Spherical Sector Volume Calculator
A spherical sector, sometimes called a spherical cone, is the solid bounded by a spherical cap and the cone running from the rim of the cap to the centre of the sphere. It is the three-dimensional equivalent of a pie slice cut from a ball. The volume depends only on the sphere radius and the height of the bounding cap, captured by a remarkably simple exact formula. This tool computes the sector volume and also reports the corresponding spherical cap volume and full sphere volume so you can compare the pieces. Keep both inputs in the same length unit.
Spherical sector volume formula
Sector volume = (2 / 3) * pi * R^2 * h
Cap volume = (pi * h^2 / 3) * (3R - h)
Full sphere volume = (4 / 3) * pi * R^3
Here R is the sphere radius and h is the height of the bounding spherical cap. The sector is the cap plus the cone from the cap rim to the sphere centre, so it equals or exceeds the cap volume.
How spherical sector volume works
- The formula is exact and follows directly from integrating the sphere in spherical coordinates.
- A spherical sector combines a spherical cap with the cone joining its rim to the sphere centre.
- When the cap height equals the radius, the sector is a hemisphere with volume two-thirds of pi times R cubed.
- The cap height cannot exceed the diameter, so any height above 2R is treated as invalid.
- The result is in cubic units matching the unit of your radius and cap height inputs.
Spherical sector volume: frequently asked questions
What is a spherical sector?
A spherical sector is the solid formed by a spherical cap together with the cone that joins the edge of the cap to the centre of the sphere. Imagine an ice-cream cone whose rounded top is a piece of a sphere: the whole cone-plus-dome shape is a spherical sector. It is sometimes called a spherical cone.
What is the formula for the volume of a spherical sector?
The volume of a spherical sector is V = (2/3) * pi * R^2 * h, where R is the radius of the sphere and h is the height of the spherical cap that bounds the sector. This compact exact formula is a standard result of solid geometry derived from integration.
How is a spherical sector different from a spherical cap?
A spherical cap is only the dome cut off by a plane. A spherical sector adds the cone connecting the rim of that cap down to the centre of the sphere, so a sector is generally larger than the cap of the same height. When the cap height equals the radius, both shapes coincide with a hemisphere.
What happens when the cap height equals the sphere radius?
When h equals R the spherical sector becomes a hemisphere. The formula gives V = (2/3) * pi * R^2 * R = (2/3) * pi * R^3, which is exactly half the full sphere volume of (4/3) * pi * R^3, confirming the hemisphere result.
What units does the result use?
The volume is in cubic units of whatever length unit you use. If the radius and cap height are entered in centimetres, the volume is in cubic centimetres. Keep both inputs in the same unit for a correct answer.
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.