A slice of a sphere
Slice a sphere with a flat plane and you get two spherical caps. Domes, bowls, contact lenses, the curved ends of some tanks, and the liquid sitting in the bottom of a spherical tank all have this shape.
The formulas
With R the radius of the sphere and h the height of the cap:
- Volume: V = π × h² × (3R − h) ÷ 3
- Base radius: a = √(h × (2R − h))
- Curved surface area: A = 2π × R × h
- Base area: π × a²
The curved area formula is remarkably simple: it equals the side area of a cylinder with the same radius and height. Archimedes proved it more than 2,200 years ago, and it was one of the results he was proudest of.
Worked example
A dome cut from a sphere of radius 10 feet, rising 4 feet:
| Result | Value |
|---|---|
| Volume | 435.6 ft³ (about 3,260 gallons) |
| Base radius | 8 ft (16 ft across) |
| Curved surface area | 251.3 sq ft |
| Base area | 201.1 sq ft |
| Share of the whole sphere | 10.4% |
Working from width and height
Often you know the width of a dome and its height, not the sphere’s radius. With base radius a and height h:
R = (a² + h²) ÷ (2h)
A dome 16 feet wide (a = 8) and 4 feet tall comes from a sphere of radius (64 + 16) ÷ 8 = 10 feet.
Practical uses
- Domes and roofs: surface area for roofing or paint, and enclosed volume for heating.
- Tanks: the volume of liquid in a spherical tank filled to height h; for a liquid above the midline, subtract the empty cap at the top from the full sphere.
- Lenses and optics: the sag of a curved lens surface.
- Cooking: the volume of a round-bottomed bowl filled to a given depth.
Checks
When h = R, the cap is a hemisphere: V = 2πR³ ÷ 3 and A = 2πR². When h = 2R, it is the full sphere: V = 4πR³ ÷ 3. If you enter a height larger than the sphere’s diameter, the calculator limits it to the full sphere.