Case 1
- Sphere radius R
- 5
- Cap height h
- 2
54.454273
Open with these valuesSpherical Cap Volume Calculator
54.454273units³
Result: 54.454273 units³A spherical cap is the dome left when a flat plane slices a sphere. From the sphere radius R and the cap height h the volume is πh²(3R − h)/3 — for R = 5 and h = 2 that is 52π/3, about 54.45 cubic units. At h = R the cap is exactly a hemisphere.
Worked examples
Case 2
- Sphere radius R
- 5
- Cap height h
- 5
261.799388
Open with these valuesCase 3
- Sphere radius R
- 10
- Cap height h
- 3
254.469005
Open with these valuesHow it's calculated
V = π × h² × (3R − h) ÷ 3, h at most 2R
- StepEnter the radius of the whole sphere the cap was cut from.
- StepEnter the cap height: the flat cut up to the top of the dome.
- ResultRead the volume in the cube of the unit you used.
Reference table
| Radius, cap height | Exact | Volume |
|---|---|---|
| 4, 1 | 11π ÷ 3 | 11.519173 |
| 5, 2 | 52π ÷ 3 | 54.454273 |
| 6, 4 | 224π ÷ 3 | 234.572251 |
| 10, 3 | 81π | 254.469005 |
| 5, 5 | 250π ÷ 3 (hemisphere) | 261.799388 |
| 5, 12 | h over 2R: whole sphere, 500π ÷ 3 | 523.598776 |
Questions
What is a spherical cap?
A spherical cap is the dome-shaped piece of a sphere lying on one side of a flat plane through it. Two numbers describe it: the radius R of the sphere it was cut from and the cap height h, measured from the flat cut up to the top of the dome. Think of the lid sliced off an orange, or liquid in the rounded bottom of a spherical tank.
How do I calculate the volume of a spherical cap?
Use V = πh²(3R − h)/3, with R the sphere radius and h the cap height. For R = 5 and h = 2 that is (π × 4/3) × 13, about 54.454273 cubic units.
I know the base radius, not the sphere radius — what then?
The flat circle at the cut has radius a = √(h(2R − h)), which rearranges to R = (a² + h²)/(2h). Work R out that way first, then enter it here.
What happens at a cap height equal to the radius or the diameter?
At h = R the plane passes through the centre and the cap is exactly a hemisphere. At h = 2R the cap is the whole sphere, 4πR³/3. Anything above 2R is not a cap at all, so this calculator holds the answer at the full sphere instead of returning nonsense.
Which units does the answer use?
Whatever you entered, cubed. Radius and cap height in centimetres give cubic centimetres, in inches cubic inches.
Sources and last check
Information, not professional advice.