Cylinder Volume Calculator

What is the volume of a cylinder?

The volume of a cylinder is the amount of space inside it. It tells you how much a cylinder can hold, such as water in a glass or concrete in a round column. A cylinder is a solid with two equal, parallel circles for its top and bottom. A curved surface joins them. A can, a drum, a pipe and a candle are all close to cylinders. Volume is measured in cubic units: cubic inches (in³), cubic feet (ft³), cubic centimeters (cm³) or cubic meters (m³). For liquids, you often change it to gallons or liters.

Parts of a cylinder

• Base: each flat end is a circle. A cylinder has two equal bases.
• Radius (r): the distance from the center of a base to its edge.
• Diameter (d): the distance across a base through its center. It equals 2r.
• Height (h): the straight distance between the two bases, measured at a right angle to them.
• Axis: the line that joins the centers of the two bases.
• Lateral surface: the curved side. If you unroll it, it becomes a rectangle.

When people say "cylinder", they usually mean a right circular cylinder. Its sides stand straight up from its round bases.

Is a cylinder a prism?

A cylinder works like a prism with a round base. A prism has two equal, parallel bases joined by flat sides. A cylinder has two equal, parallel circles joined by one curved side. That is why both use the same rule: volume = base area × height. For a prism, the base is a polygon. For a cylinder, the base is a circle.

Cylinder volume formula

The formula for the volume of a cylinder is:

Where:

• V is the volume.
• π (pi) is about 3.14159.
• r is the radius of the base.
• h is the height.

If you know the diameter instead of the radius, use this version of the cylinder equation:

Both formulas give the same answer, because r = d ÷ 2. The second one saves you a step when you measure across the cylinder.

Why the cylinder volume formula works

The volume of any prism-like solid is its base area times its height. A cylinder's base is a circle, and the area of a circle is π × r². Multiply that area by the height, and you get π × r² × h. Picture a stack of coins. Each coin is a thin circle. Stack enough of them and you get a cylinder. The volume is the area of one coin times the height of the stack.

Volume of a circle vs a cylinder

People often search for the "volume of a circle". A circle is flat, so it has area, not volume. Its area is π × r². What they usually want is the volume of a cylinder, which is a circle with height. Some people mean a ball. A ball is a sphere, and its volume is 4/3 × π × r³. (sphere links to /sphere/ only if live.)

How to find the volume of a cylinder

To calculate the volume of a cylinder by hand, follow these steps:

1. Measure the radius. If you have the diameter, divide it by 2.
2. Square the radius: multiply it by itself.
3. Multiply by π (3.14159) to get the base area.
4. Measure the height, in the same unit as the radius.
5. Multiply the base area by the height.

The answer is in cubic units. Use the same unit for every measurement. If the radius is in inches and the height is in feet, change one of them first. The cylinder calculator above does this for you.

Cylinder volume example in inches

A cylinder is 10 inches across and 12 inches tall.

1. Radius = 10 ÷ 2 = 5 inches.
2. r² = 5 × 5 = 25.
3. Base area = 3.14159 × 25 = 78.54 square inches.
4. Height = 12 inches.
5. Volume = 78.54 × 12 = 942.48 cubic inches.

To find gallons, divide by 231: 942.48 ÷ 231 = 4.08 gallons.

Cylinder volume example in centimeters

A jar has a radius of 6 cm and a height of 15 cm. V = 3.14159 × 6² × 15 = 3.14159 × 36 × 15 = 1,696.46 cm³. Since 1,000 cm³ = 1 liter, the jar holds about 1.7 liters.

Cylinder volume example in feet

A round drum is 2 feet across and 3 feet tall. V = 3.14159 × 1² × 3 = 9.42 cubic feet. One cubic foot holds 7.48 gallons, so the drum holds 9.42 × 7.48 = 70.5 gallons.

Cylinder volume example with mixed units

A tube is 18 inches across and 4 feet tall. First change the height to inches: 4 × 12 = 48 inches. V = 3.14159 × 9² × 48 = 12,214.5 cubic inches, or about 52.9 gallons. In the calculator, you can skip this step. Pick inches for the diameter and feet for the height.

Cylinder volume in gallons, liters and cubic feet

The cylinder volume formula gives cubic units. To know how much liquid a cylinder holds, change the result to gallons or liters.

Cylinder volume calculator in gallons

To find the gallons in a cylinder, measure in inches and divide the cubic inches by 231. Or measure in feet and multiply the cubic feet by 7.48.

For tanks that lie on their side or are only partly full, use the tank volume calculator. For other unit changes, use the gallon converter.

Multiply the gallons per inch by the height in inches to get the cylinder's capacity. A cylinder 24 inches across and 30 inches tall holds about 1.958 × 30 = 58.7 gallons.

Cylinder volume in liters

Measure in centimeters, then divide the cubic centimeters by 1,000. Or measure in meters, then multiply the cubic meters by 1,000.

Example: A can 8 cm across and 12 cm tall holds 3.14159 × 4² × 12 ÷ 1,000 = 0.603 liters.

Cylinder volume in cubic feet and cubic yards

For concrete columns, round pits and other large jobs, work in feet. Divide cubic feet by 27 to get cubic yards.

Example: A round concrete column 12 inches across and 8 feet tall has a radius of 0.5 feet. V = 3.14159 × 0.5² × 8 = 6.28 cubic feet, or 0.23 cubic yards.

How to measure a cylinder

Your result is only as good as your measurements. Use these tips to measure a cylinder the right way.

• Measure across the widest point. The diameter goes through the center. Measure a few times in different directions and use the largest reading.
• Use the distance around. If you cannot reach across, wrap a tape around the cylinder to get its circumference C. Then d = C ÷ 3.14159. A pipe 31.4 inches around is 10 inches across.
• Measure the inside for capacity. To know how much a container holds, measure its inside diameter and inside depth. The outside sizes include the walls.
• Measure the height straight. Hold the tape at a right angle to the base, not along a slanted side.
• Check for a taper. Many cups, buckets and flower pots are wider at the top. Those are not cylinders. Use the truncated cone calculator for them.
• Check for rounded ends. A cylinder with domed ends is a capsule. Use the capsule calculator.

How radius and height change cylinder volume

The radius matters more than the height, because the formula squares it.

• Double the height, and the volume doubles.
• Double the radius, and the volume grows 4 times.
• Double both, and the volume grows 8 times.
• Cut the radius in half, and the volume drops to one quarter.

This is why a small error in the radius gives a big error in the volume. Measure the diameter with care.

Types of cylinders and their volume formulas

Not every cylinder is a straight, solid can. Pick the type in the cylinder calculator to use the right formula.

Right circular cylinder

The sides stand at a right angle to the round bases. This is the standard cylinder, and it uses V = π × r² × h.

Oblique cylinder volume

An oblique cylinder leans to one side. Its sides are not at a right angle to the bases. A stack of coins pushed sideways is an oblique cylinder. Its volume uses the same formula, V = π × r² × h. The height must be the perpendicular height: the straight up-and-down distance between the bases, not the length of the slanted side. This follows from Cavalieri's principle. Two solids with the same cross-section area at every height have the same volume. Leaning the stack does not change any coin. If you only know the slant length s and the angle θ between the side and the base, find the height first:

Example: A cylinder with a radius of 2 inches has a slant side of 10 inches at 60° to the base. h = 10 × sin 60° = 8.66 inches. V = 3.14159 × 4 × 8.66 = 108.8 cubic inches.

Hollow cylinder volume

A hollow cylinder, also called a cylindrical shell, is a tube with a wall. Think of a pipe, a straw or a paper towel roll. To find the volume of the wall material, take the volume of the outer cylinder and subtract the inner one.

R and D are the outer radius and diameter. r and d are the inner ones.

Example: A tube with a 6-inch outer diameter, a 5-inch inner diameter and a length of 24 inches has V = 3.14159 × (36 − 25) × 24 ÷ 4 = 207.3 cubic inches of material.

To find how much a pipe holds inside, use the inner diameter in the regular cylinder formula. For more detail, see the hollow cylinder volume calculator and the pipe volume calculator. (Each links to /hollow-cylinder/ and /pipe/ only if live.)

Elliptical cylinder volume

An elliptical cylinder has an oval base instead of a circle. Many oval tanks and some cups have this shape. The area of an oval is π × a × b, where a and b are half the long width and half the short width.

Example: An oval cylinder 8 inches wide, 5 inches deep and 10 inches tall has a = 4 and b = 2.5. V = 3.14159 × 4 × 2.5 × 10 = 314.2 cubic inches.

If a = b, the oval is a circle, and the formula becomes π × r² × h.

Cylinder with rounded ends

A cylinder with a half sphere on each end is called a capsule. Propane tanks and pills have this shape. Add the volume of one full sphere to the cylinder part:

Here h is the length of the straight part only. For this shape, use the capsule calculator.

Half cylinder volume

A half cylinder is a cylinder cut in half along its length, like a trough or a tunnel roof. Its volume is half of a full cylinder.

Example: A half cylinder trough with a radius of 6 inches and a length of 20 inches holds 3.14159 × 36 × 20 ÷ 2 = 1,131 cubic inches, or 4.9 gallons.

Find the height or radius of a cylinder from its volume

You can turn the cylinder volume formula around to find a missing size. Pick "Height" or "Radius" under "Solve for" in the cylinder calculator.

How to find the height of a cylinder

Divide the volume by the base area.

Example: You want a cylinder that holds 5 gallons, and it is 10 inches across. 5 gallons = 1,155 cubic inches. h = 1,155 ÷ (3.14159 × 25) = 14.7 inches.

How to find the radius of a cylinder

Divide the volume by π times the height, then take the square root.

Example: You want a container 8 inches tall that holds 1 gallon (231 cubic inches). r = √(231 ÷ (3.14159 × 8)) = 3.03 inches. The diameter is about 6.06 inches.

Make sure the volume and the length use matching units, such as cubic inches and inches.

How to find the diameter of a cylinder

Find the radius first, then double it. If you can wrap a tape around the cylinder, divide the distance around by π instead.

Example: A 5-gallon container 12 inches tall needs a diameter of 2 × √(1,155 ÷ (3.14159 × 12)) = 11.07 inches.

Cylinder volume vs cone and sphere

A cylinder, a cone and a sphere with the same radius have a neat link between their volumes.

• Cone: a cone with the same base and height as a cylinder holds exactly one third as much. V = 1/3 × π × r² × h. (cone links to /cone/ only if live.)
• Sphere: a sphere that just fits inside a cylinder (h = 2r) holds exactly two thirds as much. V = 4/3 × π × r³.
• Hemisphere: half a sphere holds one third of a cylinder with the same radius and a height of 2r. That is the same as the cone.

The Greek mathematician Archimedes proved the sphere and cylinder rule more than 2,000 years ago. He asked for a sphere inside a cylinder to be carved on his tomb.

Cylinder volume and surface area

Volume tells you how much fits inside. Surface area tells you how much material covers the outside, such as the metal in a can or the paint on a column.

Example: For the 10 by 12 inch cylinder, the lateral surface is 2 × 3.14159 × 5 × 12 = 377 square inches. The total surface, with both ends, is 534.1 square inches.

Surface area to volume ratio of a cylinder

Divide the surface area by the volume.

A large cylinder has a lower ratio than a small one. That is why a big tank loses heat more slowly than a small one with the same shape. For the 10 by 12 inch cylinder, the ratio is 2 ÷ 5 + 2 ÷ 12 = 0.57 per inch.

The cylinder with the most volume for its surface

For a set volume, a cylinder uses the least material when its height equals its diameter (h = 2r). It looks about as tall as it is wide.

Example: To hold 1 gallon (231 cubic inches), the best cylinder has r = ∛(231 ÷ 6.2832) = 3.33 inches and h = 6.65 inches. Its total surface is 208.4 square inches. A tall, thin cylinder with a 2-inch radius needs 18.4 inches of height and 256.1 square inches of surface to hold the same gallon. That is about 23% more material. Real cans are often taller than this, because shape, grip and shelf space matter too.

Cylinder volume with calculus

Calculus shows why the cylinder volume formula is true. Slice the cylinder into thin discs of thickness dz. Each disc has the area π × r². Add up all the discs from the bottom (z = 0) to the top (z = h):

You get the same answer with the shell method. Wrap the cylinder in thin tubes of radius x and thickness dx. Each tube has the volume 2 × π × x × h × dx:

Both methods prove V = π × r² × h. The disc method also explains the cone: its slices shrink as you go up, which gives one third of the cylinder.

Weight of a cylinder

To find the weight of a cylinder, multiply its volume by the density of what it is made of or what it holds.

For water, one gallon weighs about 8.34 pounds, and one liter weighs about 1 kilogram.

Example: The 10 by 12 inch cylinder holds 4.08 gallons. Full of water, the water weighs 4.08 × 8.34 = 34 pounds, or 15.4 kilograms. For other materials, use the volume to weight calculator. It covers metals, oils, concrete and more.

Real-life cylinder volume examples

Cylinders are all around you. Use these examples to check your own results.

In an engine, the space a piston sweeps is a cylinder. Its volume is π × (bore ÷ 2)² × stroke. Four such cylinders make an engine of about 2.0 liters.

Common mistakes when calculating cylinder volume

• Using the diameter as the radius. This makes the answer 4 times too big. Halve the diameter first, or use the π × d² × h ÷ 4 formula.
• Mixing units. A radius in inches and a height in feet give a wrong answer. Change them to one unit first.
• Using the slant length for an oblique cylinder. Use the perpendicular height.
• Measuring the outside of a container. For how much a container holds, measure the inside. Walls take up space.
• Rounding π too early. Using 3 instead of 3.14159 makes the answer about 4.5% too small.
• Forgetting cubic units. The answer is in cubic inches, cubic feet or cubic centimeters, not in square units.

Where cylinder volume is used

• Home and kitchen: how much a pot, jar or glass holds.
• Water and tanks: the gallons in a round tank, drum or well.
• Construction: concrete for round columns, piers and post holes. (concrete links to /concrete/ only if live.)
• Pipes: how much water a pipe holds.
• Engines: the displacement of an engine's cylinders.
• Packaging: the size of cans, tubes and bottles.
• Science: the volume of test tubes, graduated cylinders and lab columns.

Related calculators

Sources

We checked the formulas and unit values on this page against these sources.

1. Cylinder. Wolfram MathWorld. “Cylinder volume and surface area formulas.”
2. Cavalieri's Principle. Wolfram MathWorld. “Why an oblique cylinder has the same volume as a right one.”
3. Elliptic Cylinder. Wolfram MathWorld. “The volume of a cylinder with an oval base.”
4. NIST Handbook 44, Appendix C: General Tables of Units of Measurement. National Institute of Standards and Technology. “Cubic inch, gallon and liter values.”