Spherical Cap Volume Calculator
From a sphere radius and a cap height, get the volume, the curved surface area, and the base radius — the three numbers that describe any dome sliced from a sphere.
Two inputs, three answers
Enter the sphere radius R and the cap height h and the calculator returns the volume ((πh²/3)(3R−h)), the curved surface area (2πRh), and the base radius (√(h(2R−h))) at once.
A cap can't beat its sphere
The cap height can never exceed the sphere's diameter (2R). At h = R the cap is a hemisphere; at h = 2R it is the whole sphere — anything taller is rejected.
What is a spherical cap volume calculator?
Sphere radius and cap height in, full dome out
A spherical cap volume calculator turns two measurements — the radius R of a full sphere and the cap height h — into the numbers that describe the dome sliced from that sphere by a flat plane: how much it holds (volume), the area of its curved outer skin (curved surface), and the radius of the flat circular face left by the cut (base radius). Each one is fixed once you know R and h, because every cap shares the same constant π (pi). That makes those two inputs all you need for liquid sitting in the rounded bottom of a spherical tank, architectural domes, and contact-lens or lens-segment volumes.
Enter the sphere radius and cap height in any length unit to get the volume, curved surface, and base radius instantly.
Three short formulas, all built from the sphere radius R, the cap height h, and the constant π (about 3.14159).
volume = (π × h² / 3) × (3R − h)The curved surface — the dome's outer skin, the slice of the sphere's surface that belongs to the cap — is 2 × π × R × h. The base radius, the radius of the flat circular face where the plane cut the sphere, is √(h × (2R − h)) by the intersecting-chord relationship. The volume, the space inside the dome, is (π × h² / 3) × (3R − h).
Suppose you slice a cap of height 2 from a sphere of radius 5.
Base radius
√(2 × (2 × 5 − 2)) = √(2 × 8) = √16 = 4 — the flat circular face left by the cut.
Curved surface
2 × π × 5 × 2 = 62.831853 square units — the dome's outer skin.
Volume
(π × 2² / 3) × (3 × 5 − 2) = 54.454273 cubic units — the space inside.
The three outputs answer three different everyday questions. The volume (about 54.454273 cubic units for R = 5, h = 2) is how much the cap holds — the liquid pooled in the rounded bottom of a spherical tank, the material in a dome, the fluid in a lens segment. The most useful insight is how the cap height sets the shape: when h = R the plane passes through the centre and the cap is exactly a hemisphere, half the sphere; when h = 2R the cap height reaches the full diameter and the cap is the whole sphere. The base radius (4 here) is the radius of the flat circular face the cut leaves behind — it grows as the cap deepens toward the equator, then shrinks again past it, peaking when h = R. The curved surface (about 62.831853 square units) is the dome's outer skin you would paint, plate, or glaze, and it rises in simple proportion to the cap height because it equals 2πRh. π ties it all together — the same constant links R and h to the volume, the surface, and the base of every cap, large or small.
The formulas are exact, but a couple of practical points are worth keeping in mind.
True spheres and consistent units
These formulas describe a cap sliced by a flat plane from a perfect sphere. A cap from an egg-shaped (ellipsoidal) surface, a dome with thick walls, or a tank whose ends are not truly spherical will differ from the computed value. The cap height must also stay within the sphere's diameter (h ≤ 2R), and the inputs are unit-agnostic, so the answers are only meaningful if you keep one unit throughout: a radius and height in centimetres give a volume in cubic centimetres and a curved surface in square centimetres, never a mix.