A spherical cap is the piece of a sphere cut off by a flat plane, like a dome, a bowl or a contact-lens shape. Enter the radius of the sphere it came from and the height of the cap to get the volume, the base radius and the surface area.
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R is the radius of the full sphere and h is the height of the cap from its flat base to its top. The cap cannot be taller than the sphere’s diameter. If you know the base radius a and the height h instead, first find R = (a² + h²) / (2h).
This page is the spherical cap view of the full 3D Shape Calculator, which covers 24 solids. Every shape button is still available, so you can compare solids without leaving the page.
In every formula, r is a radius, h is a height or depth, and all measurements must use the same unit. The result is in that unit cubed for volume and squared for area.
A dome is cut from a sphere of radius 6 m, and its height is 3 m. The volume is π × 3² × (18 − 3) / 3 ≈ 141.37 m³, the base radius is √(3 × 9) ≈ 5.20 m and the curved roof area is 2 × π × 6 × 3 ≈ 113.10 m².
Spherical caps describe domes, planetarium roofs, tank end caps, bowls, lenses, water droplets on a surface and the lower part of a partly filled spherical tank.
What is the difference between a spherical cap and a hemisphere? A hemisphere is the special case where the cap height equals the radius. A cap can be shallower or taller than that.
What is the maximum cap height? The cap height can be at most the sphere’s diameter 2R, at which point the cap is the whole sphere.
How do I find the sphere radius from the base radius and the height? Use R = (a² + h²) / (2h), where a is the radius of the flat base and h is the cap height.
How is this used for a partly filled sphere tank? The liquid in a spherical tank at depth h forms a spherical cap, so its volume is π × h² × (3r − h) / 3. The Tank Volume Calculator does this directly.