Spherical Cap Volume Calculator
Find the volume, curved surface area and base radius of a spherical cap from the sphere radius and cap height.
How it works
V = (π·h²/3)(3R − h) · Curved area = 2·π·R·h · Base radius = √(h(2R − h))
A spherical cap is the dome sliced off a sphere by a flat plane — the portion above (or below) the cut. Its volume depends only on the sphere’s radius R and the cap’s height h, through (πh²/3)(3R − h). The curved surface area is 2πRh, a strikingly simple result (Archimedes’ hat-box theorem), and the flat circular base has radius √(h(2R − h)). When h equals R the cap is exactly a hemisphere.
Worked example
For a sphere of radius 5 with a cap 2 units tall: volume = (π·2²/3)(3·5 − 2) = (4π/3)(13) ≈ 54.45 units³; curved area = 2·π·5·2 = 20π ≈ 62.83 units²; base radius = √(2·(10 − 2)) = √16 = 4 units.
Sources: Spherical cap — Wolfram MathWorld
Frequently asked questions
What happens when the cap height equals the radius?
When h = R the cap is a hemisphere. Substituting gives V = (πR²/3)(3R − R) = (2/3)πR³, exactly the hemisphere volume, which confirms the formula.
Why does the curved surface area not depend on where the cut is?
By Archimedes’ theorem, the curved area of a spherical cap is 2πRh regardless of latitude — two parallel cuts the same vertical distance apart always enclose the same amount of sphere surface.
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This calculator is for educational purposes. Double-check important results before acting on them.