Spherical Cap Calculator
Find the volume and surface of a spherical cap.
Results update as you type.
How it works
A spherical cap is the dome-shaped piece you get when a sphere is sliced by a flat plane, and this calculator finds its volume and surfaces. You enter the radius of the full sphere and the height of the cap, the distance from the flat cut to the top of the dome, and it returns the cap's volume, its curved surface area, and the radius of its circular flat base. The volume formula involves pi times the height squared over three, multiplied by three times the sphere radius minus the height, a neat expression that gives everything from a hemisphere down to a thin sliver. The curved surface area, the area of the domed part, is remarkably simple: two pi times the sphere radius times the cap height, a result that reflects a beautiful property of spheres. The base radius, the radius of the flat circular face where the cut was made, follows from the geometry of the slice. Spherical caps appear in many contexts: the volume of liquid in a spherical or partially filled tank, domes in architecture, contact lenses and optical elements, and portions of planets or bubbles. The cap height cannot exceed the sphere's diameter, since a cap taller than the whole sphere is impossible; when the height equals the diameter the cap becomes the entire sphere. Enter the sphere radius and cap height to compute the cap's volume, surface, and base radius.
V = (πh²/3)(3R − h) ; curved surface = 2πRh
A spherical cap is the portion of a sphere cut off by a plane. Its volume is pi times the cap height squared, divided by 3, times three times the sphere radius minus the height. The curved surface area is two pi times the sphere radius times the cap height, and the flat base radius comes from the geometry of the slice.
Source:Solid geometry
We review all our tools from time to time to ensure accuracy.
Worked example
How to use this calculator
- 1
Enter your values
Enter sphere radius (r) and cap height (h).
- 2
Adjust the options
Fine-tune any options to match your situation.
- 3
Read your result
Cap volume updates instantly as you type — no button to press.
Frequently asked questions
What is a spherical cap?
A spherical cap is the portion of a sphere that lies on one side of a plane cutting through it, forming a dome shape. It has a curved top from the sphere's surface and a flat circular base where the cut was made. A hemisphere is the special case where the cutting plane passes through the sphere's center.
How do you find the volume of a spherical cap?
The volume equals pi times the cap height squared, divided by three, times the quantity three times the sphere radius minus the cap height. This formula covers everything from a shallow slice to a full hemisphere. The calculator computes it from the sphere radius and cap height you provide, along with the surface area and base radius.
What is the curved surface area of a cap?
The curved surface area, the domed part of the cap excluding the flat base, is two pi times the sphere radius times the cap height. A striking feature is that this depends only on the sphere radius and the height, not the base radius, reflecting Archimedes' result that equal heights of a sphere have equal curved surface areas.
How is the base radius determined?
The base radius is the radius of the flat circular face created by the cutting plane. It is found from the geometry of the slice as the square root of the cap height times the quantity twice the sphere radius minus the cap height. The calculator reports it so you know the size of the flat face of the dome.
Where are spherical cap calculations used?
They are used to find the volume of liquid in spherical tanks or partially filled round containers, to design architectural domes, to model contact lenses and optical surfaces, and to compute portions of spheres in science and engineering. Any time a sphere is cut by a plane, the spherical cap formulas give the measurements of the resulting dome.