Volume Calculator

About 3D volume calculations

Volume is the amount of three-dimensional space enclosed by a solid figure, measured in cubic units (cubic centimetres, cubic metres, cubic inches, cubic feet). Every closed 3D shape has an exact volume formula derived from integral calculus, but for the eight shapes on this page the formulas reduce to simple algebra. The eight covered (cube, rectangular box, sphere, cylinder, cone, rectangular pyramid, ellipsoid, capsule) handle 95 percent of real-world volume problems: rooms, water tanks, swimming pools, packaging, fuel cans, pills, propane cylinders, footballs, planets approximated as ellipsoids, and prilled-tip capsule warheads.

Volume problems arise everywhere. Shipping a parcel uses dimensional weight (length x width x height divided by 5000 for international air freight). Concrete pours need cubic metres of mix. Aquariums and pool fill times depend on capacity. Pharmaceutical dosing converts tablet volume to mass. The capsule shape (cylinder with hemispherical end caps) shows up in propane tanks, vitamin capsules, and sailboat hulls. The ellipsoid covers eggs, melons, American footballs, and approximations of human organs in radiology.

How the formulas work

Every formula on this calculator can be derived by slicing the solid into thin horizontal disks (or boxes), computing the area of each slice, and integrating that area along the height axis. For shapes with constant cross-section (cube, rectangular box, cylinder) the integral collapses to area times height. For tapered shapes (cone, pyramid) the cross-section shrinks linearly so the integral picks up a factor of one-third.

Cube           V = s^3
Rectangular box V = L x W x H
Sphere         V = (4/3) x pi x r^3
Cylinder       V = pi x r^2 x h
Cone           V = (1/3) x pi x r^2 x h
Pyramid (rect) V = (1/3) x L x W x H
Ellipsoid      V = (4/3) x pi x a x b x c
Capsule        V = pi x r^2 x h + (4/3) x pi x r^3

• s = side length of the cube (all six faces identical).
• r = radius (half the diameter) for sphere, cylinder, cone, capsule.
• h = height (cylinder, cone, pyramid) or cylinder-section height (capsule).
• a, b, c = the three semi-axes of an ellipsoid (radius along each axis).

Worked example: cylindrical water tank

You need to size a vertical rainwater tank for a 4-person household. Each person uses 80 litres per day, you want 7 days of buffer, and you have a 1.5 m diameter spot on the roof.

1. Target capacity: 4 people x 80 L x 7 days = 2,240 litres = 2.24 cubic metres.
2. Radius: 1.5 m diameter, so r = 0.75 m.
3. Base area: pi x r squared = 3.1416 x 0.5625 = 1.767 square metres.
4. Required height: Volume divided by base area = 2.24 / 1.767 = 1.27 metres.
5. Round up: Buy a 1.5 m tall, 1.5 m diameter tank with volume pi x 0.75 squared x 1.5 = 2.65 cubic metres (2,650 L), giving a comfortable margin.

Volume unit conversions

Pitfalls to avoid

• Confusing radius and diameter. Sphere, cylinder, and capsule formulas take the radius (half the diameter). Plugging in the diameter gives 8x the correct answer for a sphere.
• Mixing units mid-formula. A cylinder calculated as pi x (5 cm) squared x (2 m) returns nonsense. Convert all dimensions to the same unit before computing.
• Forgetting wall thickness. Container capacity uses internal dimensions; volume calculators give external geometry unless you subtract wall thickness from each dimension.
• Treating an oval as a sphere. A rugby ball or an egg is an ellipsoid (three different semi-axes). Using sphere volume understates the long-axis case.
• Cone tip versus base orientation. The formula does not care which end is up, but if you want partial fill (cone half full of liquid), the volume from the tip is r squared x h cubed proportional, not linear.
• Rounding errors in pi. Using pi = 3.14 for a 1 cubic metre sphere costs you 0.05 percent accuracy; for engineering use pi to at least 5 decimals (3.14159) or the language constant.

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Frequently asked questions

Last updated 2026-05-28.