How to use the spherical cap and zone calculator
- Pick what you know: sphere radius and cap height, sphere radius and base radius, cap height and base radius, or a zone between two planes.
- Enter the lengths in any one unit.
- Read the volume and curved surface area, then the base radius, total surface and share of the sphere.
- Use the check line to confirm that the cap and the rest of the sphere add up to the whole.
Formula
Cap volume = pi h^2 (3R - h) / 3
Curved area = 2 pi R h; base radius a = sqrt(h (2R - h))
Sphere radius from a cap: R = (a^2 + h^2) / (2h)
Zone volume = pi (R^2 h - (t_high^3 - t_low^3) / 3); zone area = 2 pi R h
R is the sphere radius, h the height of the cap or zone, a the radius of the flat circular face and t the signed distance of a plane from the center.
Cap and zone of a 10 unit sphere
A cap of height 4 cut from a sphere of radius 10 has volume pi x 16 x 26 / 3 = 435.6342, a curved area of 2 pi x 10 x 4 = 251.3274 and a flat face of radius sqrt(4 x 16) = 8. A cap of height 2 with a base radius of 6 gives R = (36 + 4) / 4 = 10 and a volume of 117.2861. A cap whose height equals the radius is a hemisphere, with volume 2094.3951.
A zone of the same sphere between planes 3 below and 6 above the center is 9 tall. Its curved area is 2 pi x 10 x 9 = 565.4867 and its volume is 2572.9644, with end circles of radii 9.539392 and 8.
Caps, zones and other slices
A cap is the piece cut off a sphere by one plane. A zone is the band between two parallel planes. A cap is a zone with one plane at the pole. Both have a curved area that depends only on the radius and the height, a result known since Archimedes, so equal slices of any position on a sphere have equal curved area.
The same shapes show up as partly filled round tanks. For tanks filled to a depth see the tank volume calculator, for pipes use the pipe volume calculator, and to compare circles see the circle overlap area calculator.
Smaller or larger cap
When you give the base radius, two caps share that flat face: the smaller one below the plane and the larger one above it. The page lets you choose either. With a base radius equal to the sphere radius both are hemispheres.
Assumptions and limits
- The sphere is perfect and all lengths are in one unit. Volumes are in that unit cubed and areas in that unit squared.
- A cap height can be at most the sphere diameter, and a plane must lie between the poles.
- The base radius cannot exceed the sphere radius when it is combined with the sphere radius.
- Zone planes are given as signed distances from the center: negative below, positive above.
Frequently asked questions
How do I find the volume of a spherical cap?
Use V = pi h^2 (3R - h) / 3 with the sphere radius R and the cap height h. If you know the base radius a instead, first find h = R - sqrt(R^2 - a^2) for the smaller cap.
How do I find the sphere radius from a cap?
Measure the height h and the radius a of the flat face. The sphere radius is R = (a^2 + h^2) / (2h), the same relation that describes a circle through three points.
What is the surface area of a zone?
It is 2 pi R h, the radius times the height times 2 pi. The curved area depends on the height of the band only, not on where on the sphere it sits.
Does the total surface include the flat face?
The page gives the curved area and, separately, the total with the flat circular face added, because it depends on whether you need the open shell or the closed solid.