About the Spherical Cap Calculator
A spherical cap is the piece of a sphere cut off by a flat plane — a dome, a lens, the liquid in the bottom of a round flask, or the top of a ball sliced off with a knife. This spherical cap calculator finds its volume, the curved (dome) surface area, the area of the flat circular base and the total surface area, from any two of the three defining lengths: the sphere radius R, the cap height h and the base radius a.
It is useful for estimating the volume of domed roofs, bowls, contact lenses and storage-tank heads, the amount of material in a dome shell, or how much liquid sits in a spherical vessel filled to a certain depth. A height up to R gives a cap no bigger than a hemisphere; a height between R and 2R gives the larger part of the sphere.
Use the same length unit for every input; areas are returned in square units and volumes in cubic units. With centimetres, the volume in cm³ equals millilitres.
With the default inputs, the cap volume is 54.4543 cu units. Change any value above to recalculate instantly.
How to use the spherical cap calculator
- 1Choose which two lengths you know.
- 2Enter them in the same unit.
- 3If you gave R and a, choose the smaller or larger cap.
- 4Read the cap volume, curved area, base area and total area.
Formula and method
The volume of a cap of height h cut from a sphere of radius R is πh²(3R − h)/3, which comes from integrating circular slices from the top of the sphere down to depth h. Its curved surface area is remarkably simple: 2πRh, the same as the side of a cylinder of radius R and height h (Archimedes’ hat-box theorem). The flat base is a circle of radius a, where a² = h(2R − h) from the Pythagorean theorem.
If you enter the base radius and height, the sphere radius is recovered from R = (a² + h²)/(2h). If you enter R and a, the height is R − √(R² − a²) for the smaller cap or R + √(R² − a²) for the larger one. The half-angle is the angle at the sphere’s centre between the axis and the rim of the cap, arccos((R − h)/R).
- R
- Radius of the whole sphere
- h
- Height of the cap
- a
- Radius of the cap’s circular base
- V
- Cap volume
Worked examples
Cap of height 2 from a sphere of radius 5
V = π × 4 × (15 − 2) ÷ 3 ≈ 54.45. The base radius is √(2 × 8) = 4, the dome area is 2π × 5 × 2 ≈ 62.83 and the cap holds 10.4% of the sphere.
Dome 12 m across and 2 m high
R = (36 + 4) ÷ 4 = 10 m. The dome encloses π × 2 × (108 + 4) ÷ 6 ≈ 117.3 m³ and its roof surface is π × (36 + 4) ≈ 125.7 m².
Sphere radius 10, base radius 8 — both pieces
h = 10 − √(100 − 64) = 4, so the smaller cap has V = π × 16 × 26 ÷ 3 ≈ 435.63. Choosing the larger piece gives h = 16 and V ≈ 3,753.16.
The larger piece of the same sphere
h = 10 + 6 = 16 and V = π × 256 × 14 ÷ 3 ≈ 3,753.16. Together with the small cap (435.63) this adds up to the full sphere volume of about 4,188.79.
Frequently asked questions
What is the formula for the volume of a spherical cap?+
V = πh²(3R − h)/3, where R is the sphere radius and h the cap height. If you know the base radius a instead of R, use V = πh(3a² + h²)/6.
How do you find the surface area of a spherical cap?+
The curved area is 2πRh, or equivalently π(a² + h²). Add the flat base πa² for the total surface area of a solid cap.
How do I calculate the volume of a dome?+
Measure its base radius a (half the span) and height h, then use V = πh(3a² + h²)/6. A dome 12 m across and 2 m high encloses about 117.3 m³.
How do I find the sphere radius from a cap?+
Use R = (a² + h²)/(2h). This is handy for lenses and domes where you can measure the span and rise but not the full sphere.
Is a hemisphere a spherical cap?+
Yes — it is the special case h = R. Then V = πR²(2R)/3 = (2/3)πR³ and the curved area is 2πR², exactly half the sphere.