Spherical Sector Volume Calculator
From the sphere radius and the cap height, get the volume of a spherical sector — the cone-plus-cap solid swept from a sphere's centre.
Two inputs, one answer
Enter the sphere radius R and the cap height h, and the calculator returns the sector volume (2/3)πR²h.
Cap can't exceed the sphere
The cap height h must be no larger than the sphere's diameter 2R — a cap cannot be taller than the whole sphere, so larger values return no result.
What is a spherical sector volume calculator?
Radius and cap height in, volume out
A spherical sector is the solid you get by joining a spherical cap to the centre of its sphere — picture an ice-cream-cone shape whose rounded top is a slice of a sphere's surface and whose point sits at the sphere's centre. It is the region swept out when a radius rotates around to trace a circular patch on the surface. This calculator turns the sphere radius R and the cap height h into the sector's volume, the space it encloses. It is the tool for optics (lens and reflector caps), antenna and dish geometry, tank and dome design, and any geometry problem where a wedge of a sphere appears.
Enter the sphere radius and the cap height in any length unit to get the sector volume instantly.
One formula, built from the sphere radius R, the cap height h, and π (≈ 3.14159).
volume = (2/3) × π × R² × hThe volume grows with the square of the sphere radius and in direct proportion to the cap height: a taller cap on the same sphere sweeps a bigger sector. When the cap height reaches the full diameter (h = 2R), the formula gives (4/3)πR³ — the volume of the entire sphere — because the sector has grown to fill it.
Suppose you have a sphere of radius 5 and a cap height of 2.
Square the radius
R² = 5² = 25 — this scales the sector with the size of the sphere.
Apply the formula
(2/3) × π × 25 × 2 = 104.719755 cubic units — the space the spherical sector encloses.
The volume (about 104.719755 cubic units for R = 5, h = 2) is how much space the spherical sector encloses — the cone reaching out from the sphere's centre plus the rounded cap on top. The single most useful insight is the proportionality: volume rises with the square of the radius but only linearly with the cap height, so a small change in R moves the answer far more than the same change in h. A handy sanity check is the limiting case — set h = 2R and the formula returns (4/3)πR³, the volume of the whole sphere, confirming that the sector has grown to fill it. For our example, the sector's 104.72 cubic units is a fifth of the full sphere's 523.6 cubic units, matching h being one-fifth of the diameter. Because the result is unit-agnostic, the same number describes a tiny lens or a planetary dome — only the unit of the inputs changes the meaning.
The formula is exact, but a couple of practical points are worth keeping in mind.
Cap height limits and consistent units
This formula describes a true spherical sector of a perfect sphere, with the cap height measured from the cap's flat base circle to the top of the dome. The cap height cannot exceed the sphere's diameter (h ≤ 2R) — a larger value is geometrically impossible and returns no result. The radius and height are also unit-agnostic, so the answer is only meaningful if you keep one unit throughout: a radius in centimetres gives a volume in cubic centimetres, never a mix of units.