Case 1
- Sphere radius R
- 5
- Cap height h
- 2
62.831853
Open with these valuesSpherical Cap Surface Area Calculator
62.831853units²
Result: 62.831853 units²The curved skin of a spherical cap is 2πRh, and nothing else: it depends on the sphere radius and the cap height, never on where along the sphere the slice sits. R = 5 and h = 2 give 20π, about 62.83 square units. The flat circular base is not included; add πa² for that.
Worked examples
Case 2
- Sphere radius R
- 10
- Cap height h
- 3
188.495559
Open with these valuesCase 3
- Sphere radius R
- 3
- Cap height h
- 1.5
28.274334
Open with these valuesHow it's calculated
S = 2 × π × R × h, 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 curved area; add πa² if you also need the flat base.
Reference table
| Radius, cap height | Exact | With flat base | Curved surface |
|---|---|---|---|
| 3, 1.5 | 9π | 49.480084 | 28.274334 |
| 5, 2 | 20π | 113.097336 | 62.831853 |
| 5, 5 | 50π (hemisphere) | 235.619449 | 157.079633 |
| 10, 3 | 60π | 348.716785 | 188.495559 |
| 5, 10 | 100π (whole sphere) | 314.159265 | 314.159265 |
Questions
How do I calculate the curved surface area of a spherical cap?
Multiply 2π by the sphere radius and the cap height: S = 2 × π × R × h. For R = 5 and h = 2 that is 62.831853 square units. Remarkably it depends only on R and h, not on where the slice sits.
Is the flat base included?
No. This calculator returns the dome alone, which is what MathWorld calls the surface area of a spherical cap. For a closed lid add the base circle, π × a², where a = √(h(2R − h)) — the third column of the table above shows that total.
What is the base radius of a spherical cap?
The base radius a is the radius of the flat circle left by the cut, a = √(h(2R − h)). For R = 5 and h = 2 it is √16 = 4. It is largest at a hemisphere, h = R, and shrinks back to zero as the cap grows into the whole sphere.
What happens at a cap height equal to the diameter?
At h = 2R the cap is the whole sphere and 2πRh becomes 4πR², the familiar sphere surface. A cap cannot be taller than that, so this calculator holds the answer there instead of returning nonsense.
Which units does the answer use?
Whatever you entered, squared. Radius and cap height in centimetres give square centimetres, in inches square inches.
Sources and last check
Information, not professional advice.