Slice a sphere with a flat plane and each piece is a spherical cap. Domes, lenses, contact lenses, bowls, the liquid in a round tank and the polar region of a planet are all caps. Three measurements describe one — the sphere’s radius R, the cap’s height h, and the radius a of its flat circular base — and any two of them fix the third. This calculator works from whichever pair you know and returns the volume, the curved and total surface areas, and the angle the cap spans.
How to use the spherical cap calculator
- Choose the pair you know: sphere radius and cap height, base radius and cap height, or sphere radius and base radius.
- Enter the two values and pick the unit.
- Read the cap’s volume on the tape, followed by the curved (dome) area, the base area, the total area, all three dimensions, the polar angle, the cap’s share of the whole sphere and its capacity in liters and gallons.
Spherical cap formulas
| Quantity | Formula |
|---|---|
| Base radius | a = √(h(2R − h)) |
| Sphere radius | R = (a2 + h2) ÷ (2h) |
| Height (smaller cap) | h = R − √(R2 − a2) |
| Volume | V = πh2(3R − h) ÷ 3 = πh(3a2 + h2) ÷ 6 |
| Curved area | A = 2πRh = π(a2 + h2) |
| Base area | πa2 |
The base-radius relation is the Pythagorean theorem in the cross-section: the sphere’s center, the base’s center and a point on the base rim form a right triangle with legs a and R − h and hypotenuse R.
Worked example
A dome is cut from a sphere of radius 10 cm, with a height of 4 cm.
Base radius: a = √(4 × (20 − 4)) = √64 = 8 cm, so the dome is 16 cm across.
Volume: V = π × 42 × (30 − 4) ÷ 3 = 416π/3 ≈ 435.6 cm³.
Check: π × 4 × (3 × 64 + 16) ÷ 6 = π × 4 × 208 ÷ 6 ≈ 435.6 cm³ ✓.
Curved area: 2π × 10 × 4 = 80π ≈ 251.3 cm²; with the 201.1 cm² base the total is about 452.4 cm².
Angle: cos θ = (10 − 4)/10 = 0.6, so the cap spans 53.13° from the sphere's center to its rim. It holds about 10.4% of the sphere's volume.
Special cases and checks
- h = R gives a hemisphere: V = (2/3)πR3 and curved area 2πR2.
- h = 2R gives the whole sphere: V = (4/3)πR3.
- Small h gives a shallow dome whose volume is roughly πa2h/2, half of the cylinder with the same base and height.
- The same base, two caps. A given base circle cuts a sphere into a small cap and a large one; their heights add up to 2R. When you enter R and a, the calculator returns the smaller cap and tells you the larger one’s height.
Practical uses
- Domes and roofs: estimate the enclosed air volume for heating, or the dome’s surface for paint or panels, from its span and rise.
- Spherical tanks: the volume of liquid at depth h is a cap volume, so a dipstick reading converts directly to gallons.
- Optics: lenses and contact lenses are caps; their sag (height) comes from the curvature radius and diameter.
- Earth science: the area of Earth north of a given latitude is a cap area, 2πRh.
Related tools
For whole spheres and half-spheres, use the sphere calculator and the hemisphere calculator. The cap’s flat base is a circle, covered by the circle calculator, and squashed domes are better modeled with the ellipsoid calculator.
Frequently asked questions
What is the volume of a spherical cap?
V = πh²(3R − h)/3, where R is the sphere's radius and h the cap's height. Equivalently V = πh(3a² + h²)/6 using the base radius a. A cap 4 cm high on a 10 cm sphere holds about 435.6 cm³.
What is the curved surface area of a cap?
A = 2πRh. Remarkably, it depends only on the height, not on where the slice is taken: any band of height h on a sphere of radius R has the same area, a result known since Archimedes.
How do I find the sphere's radius from a dome's width and height?
With base radius a and height h, R = (a² + h²)/(2h). A dome 16 m wide (a = 8) and 4 m tall comes from a sphere of radius (64 + 16)/8 = 10 m.
How do I find the volume of liquid in a spherical tank?
The liquid forms a cap whose height is the liquid depth, so V = πh²(3R − h)/3. If the tank is more than half full, the formula still works with h up to the full diameter 2R.