Cylinder Volume Calculator

The volume of a cylinder formula

A cylinder is a circle pushed straight up. The circle at the bottom has area πr², and every slice of the cylinder above it has that same area, so the volume is the base area times the height. Think of a stack of coins: one coin's area times the height of the stack.

For a cylinder with radius 3 cm and height 5 cm:

1. Square the radius: 3² = 9. The base has area 9π cm².
2. Multiply by the height: 9 × 5 = 45, so V = 45π cm³. That is the answer "in terms of π".
3. Multiply by π for a decimal: 45 × 3.14159 ≈ 141.37 cm³.

Show r = 3 cm, h = 5 cm in the calculator

Volume of a cylinder with the diameter

Most real objects are measured across, which gives the diameter. Halve it to get the radius, or put r = d ÷ 2 into the formula to get V = πd²h ÷ 4. Since π ÷ 4 ≈ 0.7854, trade books write it as V = 0.7854 × d² × h. A drinks can about 6.6 cm across and 12.2 cm tall has π × 6.6² × 12.2 ÷ 4 ≈ 417.39 cm³. A real can holds less than that because its top and bottom are narrower and it is not filled to the brim.

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The most common mistake is to use the diameter as if it were the radius. Because the radius is squared, that makes the answer four times too big.

Cylinder volume in litres and gallons

The formula gives cubic units: cm³ if you measured in centimetres, in³ if you measured in inches. To get a capacity, convert with these exact or standard values:

A water tank with radius 30 cm and height 100 cm holds π × 30² × 100 = 90,000π ≈ 282,743.34 cm³, which is about 282.74 litres. A bucket 6 inches in radius and 12 inches tall holds 432π ≈ 1,357.17 in³, or about 5.8752 US gallons.

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Surface area of a cylinder

Cut a cylinder along its side and unroll it, and you get a flat net: one rectangle and two circles. The rectangle is as wide as the circle's circumference, 2πr, and as tall as the cylinder, h.

Indian and British textbooks call the curved part the CSA (curved surface area) and the whole the TSA (total surface area). For r = 3 and h = 5 the ends give 18π and the side gives 30π, so the total is 48π ≈ 150.8 square units. A cylinder with no lid, such as a cup or an open tank, has one end less: πr² + 2πrh.

Volume of a hollow cylinder

A pipe or tube is a big cylinder with a smaller one taken out. Its material has volume V = πR²h − πr²h = π(R² − r²)h, where R is the outer radius and r the inner one. The end of the pipe is a ring with area π(R² − r²).

Take a pipe 60 mm across the outside and 50 mm across the inside, 2 m long. The radii are 30 mm and 25 mm, so the material is π × (900 − 625) × 2000 = 550,000π ≈ 1,727,876 mm³, about 1.7279 litres. The hole inside holds π × 25² × 2000 ≈ 3.927 litres of water.

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Liquid in a partly filled tank

In a tank standing upright, the liquid is a shorter cylinder, so its volume is πr² times the depth. A tank lying on its side is harder. Seen from the end, the liquid fills a slice of the circle called a circular segment, and its area depends on the depth y:

A tank 120 cm across and 250 cm long with liquid 45 cm deep holds about 968.45 litres. That is about 34.25% of the tank, even though the depth is 37.5% of the diameter, because the tank is narrow at the bottom.

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Finding the height or radius from the volume

Rearrange V = πr²h. For the height, divide the volume by the base area: h = V ÷ (πr²). For the radius, divide by πh and take the square root: r = √(V ÷ (πh)). Make sure the volume is in the cube of your length unit first, so 2 litres becomes 2,000 cm³. A container with a radius of 5 cm needs to be 2000 ÷ (25π) = 80/π ≈ 25.4648 cm tall to hold 2 litres.

Find the height for 2 litres

Cylinder, cone and sphere

A cone with the same base and height as a cylinder holds exactly a third of it, V = ⅓πr²h. A sphere that just fits inside a cylinder (so h = 2r) holds two thirds of it, V = ⁴⁄₃πr³. Archimedes worked out that second fact more than 2,000 years ago. For r = 3 cm and h = 6 cm the three volumes are:

Oblique cylinders

A cylinder that leans to one side, like a stack of coins pushed over at an angle, has the same volume as a straight one, V = πr²h, as long as h is the straight-up height and not the length of the slanted side. Every slice is still a circle of area πr², and there are just as many slices. This is Cavalieri's principle. The surface area does change, because the side is longer.

Common mistakes

• Using the diameter as the radius. The answer comes out four times too big.
• Mixing units, such as a radius in centimetres and a height in metres. Convert first.
• Squaring the height instead of the radius. Only r is squared in πr²h.
• Reading cm³ as litres. 1,000 cm³ make one litre, so 141 cm³ is 0.141 L.
• Measuring a tank on the outside. The capacity depends on the inside size, without the wall.