How this instrument works
A cylinder's volume comes from the oldest rule in solid geometry: base area times height. The base here is a circle, so its area is πr², and multiplying by the height h sweeps that flat circle straight through space to fill the whole solid — V = πr²h. Every right prism shares this same logic; only the shape of the base changes which area formula gets plugged in.
The formula falls out of stacking. Picture slicing the cylinder into thousands of paper-thin disks, each one a circle of area πr² and thickness approaching zero, then stacking h units of those disks. Their combined volume is πr² multiplied by h, because every slice contributes the identical base area no matter where along the height you cut it. That constant cross-section is also why a tilted cylinder holds precisely the same volume as an upright one of matching radius and height, a fact geometers call Cavalieri's principle.
A genuinely surprising consequence sits inside this formula's history. Fit a sphere of radius r snugly inside a cylinder of radius r and height 2r, and the sphere's volume turns out to be exactly two-thirds of the cylinder's; a cone sharing that same base and height takes exactly one-third, so the three volumes stand in the ratio 1:2:3. Archimedes regarded this discovery as his finest result and asked for the sphere-in-cylinder figure to be carved on his tomb.
- Enter the cylinder's radius in the Radius field — the distance from the central axis out to the curved wall.
- Enter the cylinder's height in the Height field — the straight-line distance between its two flat circular ends.
- Read the result in the Volume field; the sheet multiplies π by the squared radius and by the height the instant you type.
- Change either figure and watch Volume update, so you can see directly how much more a taller cylinder holds versus a wider one.
Worked example — radius 3, height 5
Take a cylindrical tank with a radius of 3 units and a height of 5 units — the figures this sheet uses by default. Squaring the radius gives r² = 9, and π times that base area works out to roughly 28.27433388 square units. Extruding that circle straight up through all 5 units of height gives V = πr²h = π × 9 × 5 = 45π ≈ 141.37166941 cubic units, the exact figure the Volume field returns for these inputs.
The asymmetry between radius and height shows up immediately if either figure moves. Halve the height to 2.5 while keeping the radius at 3, and volume falls to exactly half, about 70.685835 cubic units, because h enters the formula only once. Halve the radius to 1.5 instead while keeping the height at 5, and volume drops to a quarter of the original, about 35.342917 cubic units, because r is squared — a change in radius always carries twice the leverage that the same change in height does.
Questions
What is the formula for the volume of a cylinder?
V = πr²h, where r is the radius of the circular base and h is the height between the two circular ends. The πr² term is the base's area; multiplying by h extrudes that flat circle through the full height, the same base-times-height logic used for any right prism.
How is the cylinder volume formula actually derived?
Slice the cylinder into thin parallel disks and stack them back up. Each disk has area πr², and because that area never changes as you move up the axis, the total volume is simply πr² added h times over — or, in calculus terms, the constant πr² integrated from 0 to h, which gives πr²h directly.
Does doubling the radius or doubling the height do more to the volume?
Doubling the radius does far more: since r is squared in πr²h, doubling it quadruples the volume, while doubling the height only doubles it. Halving works the same way in reverse — a radius cut in half divides the volume by four, but a height cut in half only divides it by two.
How is a cylinder's volume related to a sphere's or a cone's?
A sphere of radius r fits exactly inside a cylinder of radius r and height 2r with two-thirds of the cylinder's volume, and a cone sharing that same base and height holds exactly one-third — cone, sphere, and cylinder sit in the ratio 1:2:3. Archimedes proved this and regarded it as his best result.
Does the volume formula still work if the cylinder leans to one side?
Yes, provided the height is still measured as the straight perpendicular distance between the two circular ends rather than along the slanted side. Every horizontal slice through a leaning cylinder is still a circle of area πr², so Cavalieri's principle guarantees the same total volume as an upright cylinder with matching radius and height.
What volume does the formula give for a radius or height of zero?
Exactly zero either way. A zero radius collapses the circular base to a single point, leaving no area to extrude; a zero height leaves a flat disk with no thickness to fill. Both are legitimate edge cases the formula handles cleanly, without needing any special-case rule.