Sphere Cap Volume Calculator
Dome volume, base, and curve from R and h.
What it does
Domes, lenses, and tank heads are sphere caps. Slicing a sphere at one height leaves a cap with its own volume, flat base circle, and curved skin. Radius plus cut height fix all three numbers at once.
The tool above runs the trio. With sphere radius 10 cm and cap 4 cm, volume is 435.6 cm3 with an 8 cm base and 251.33 cm2 of curve. With radius 6 cm and cap 2 cm, volume is 67 cm3. The imperial mirror at 3.9 in with 1.6 in gives 27.1 in3 in the same proportions.
Height limits keep inputs honest. Cap height runs from zero to the full diameter, where the cap becomes the whole sphere at 4/3 pi R cubed. Base radius follows from the right triangle inside, which is why shallow caps spread wide while staying thin.
This page models solid caps on true spheres. It does not handle ellipsoids, wall thickness, or liquid free surfaces. For tank contents, add outage geometry after.
How to use
- Pick metric (cm) or imperial (in).
- Enter the sphere radius.
- Enter the cap height within the diameter.
- Read volume first.
- Read base radius, then curved area.
- Check full sphere at h = 2R.
Formula
- R
- Sphere radius in cm or in.
- h
- Cap height from tip to cut plane.
- a
- Radius of the flat cut circle.
Spherical cap with height h on sphere radius R. At zero height all outputs are zero; at full diameter the volume is the whole sphere. Hollow shells need thickness math.
Worked example
Example: cap 4 cm on R 10 cm
Defaults: metric sphere radius 10 cm with a 4 cm cap. Imperial mirror is 3.9 in with 1.6 in.
- Inputs (metric default): sphere radius 10 cm with cap 4 cm.
- Volume: pi times 16 times 26 / 3 is 435.6 cm3.
- Base: root of 4 times 16 is 8 cm.
- Curved: 2 pi times 10 times 4 is 251.33 cm2.
- Imperial mirror (3.9 in, 1.6 in) gives 27.1 in3. Tool shows the same.
Reference table
| Sphere and cap | Volume | Base |
|---|---|---|
| R10 h4 cm | 435.6 cm3 | 8 cm |
| R6 h2 cm | 67 cm3 | 4.47 cm |
| R15 h7.5 cm | 2208.9 cm3 | 12.99 cm |
| R3.9 h1.6 in | 27.1 in3 | 3.15 in |
| R5 h5 cm | 261.8 cm3 | 5 cm |
Limits
- True spheres only. Ellipsoids excluded.
- Solid caps. Shells need thickness.
- Height within diameter enforced.
- No liquid surface modeling.
Common mistakes
- Using diameter as radius, 8x on volume.
- Allowing height past the diameter.
- Mixing cm with inches.
- Reading base diameter as radius.
- Forgetting the divide by 3.
Try it yourself
Quick checks. Answers sit tucked under each question.
Sphere radius 6 cm with cap 2 cm?
Volume is pi times 4 times 16 / 3 is 67 cm3 with base 4.47 cm and curve 75.4 cm2.
Sphere radius 15 cm with cap 7.5 cm?
Volume is pi times 56.25 times 37.5 / 3 is 2208.9 cm3. Base is 12.99 cm.
Questions
What is the cap volume formula?
V = pi h squared (3R minus h) / 3. Here 435.6 cm3.
How is base radius figured?
Root of h(2R minus h). Here 8 cm.
How is curved area figured?
2 pi R h. Here 251.33 cm2.
What happens at full height?
At h = 2R the cap is the whole sphere 4/3 pi R cubed.
Does this do ellipsoid domes?
No. Ellipsoid caps need other methods.
What are the input limits?
Yes. Height must stay within zero and diameter.