Cylinder Volume Calculator

Cylinder volume calculator at a glance#

What is the formula for the volume of a cylinder?#

The formula for the volume of a cylinder is V = π × r² × h, where r represents the radius of the circular base, and h is the height. In words, you calculate the area of the base circle and multiply it by the height.

How do you find the volume of a cylinder from the diameter?#

First, halve the diameter to determine the radius, and then apply V = π × r² × h. Alternatively, you can use the diameter directly with V = π × d² × h ÷ 4. If you square the diameter without dividing by 4, your result will be four times larger than it should be.

What is the volume of a cylinder with a base area of 10 square units and a height of 7 units?#

The volume in this case is 70 cubic units. If the base area is already known, the formula simplifies to V = base area × height, and 10 × 7 = 70.

What cylinder volume means#

A cylinder is a series of identical circles stacked upon each other. Each slice, no matter how thin, shares the same area as the base, which means you can find the volume by multiplying the base area by the cylinder's height. OpenStax Prealgebra 2e provides this formula in both forms, V = π × r² × h and V = B × h, where B represents the area of the base.

Let's consider a water tank that measures 4 feet in diameter and 6 feet in height. The radius, being half the diameter, is 2 feet.

First, calculate the base area using the formula π × 2² = 4π, which is approximately 12.57 square feet. Next, multiply this area by the height, using 4π × 6 = 24π, to find a volume of about 75.40 cubic feet. If you need to convert this, 75.40 cubic feet equals about 564 US gallons or approximately 2,135 liters.

The volume will be in cubic units based on the measurement unit used. Worked by hand, if you measure in feet, you'll get cubic feet (the calculator always answers in cubic meters). If you use inches, you'll obtain cubic inches. Keep the radius and height in the same unit before multiplying.

Radius vs diameter: the most common mistake#

Most errors with cylinders arise from using the diameter when the formula actually requires the radius. Tanks, pipes, and cans are typically measured by their width, meaning the diameter. If you take the tank's width of 4 feet and mistakenly use it as the radius, you end up with π × 4² × 6 = 96π, approximately 301.6 cubic feet. This is four times the correct volume of 75.40 cubic feet, because when you double the radius, the base circle becomes four times as large.

Homework problems often present you with a diameter to illustrate this point. Consider a cylinder 18.5 meters tall with a base diameter of 5.2 meters. Here, the radius is 2.6 meters, so the volume calculates to π × 2.6² × 18.5, which is roughly 392.9 cubic meters rounded to the nearest tenth. If the problem provides the radius directly, like 8.9 meters with a height of 16.5 meters, use that as given: π × 8.9² × 16.5 is about 4,106.0 cubic meters.

How to use the calculator#

First, select the two values you know: you can choose between radius and height, radius and volume, radius and lateral area, height and lateral area, or height and volume. Each value has its own unit menu, so you might have the radius in inches and the height in feet. You also have the option to adjust the value of π.

The calculator will provide seven measurements: radius, height, volume, lateral surface area, top area, base area, and total surface area. Regardless of the units you input, the results are displayed in meters, square meters, and cubic meters. For the tank above, a radius of 2 feet and a height of 6 feet results in a volume of approximately 2.135 cubic meters. To convert cubic meters to cubic feet, multiply by 35.315, or to US gallons, multiply by 264.17.

The other measurements are derived from these formulas:

• Lateral, or side, area: L = 2 × π × r × h
• Top or base area: π × r²
• Total surface area: A = 2 × π × r × (h + r)
• Height from volume: h = V ÷ (π × r²)
• Radius from volume: r = √(V ÷ (π × h))

For the tank, the side area is 2 × π × 2 × 6 = 24π, approximately 75.40 square feet. This matches the volume figure only because the radius is 2 here, with one in square feet and the other in cubic feet. The total surface area is 2 × π × 2 × (6 + 2) = 32π, around 100.53 square feet.

Cylinder volume in gallons and liters#

Tanks, drums, and pipes are typically measured by their capacity. NIST Handbook 44 specifies the US gallon as precisely 231 cubic inches and indicates 1 cubic foot is equivalent to 7.480519 gallons. These conversions cover most situations.

Convert cubic feet to US gallons by multiplying by 7.4805. For cubic inches, divide by 231 to get US gallons. Calculating liters can be done by multiplying cubic feet by 28.317 or cubic meters by 1,000.

This means the tank in the example holds about 564 gallons. For a straightforward approximation of an upright tank, find the gallons per foot of height, then multiply by the water depth. The table below provides this value for standard inside diameters and also indicates the water contained in each foot of pipe.

Pipes and hollow cylinders#

Water is contained within a pipe's inner diameter, so use the inner measurement for capacity. A pipe with a 2 inch inside diameter contains approximately 0.163 gallons per foot. So 100 feet of such pipe holds around 16.3 gallons.

To calculate the material in a tube's wall, subtract the inner cylinder's volume from the outer one: V = π × (R² − r²) × h, where R is the outer radius and r is the inner radius. This calculator is designed for solid cylinders. For a hollow version using two diameters, use the tube shape in the volume calculator.

Sources#

• OpenStax, Prealgebra 2e, section 9.6: Solve Geometry Applications: Volume and Surface Area
• NIST Handbook 44 (2024), Appendix C: General Tables of Units of Measurement

Frequently asked questions#

How many gallons are in a cylinder?#

First, find the volume in cubic feet with V = π × r² × h, then multiply by 7.4805. A tank that measures 4 feet across and 6 feet tall holds roughly 75.40 cubic feet, which is about 564 gallons. If your measurements are in inches, divide the cubic inches by 231 instead.

How do you find the height of a cylinder from its volume?#

To solve for height, divide the volume by the base area: h = V ÷ (π × r²). For a tank with a radius of 1 foot designed to hold 100 gallons (about 13.37 cubic feet), you would require a height of about 4.26 feet. The calculator's radius and volume option does this too: enter 13.37 ft³ and it returns the height in meters, about 1.30 m.

How do you find the cubic yards in a cylinder?#

Determine the volume in cubic feet first, then divide by 27, because one cubic yard equals 27 cubic feet.

Is the volume of a cone one third of a cylinder?#

Yes. A cone with the same base and height as a cylinder holds exactly one third as much, V = π × r² × h ÷ 3. The volume calculator covers cones, spheres, and nine other shapes.

Why does the formula square the radius?#

Because the base is a circle, and the area of a circle is π × r². Doubling the radius makes the circle four times larger. So, a can that's twice as wide holds four times as much at the same height.

Does a cylinder lying on its side have the same volume?#

Yes, the total volume does not change with position. What changes is how the depth relates to the contents. A horizontal tank filled to half its depth is half full. But at other depths, the gallons do not rise evenly with the depth, so this calculator gives the full volume only.