height is found by Pythagoras from the radius and vertical height, and it is what the curved surface actually needs: the lateral area is pi times radius times slant height. Unrolled flat, that curved surface becomes a sector of a circle whose radius is the slant height — which is precisely how sheet-metal cones and paper funnels are marked out and cut.
Conical geometry appears wherever loose material piles up. A stockpile of sand, gravel or grain settles into a cone whose sides sit at the material's angle of repose, so measuring the base circumference and knowing that angle is enough to estimate tonnage without touching the pile. Hopper design uses the same relationships in reverse, sizing the cone so material flows rather than bridging.
V = (πr²h)/3; slant l = √(r² + h²); S = πr(r + l)Source: Archimedes' ratio of cone to cylinder; see Euclid, Elements Book XII, Proposition 10.
Worked example
A gravel stockpile has a base circumference of 25 m and a peak height of 3.2 m.
- Radius: 25 / (2π) = 3.98 m.
- Volume: π × 3.98² × 3.2 / 3 = π × 15.84 × 3.2 / 3 = 53.1 m³.
- Slant height: √(3.98² + 3.2²) = 5.11 m.
- At a bulk density of 1.6 t/m³: 53.1 × 1.6 = 85 tonnes.
About 53 m³ of gravel, roughly 85 tonnes at typical bulk density — accurate enough for a stock check.
Frequently asked questions
What if the tip is cut off?+
That is a frustum, with volume πh(R² + Rr + r²)/3 where R and r are the two radii. It is not simply the difference of two cone volumes unless you know the missing apex height.
Why is the slant height longer than the vertical height?+
It is the hypotenuse of the right triangle formed by the radius and the vertical height, so it always exceeds both.
Do I include the base in surface area?+
Only if the base is a real surface. An open funnel or a stockpile needs the lateral area alone, πrl.
Can I estimate a stockpile without climbing it?+
Yes — measure the base circumference and use the material's angle of repose to infer height, since tan(angle) equals height over radius.
Last reviewed August 31, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.