V = πr²h/3, using the VERTICAL height. A 3-4 cone has volume 12π and slant height 5. The lateral area is πrl = 15π, and it uses the slant instead. Two formulas, two heights, and swapping them is the error this shape is known for.
The two heights
A cone has two distances from base to apex, and they are never the same number.
The vertical height goes straight up from the centre of the base. This is what the volume formula wants.
The slant height goes up the sloping side, from the edge of the base to the tip. This is what the lateral area formula wants.
They are related by Pythagoras: l = √(r² + h²), so the slant is always the longer. For a 3-4 cone the vertical height is 4 and the slant is 5.
Substituting the slant into the volume formula therefore always overstates the volume — by 25% in that example. Nothing complains: πr²l/3 is a perfectly valid arithmetic expression, it just does not mean anything. This page shows both numbers and says which formula wants which, and it prints what the wrong answer would have been.
Why a third
A cone holds exactly a third of the cylinder with the same base and height. Not approximately — exactly, and for every cone.
The same third applies to a pyramid and its prism, whatever the base shape. That generality is the clue: the factor has nothing to do with circles.
It comes from Cavalieri’s principle. Two solids with matching cross-sectional areas at every height have the same volume. A cone’s cross-section at height x is a circle scaled by (1 − x/h), so its area carries a squared factor — and integrating that square across the height produces the third.
The physical demonstration is worth doing: a conical vessel filled with water empties exactly three times into the matching cylinder. Democritus is credited with knowing this around 400 BC, and Eudoxus with proving it.
Unrolling the surface
Cut a cone up one side and flatten it, and you get a sector of a circle whose radius is the slant height.
That is where πrl comes from. The sector’s arc has to match the base circumference, 2πr, and a sector of radius l with that arc length has area πrl. No memorisation required.
It is also directly practical. To make a paper cone, cut a circle of radius l and remove a wedge — and this page tells you exactly what angle to keep: 360° × r/l.
This unrolling is why a cone is a developable surface: it can be made from flat material without stretching. A sphere cannot, which is why every flat map of the Earth distorts something, and why cone-based projections are used for mid-latitude maps.
The frustum
Slice the tip off a cone parallel to the base and what remains is a frustum — a bucket, a lampshade, a paper cup.
Its volume is πh(R² + Rr + r²)/3, and that middle Rr term is the one people drop.
Using the average of the two end areas instead understates the volume, because the sides slope and the cross-section grows faster than linearly between them. The correct expression is more than an average.
The formula contains both simpler shapes. Set r = 0 and it becomes a cone’s. Set R = r and it becomes a cylinder’s. Both are special cases, which is a good sign the expression is the right one.
Where it gets used
Storage and handling. Hoppers, silos and funnels are conical so material flows out under its own weight rather than bridging.
Construction. Conical roofs, spires and turrets. The lateral area is what determines the material, so the slant height is the measurement that matters.
Volume estimation. A pile of gravel, sand or grain forms a cone whose angle depends on the material, so the volume can be estimated from the base diameter alone.
Optics and acoustics. Light and sound cones, field of view, and the loudspeaker cone that gives the component its name.
Sources and methodology
The cone formulas are classical results; these are the references.
Method. The page asks for the vertical height and derives the slant, so the two cannot be confused on the way in — and both are displayed with the formula that needs each. Volumes are checked against numerical integration by discs, which is independent of the πr²h/3 formula entirely, and the suite asserts on five hundred generated cones that the slant always exceeds the vertical height, so substituting it always overstates the volume. That engine is verified on every change against 96 hand-written assertions, including that cone volumes match disc integration, that a cone is exactly a third of its cylinder on four hundred generated pairs, and that substituting the slant for the height always overstates the volume. The count and the per-case breakdown are published on the formula verification page.
Read the guide
A cone is exactly a third of the cylinder with the same base and height — the Cylinder Calculator handles that one.