Cone Volume Calculator
Calculate the volume of a cone from its radius and height.
About
Cone Volume Calculator
The volume of a cone is V = 1⁄3 πr²h: exactly one third of the cylinder that would enclose it. Enter a radius and a perpendicular height above and the calculator shows the result with its working.
Why one third
Stand a cone inside a cylinder of the same radius and the same height and the cone occupies precisely a third of the space. This is not an approximation, and it holds for every cone regardless of how wide or narrow.
The same one-third relationship applies to pyramids: any pyramid is one third of the prism with the same base and height. A cone is simply a pyramid whose base has been smoothed into a circle, so it inherits the rule. Anyone who has filled a conical measure three times to fill a cylindrical one of matching dimensions has demonstrated it.
The reason is that a cone's cross-section shrinks linearly from the base to the point, so its area shrinks with the square of the remaining fraction. Integrating that square over the height produces the factor of a third.
A worked example
Take a radius of 2 units and a height of 3 units.
- Square the radius: 2² = 4
- Multiply by π: 4π for the base area
- Multiply by height: 4π × 3 = 12π
- Divide by three: 4π, which is 12.5664 cubic units
Height, not slant height
This is the error that ruins most cone calculations. The formula needs the perpendicular height from the base to the tip, measured straight up the middle. The slant height runs along the sloping surface from the rim to the tip, and it is always longer.
The two are related by Pythagoras: l² = r² + h², where l is the slant height. If you have measured the slope, recover the true height with h = √(l² − r²). A cone with a 3 unit radius and a 5 unit slant has a real height of √(25 − 9) = 4, and using 5 instead would overstate the volume by a quarter.
Slant height is the right measurement for surface area, which is πr(r + l), so both figures have their uses. Keeping them straight is what matters.
Practical sizes
A waffle ice cream cone roughly 6 cm across the top and 12 cm deep has a radius of 3 cm, giving 1⁄3 π × 9 × 12 = 113.1 cm³, or about 113 ml. That is the capacity to the rim, which is why a single scoop sits above the cone rather than inside it.
A conical pile of gravel or grain follows the same formula, and it is how bulk material is estimated in a yard. Measure the circumference around the base, divide by 2π for the radius, measure the height to the peak, and apply the formula. The angle the material naturally settles at, its angle of repose, sets the relationship between the two and is around 30 to 45 degrees for most aggregates.
Truncated cones
Cut the tip off a cone and you have a frustum, which is the shape of most buckets, plant pots and lampshades. Its volume is 1⁄3 πh(R² + Rr + r²), where R and r are the two radii. It reduces to the cone formula when r is zero, which is a useful check that you have remembered it correctly.
Where it gets used
Bulk materials. Stockpiles of sand, salt and grain form cones, and their volume is estimated this way before being converted to tonnes using the material's bulk density.
Process equipment. Hoppers and silo bottoms are conical so that material flows out under gravity without bridging in corners.
Volcanology. The volume of a cinder cone, and so the quantity of material a vent has produced, comes from this formula applied to survey data.
Traffic and marine markers. Cones and buoys are specified by base diameter and height, and the volume determines material use and buoyancy.
Surface area and material
A cone's curved surface is πrl, where l is the slant height, and adding the circular base gives a total of πr(r + l). The curved surface unrolls into a sector of a circle, which is how a flat sheet becomes a cone: cut a sector, roll it, join the edges.
That unrolling is the basis of sheet metal work and paper cone patterns. The sector's radius is the slant height, and the angle it needs is 360 × r/l degrees. A cone with a 3 unit radius and a 5 unit slant needs a sector of 360 × 3/5 = 216 degrees.
Angle of repose, and estimating stockpiles
Loose material poured onto a flat surface forms a cone whose slope is set by the material rather than by how it was poured. That slope is the angle of repose, and knowing it gives you the height from the base radius without climbing the pile.
| Material | Angle of repose | Height for a 5 m base radius |
|---|---|---|
| Dry sand | 34° | 3.37 m |
| Gravel | 40° | 4.20 m |
| Wheat | 27° | 2.55 m |
| Wet clay | 45° | 5.00 m |
Height is the base radius multiplied by the tangent of the angle. A gravel pile 5 m in base radius stands about 4.20 m tall and holds 1⁄3 π × 25 × 4.20 = 110 m³, which at a bulk density near 1.68 tonnes per cubic metre is roughly 185 tonnes.
Cones and pyramids follow one rule
Every cone and every pyramid is one third of the prism sharing its base and height. The base can be a circle, a square, a triangle or an irregular outline, and the third still holds. That covers conical hoppers, square pyramids, tetrahedra and the tapered end of any container, so it is worth remembering in place of a list of separate formulas.
A frustum worked through
Most tapered containers are frustums rather than cones, so it is worth seeing one done. Take a plant pot 30 cm across the top, 20 cm across the base and 25 cm deep. The radii are 15 cm and 10 cm.
- Square and cross-multiply the radii: R² = 225, Rr = 150, r² = 100
- Add them: 225 + 150 + 100 = 475
- Multiply by π and by the height: π × 475 × 25 = 37,306 cm³
- Divide by three: 12,435 cm³, which is about 12.4 litres
Treating the same pot as a cylinder of average radius 12.5 cm would give 12,272 cm³, close here but drifting further apart as the taper steepens. Treating it as a full cone from the top radius would give 5890 cm³, less than half the true figure.
Checking a cone result
A cone holds a third of its enclosing cylinder, and that cylinder holds about 79% of its enclosing box. So a cone occupies roughly 26% of the box around it. Any answer far from a quarter of the bounding box is worth a second look.
Common mistakes
Using slant height as height. The single most frequent error here. Convert with h = √(l² − r²) first.
Forgetting the third. Leaving it out gives the enclosing cylinder, which is three times too large.
Using diameter as radius. Quadruples the answer, since the radius is squared.
Treating a bucket as a cone. Most buckets are frustums, and the tapered shape needs the frustum formula rather than the cone one.
Common questions
Frequently asked questions
V = 1/3 πr²h. A cone of radius 2 and height 3 has a volume of 4π, which is 12.5664 cubic units.
A cone's cross-section shrinks linearly toward the tip, so its area shrinks with the square of the remaining fraction. Integrating that over the height gives exactly one third. The same rule makes any pyramid one third of its prism.
Height is the perpendicular distance from base to tip. Slant height runs along the sloping surface and is always longer. Convert with h = √(l² − r²). Volume needs the height; surface area needs the slant.
Most are frustums, not cones, because the tip is cut off. Use V = 1/3 πh(R² + Rr + r²), where R and r are the radii of the two open ends.
A waffle cone 6 cm across and 12 cm deep holds about 113 ml to the rim, from 1/3 π × 3² × 12.
Measure the circumference around the base and divide by 2π for the radius, measure the height at the peak, then apply the formula. Multiply by bulk density to convert to tonnes.
It quadruples, since the radius is squared. Doubling the height only doubles it.
Yes, provided you use the perpendicular height. Cavalieri's principle means a tilted cone holds the same as an upright one of the same base and height.