How this instrument works
A cone's volume is V = ⅓πr²h, and the one-third is not a rounding convenience — it falls straight out of how the cone narrows. Slice the solid into thin discs parallel to the base: the disc at height t measured from the apex has shrunk to radius r×(t/h), so its area shrinks with the square of that fraction, not in step with it. Summing (or integrating) that quadratic taper from the apex down to the full base collects exactly a third of what a plain stack of full-radius discs — a cylinder — would give over the same height.
The result predates integral calculus by roughly two thousand years. Democritus is credited with guessing the one-third ratio around the fifth century BCE, picturing a cone as infinitely many stacked discs, but a guess is not a proof, and his method could not rule out some other constant entirely. Eudoxus of Cnidus supplied the rigorous argument a century later using the method of exhaustion, and Euclid recorded it as Proposition 10 of Book XII of the Elements — a piece of geometry that survived intact until calculus reproduced the same answer by different machinery.
One consequence trips up nearly everyone the first time: radius and height do not pull equal weight. Because volume depends on r² but only on h to the first power, doubling the height merely doubles the volume, while doubling the radius quadruples it — a squat, wide cone can dwarf a tall, narrow one built from the same amount of material. The formula is also more forgiving than it looks: an oblique cone, tilted so its apex sits off to one side, holds the identical volume as an upright cone sharing its base and perpendicular height, a fact guaranteed by Cavalieri's principle rather than by any symmetry of the shape.
- Enter the cone's Base radius — the distance from the centre of the circular base out to its rim, in any length unit.
- Enter the Height — the perpendicular distance straight up from the base's centre to the apex, not the slanted edge running down the side.
- Read Volume for the result of V = ⅓πr²h, reported in your input unit cubed.
- To check it by hand, work out the base area πr², multiply by Height, and divide by three — it should land on the same Volume shown.
Worked example — a cone with radius 3 and height 4
Take a cone with base radius 3 units and height 4 units — a scoop cone or a small traffic cone are both roughly this shape. The base area is πr² = π × 9, about 28.274334 square units, and one third of that times the height gives Volume = ⅓ × 28.274334 × 4 = 37.69911184307752, the exact figure this sheet returns for r = 3, h = 4.
The cylinder comparison makes the one-third factor concrete: a cylinder sharing that same radius and height holds πr²h = 28.274334 × 4 = 113.09733552923255 cubic units — exactly three times the cone's 37.69911184307752. Fill the cone with water and tip it into the cylinder three times, and the cylinder finishes exactly full on the third pour, a demonstration lecture halls have used for well over a century.
Questions
What is the formula for the volume of a cone?
V = ⅓πr²h, where r is the base radius and h is the perpendicular height from base to apex. That is exactly one third of πr²h, the volume of a cylinder sharing the same circular base and the same height — fill the cone three times and it empties exactly to the cylinder's brim.
Why does the cone volume formula include a factor of one third?
Because a cone's cross-sectional area shrinks with the square of the distance from the apex, not in a straight line. A disc sliced at height t from the apex has radius r×(t/h), so its area is πr²(t/h)². Summing that quadratic taper across the full height, rather than a constant full-radius disc, collects exactly a third of a cylinder's total.
Who first proved the one-third ratio, and how?
Eudoxus of Cnidus proved it around 350 BCE with the method of exhaustion, roughly two thousand years before integral calculus reached the identical result by different means. Euclid recorded the proof as Proposition 10 of Book XII of the Elements. Democritus had guessed the same ratio earlier by imagining a cone as stacked discs, but without Eudoxus's rigor it remained an unproven conjecture.
What is the most common mistake when computing cone volume?
Forgetting the ⅓ factor and computing πr²h outright — that number is the surrounding cylinder's volume, three times too large. The next most common slip is plugging in the slant height, the length along the cone's angled side, where h must instead be the perpendicular distance from base to apex.
Does doubling the radius or doubling the height add more volume?
Doubling the radius adds far more. Volume scales with r² but only with h to the first power, so doubling the height merely doubles the volume while doubling the radius quadruples it. A cone half as tall but twice as wide as another can still hold twice the volume, which runs against most people's first guess.
Does this formula still work for a cone that leans to one side?
Yes — an oblique cone, with its apex offset rather than sitting directly above the base's centre, holds exactly the same volume as an upright cone sharing its base and perpendicular height. Cavalieri's principle guarantees this: matching horizontal slices have identical area at every height, so the running total comes out identical regardless of the lean.