Hong Life CalculatorsUnderstanding the spherical cap and hemisphere calculator — concept and principle
A spherical cap is the piece of a ball cut off by a flat plane, like the top of a dome or the liquid in the bottom of a round flask. It is defined by the sphere radius r and the cap height h, measured from the cutting plane to the top of the cap. When h equals r the cap is a hemisphere. The calculator returns the cap volume, the curved surface area and the radius of the flat circular base, which helps with domes, lenses, spherical tank levels and rounded lids.
How it is calculated
- Cap volume: V = πh²(3r − h) ÷ 3.
- Curved surface: A = 2πrh (the flat base is not included).
- Base radius: a = √(h(2r − h)).
h must lie between 0 and 2r. With h = r you get the hemisphere: V = 2πr³/3 and curved area 2πr². With h = 2r you get the whole sphere, 4πr³/3.
The principle
Place the sphere's centre at the origin. A horizontal slice at height y is a disc of radius √(r² − y²), so its area is π(r² − y²). Adding all slices from y = r − h up to r, that is, integrating, gives πh²(3r − h)/3. The base radius follows from the right triangle formed by the sphere radius, the distance r − h from the centre to the cutting plane and the base radius: a² = r² − (r − h)² = h(2r − h).
The surface result is a famous discovery of Archimedes: a band of a sphere has the same area as the matching band on the cylinder wrapped around it, so the curved area depends only on h, not on where the cut is made.
Worked example
- Inputs: r = 5, h = 2.
- Volume: π × 2² × (3 × 5 − 2) ÷ 3 = π × 4 × 13 ÷ 3 = 52π/3 ≈ 54.45427266.
- Curved surface: 2π × 5 × 2 = 20π ≈ 62.83185307.
- Base radius: √(2 × (10 − 2)) = √16 = 4.
- Check: a 3-4-5 triangle, since the plane is 5 − 2 = 3 from the centre.
| Cap height | Cap volume |
|---|---|
| 0.5 | 3.796091123 |
| 1 | 14.66076572 |
| 1.5 | 31.80862562 |
| 2 | 54.45427266 |
| 2.5 | 81.81230869 |
| 3 | 113.0973355 |
| 4 | 184.306769 |
How to read the result
The full sphere of radius 5 has volume about 523.6, so this cap holds about 10% of it. Volume is not proportional to height: a cap of height r/2 holds only 5/16 (about 31%) of the hemisphere's volume, because the narrow top holds little. This is why the level in a spherical tank does not rise evenly as it fills. The curved area, on the other hand, grows exactly in proportion to h.
Common mistakes
- Entering the base radius a instead of the sphere radius r. If you know a and h, first find r = (a² + h²) ÷ (2h).
- Measuring h from the centre of the sphere instead of from the cutting plane.
- Expecting the curved surface to include the flat base; add πa² for the total.
Frequently asked questions
How do I find the sphere radius from a dome's width and height?
Use r = (a² + h²) ÷ (2h), where a is half the width. For a = 4 and h = 2, r = 20 ÷ 4 = 5.
What is the volume of a hemisphere of radius 5?
2πr³/3 = 2π × 125 ÷ 3 ≈ 261.8.
How full is a spherical tank at a given liquid depth?
The liquid is a cap with h equal to the depth, so its volume divided by 4πr³/3 gives the fraction full.
Sources: Archimedes, On the Sphere and Cylinder, Book I (surface of a spherical segment) · D. Zwillinger (ed.), CRC Standard Mathematical Tables and Formulae, CRC Press