Spherical Cap Volume Calculator
A spherical cap is the dome you get when a single flat plane slices a sphere. It appears whenever you measure the liquid in a partly filled spherical tank, the curve of a contact lens, or the dome of a building. Its volume depends only on the radius of the original sphere and the height of the cap. This tool applies the exact formula derived by integrating the sphere's cross-sections, and also reports the base radius of the circular cut and the full sphere volume for comparison. Keep both inputs in the same length unit.
Spherical cap volume formula
Cap volume = (pi * h^2 / 3) * (3R - h)
Base radius = sqrt(h * (2R - h))
Full sphere volume = (4 / 3) * pi * R^3
Here R is the radius of the sphere and h is the cap height measured from the cutting plane to the top of the dome. The base radius is the radius of the circular cross-section where the plane cuts the sphere.
How spherical cap volume works
- The formula is exact, derived by integrating the area of circular cross-sections of the sphere from the cutting plane to the dome top.
- When the cap height equals the radius, the cap is a hemisphere with volume two-thirds of pi times R cubed.
- When the cap height equals the full diameter (2R), the cap is the entire sphere.
- The cap height cannot exceed the diameter, so this calculator marks any height above 2R as invalid.
- The result is in cubic units matching the unit of your radius and height inputs.
Spherical cap volume: frequently asked questions
What is a spherical cap?
A spherical cap is the portion of a sphere cut off by a single flat plane. Picture slicing the top off a ball: the dome-shaped piece you remove is the spherical cap. It is defined by the radius of the original sphere and the height of the cap, measured perpendicular from the cutting plane to the top of the dome.
What is the formula for the volume of a spherical cap?
The volume of a spherical cap is V = (pi * h^2 / 3) * (3R - h), where R is the radius of the sphere and h is the height of the cap. This exact formula comes from integrating the cross-sectional area of the sphere between the cutting plane and the top, a standard result in solid geometry.
What happens when the cap height equals the sphere radius?
When the cap height h equals the sphere radius R, the cap is exactly a hemisphere. The formula then gives V = (pi * R^2 / 3) * (3R - R) = (2/3) * pi * R^3, which is the well known volume of a hemisphere, half of the full sphere volume of (4/3) * pi * R^3.
Can the cap height be larger than the sphere radius?
Yes, up to the full diameter. A cap height between R and 2R describes a cap that is more than half the sphere. When h equals 2R the cap is the entire sphere and the formula reduces to (4/3) * pi * R^3. This calculator flags any height greater than the full diameter as invalid.
What units does the result use?
The volume is in cubic units of whatever length unit you use for radius and height. If both are in metres, the volume is in cubic metres. Keep the radius and the cap height in the same unit for a correct result.
Official sources
- NASA Glenn Research Center: Volume of solids reference.
- National Institute of Standards and Technology: SI Units and measurement.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.