Cylinder volume formula

What the cylinder volume formula means

A cylinder is a circle pushed straight along a line. Every slice cut parallel to the base is the same circle, so the volume is simply that circle's area repeated over the whole height. The area of a circle is πr²; stack it to a height h and you get V = πr²h. This is the same rule as for any prism, "base area times height", applied to a round base.

Two consequences are worth remembering. Volume grows with the square of the radius: double the radius and the cylinder holds four times as much, while doubling the height only doubles it. And because only the base and the perpendicular height matter, a slanted cylinder with the same base and the same vertical height holds exactly as much as a straight one, which is explained on the page about the volume of an oblique cylinder. For a proof by stacking thin disks or by integration, see why the volume of a cylinder is πr²h.

Three equivalent forms of the formula

The diameter form is handy when you measure a pipe or a can with a ruler across the top; the page on the volume of a cylinder with the diameter works through it. The circumference form suits large tanks and tree trunks, where a tape wrapped around the outside is easier than reaching the center. If you know the volume and need a dimension instead, solve the formula backwards: the radius of a cylinder is √(V ÷ πh), and the height of a cylinder is V ÷ πr².

Volume of a cylinder: examples with real objects

Each line below is computed with the formula above and full-precision π, then rounded. Measurements are inside dimensions when the question is "how much does it hold".

Take the water tank: the radius is 4 ÷ 2 = 2 ft, the base area is π × 2² = 12.566 ft², and the volume is 12.566 × 6 = 75.398 ft³. One cubic foot holds 7.48052 US gallons, so the tank holds about 564 gallons when full. For tanks that are only partly filled, especially tanks lying on their side, use the cylindrical tank volume calculator; for runs of pipe, the pipe volume calculator gives gallons per foot.

From cubic units to gallons and liters

The formula gives a volume in the cube of the unit you measured in. Keep both measurements in the same unit before multiplying: a radius in inches and a height in feet will give a meaningless number. Then convert the result with an exact factor.

Dedicated pages cover each case with its own table of common sizes: cylinder volume in gallons, cylinder volume in liters and cubic feet of a cylinder. To turn a volume into a weight, multiply by the density of the material with the cylinder weight calculator.

Common mistakes with the volume of a cylinder

• Using the diameter as the radius. The answer comes out four times too big, because the radius is squared.
• Mixing units. A 6 in radius and a 2 ft height must become 0.5 ft and 2 ft (or 6 in and 24 in) before you multiply.
• Squaring the wrong thing. Only the radius is squared: π × r² × h, not (π × r × h)².
• Rounding π too early. Using 3.14 instead of π changes the result by about 0.05 %. That is fine for homework, but over a 10,000 gallon tank it is about 5 gallons.
• Outside instead of inside dimensions. For capacity, measure inside the walls. The walls of a pipe or a tank take up room; the hollow cylinder volume page computes the material and the bore separately.
• Confusing volume and surface area. Volume is the space inside (cubic units). The surface area of a cylinder, 2πr² + 2πrh, is the material needed to cover it (square units).