Spherical Cap Volume
Calculate the volume of a spherical cap using V = pi h squared (3R-h) / 3.
Includes step-by-step examples and diagrams.
The Formula
The spherical cap volume formula calculates the volume of a portion of a sphere that is cut off by a plane. Imagine slicing through a ball with a flat knife. The smaller piece you cut off is a spherical cap. The formula depends on two measurements: the height of the cap (h) and the radius of the original sphere (R).
This formula is derived using integral calculus by revolving a circular arc around an axis and summing infinitesimal disks. Despite the complex derivation, the final formula is compact and easy to apply. It reduces to the full sphere volume formula when h equals 2R (the full diameter), giving V = 4πR³/3 as expected.
Spherical caps appear frequently in engineering and science. Dome structures in architecture are spherical caps. The meniscus of a liquid in a cylindrical container forms a spherical cap shape. In optics, the curved surface of a lens is often a spherical cap. Astronomers use spherical cap calculations to determine what portion of the celestial sphere is visible from a given location.
An alternative form of the formula uses the base radius (a) of the cap instead of the sphere radius: V = πh(3a² + h²) / 6. Both forms are equivalent because a² = 2Rh − h². Choose whichever form matches the measurements you have available.
Variables
| Symbol | Meaning |
|---|---|
| V | Volume of the spherical cap |
| h | Height of the cap (distance from base to top) |
| R | Radius of the original sphere |
| a | Radius of the cap's circular base (alternative form) |
Example 1
Find the volume of a spherical cap with height 3 cm on a sphere of radius 10 cm.
V = π(3)2(3 × 10 − 3) / 3
V = π × 9 × 27 / 3 = π × 81
V ≈ 254.47 cm³
Example 2
A dome is a spherical cap with height 5 m on a sphere of radius 12 m. Find the enclosed volume.
V = π(5)2(3 × 12 − 5) / 3
V = π × 25 × 31 / 3 = 775π / 3
V ≈ 811.05 m³
When to Use It
Use the spherical cap formula whenever you need to find the volume of a dome-shaped or bowl-shaped portion of a sphere.
- Architecture: calculating the volume enclosed by a dome roof
- Manufacturing: measuring the volume of convex or concave lens surfaces
- Food science: estimating the volume of rounded food portions
- Civil engineering: calculating liquid volume in spherical tanks filled to a certain level
- Astronomy: determining visible sky coverage at a given elevation angle