Cylinder Volume Calculator
How much does the tank, the pipe, the well or the can actually hold, in honest cubic measure? Enter the radius and the height, and this calculator returns the cylinder’s volume at once — the base circle’s area, stacked unchanged to the full height of the vessel.
The Stacked Circle
A cylinder is a circle extruded: every horizontal slice is the same disc of area πr², and the volume is that disc stacked through the height — V = πr²h, the prism principle (base area times height) in its roundest instance. The formula's simplicity is deceptive in one place only: the radius is squared, so errors in it compound quadratically (a 10% radius over-read inflates the volume 21%), which is why the calculator's note leads with the diameter trap — the diameter is what tapes measure across an opening, and entering it unhalved quadruples the answer (2², the square exacting its tax). Capacities of tanks, wells, silos, pipes and pistons all reduce to this single line of mathematics; the page prints it to four decimals, in the cube of whatever unit the reader honestly measured in. Four decimals is itself a calibrated courtesy: field measurements of a tank's radius rarely justify two, but chained calculations (volume into weight via density, into cost via price) inherit the intermediate precision gratefully — the figure is printed finer than the tape because the arithmetic downstream of it deserves better than the tape's own rounding.
Worked Example: Radius 3, Height 7
The defaults: π × 9 × 7 = 63π = 197.92 cubic units (in litres, if the units were decimetres; in millilitres, if centimetres — the conversion habits are in the note). The example pairs with the surface page's identical dimensions to teach the pair's relationship: the same can that wears 188.50 of surface holds 197.92 of volume, and the ratio between them (surface per unit volume) is the engineer's efficiency gauge for vessels — it falls as vessels grow (the square-cube law again: volume outruns surface with scale, which is why tank farms use few large cylinders and why small animals lose heat faster than large ones, their surface-to-volume ratios inverted by the same mathematics).
The Well and the Rain Barrel: Field Arithmetic
Two rural fixtures run on this page weekly somewhere: the well, whose static water volume is the cylinder of its bore (a 15 cm bore holding a 12 m water column stores π × 0.075² × 12 ≈ 212 litres, the figure a pump installer sizes from) and the rain barrel under a downspout, whose harvest from a roof is the roof's area times the rainfall — after which the barrel's own cylinder decides how much of the storm it keeps (a 200-litre barrel under a 60 m² roof fills from just 3.3 mm of rain, a conversion that reorders gardeners' sense of their roofs). Both computations are this formula and one multiplication of context; the page's purity is what lets it serve the well-digger and the gardener without either noticing the other, the mark of a genuinely general tool — geometry first, biography supplied by the reader.
Pipes, Bores and Displacements
The working forms multiply: a pipe's capacity per run of length is this volume at the inner radius; an engine's displacement is this volume (at the bore's radius and the stroke's height) times the cylinder count — the "2.0-litre" on a badge is four of these pages summed; a well or borehole's water volume, a post hole's concrete order, a rain barrel's rating — all the same disc, stacked. Partial fills of horizontal cylinders leave the formula (segments again), and the calculator's scope statement leaves them out honestly; but a vertical vessel's fill is a pure proportion (half the height, half the volume), a linearity the vertical orientation alone grants and the FAQ records for the tank-gauging reader. Upright vessels earn their keep in that linearity every time a dipstick is read at a depot.
The Square-Cube Law in the Tank Yard
Scale any vessel up by a factor k and its volume grows by k³ while its surface (and roughly its steel bill) grows by k² — the square-cube law that runs this page and its surface sibling as a pair. Its consequences built the industrial landscape: storage migrates to ever-larger tanks because capacity outruns cost with size; heat loss per litre falls as tanks grow (surface per volume again), which is why breweries and dairies cluster around a few vast cylinders rather than many small ones; and its biological original — Galileo's observation that a bone's cross-section must outgrow its owner's weight — is the same arithmetic wearing fur. The reader holding both this page's number and the surface page's for a candidate tank holds the law in its working form: divide them, and the quotient is the vessel's efficiency, improving with every scale the foundations allow.
Related Tools
More from the Math Calculators hub: the Surface Area of a Cylinder Calculator — the same vessel's skin — and the Cone Volume Calculator, the cylinder's third-sized cousin.
Frequently Asked Questions
How do I get litres or gallons from the answer?
By the unit ladder the note posts: cubic centimetres to litres, divide by 1,000; cubic metres to litres, multiply by 1,000; cubic inches to US gallons, divide by 231; cubic feet to gallons, multiply by about 7.48052. The calculator stays in pure cubic units because the reader's ladder rung is unknowable in advance, and a hard-coded conversion would be wrong for half its audience — the volume is the invariant, the units are the reader's postal address. Keep the input unit consistent across radius and height, or the ladder's first rung is already missing.
What about a cylinder with an elliptical base?
The prism principle survives: base area times height, with the ellipse's area πab (a and b its semi-axes) replacing πr² — so an elliptical tank of semi-axes 3 and 2 and height 7 holds π × 6 × 7 = 131.95. The calculator is built for the circular case its title names; the elliptical sibling is one substitution away for the reader who meets it, stated here because oval tanks and ducts are commoner than textbooks admit, and their owners deserve the bridge.
How full is a half-height vertical tank, really?
Exactly half — the vertical cylinder's cross-section never changes, so volume is strictly proportional to height, and a sight-glass at mid-height means half capacity, a mercy the horizontal orientation refuses (its mid-height is also half, by symmetry, but every other mark is off-proportion and needs segment tables). Tank farms stand their cylinders upright partly for this gauging honesty. The calculator prices the full vessel; proportions of it are the reader's arithmetic, linear and safe.
Why does the radius matter so much more than the height?
The square: volume grows linearly in h but quadratically in r, so a 10% taller tank gains 10% while a 10% wider one gains 21% — and a mis-measured radius (the diameter trap) quadruples instead of doubling. Design exploits the asymmetry (widen before you heighten, where foundations allow), and measurement should fear it proportionally: check the radius twice, by two diameters at right angles if the vessel might be out of round, and average the readings — the formula assumes the circle your tape must verify.
What is the volume of a cylinder segment or wedge?
Partial shapes leave the single formula: the horizontal fill segment integrates the circle's slices (a standard but heavier computation), and the Steinmetz-style wedge has its own closed form. The calculator names its scope — whole right circular cylinders — rather than approximating segments badly, in keeping with this hub's rule that a tool's silence about what it cannot do is a defect, while a clear signpost is a service. The prism principle will still price any full extrusion the reader meets, whatever its base shape.
How do I find the height from a required volume?
Divide the volume by the base area: h = V ÷ πr² — the design direction, used whenever a capacity is specified and the vessel must be sized to it (a 1,000-litre tank on a 0.5 m radius footprint stands 1000 ÷ (π × 250,000 cm²…) — in consistent units, h = 1.0 m³ ÷ 0.785 m² ≈ 1.27 m). The calculator runs the survey direction; the inversion is printed so its users can size forward and verify backward, round-tripping the answer through this page's fields as the standing audit of any vessel proposal.
What changes for a gas cylinder?
The geometry is identical — the internal volume is this page's — but the CONTENTS obey the gas laws: a cylinder's litres of water capacity become hundreds of litres of gas at working pressure, by the pressure ratio (Boyle's law doing the multiplying, temperature correcting it). Divers and welders live by the product of this page's number and their pressure arithmetic; the calculator prices the vessel, the physics of compression prices the fill, and confusing the two numbers is the classic beginner's error the distinction here exists to prevent.
Who first stated the prism principle?
It is Euclid's Book XI–XII in spirit (cylinders as the round prisms, their ratio to cones proved by Eudoxus' exhaustion), made computational by the same Archimedean treatise as the neighbouring pages. The stacked-disc picture becomes literal calculus in Cavalieri's seventeenth-century indivisibles — slices of zero thickness, summed — which is to say this page's one-line formula is the integral calculus wearing its work clothes, two millennia early.