Spherical Cap Volume Calculator
The dome left when a plane slices a sphere — tank heads, viewing ports, domed roofs and the liquid pooled in a spherical vessel. Enter the height and either radius.
- ✓ Two input modes
- ✓ Formula shown
- ✓ Curved area included
Spherical cap volume
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Quick answer: a spherical cap's volume is V = (1/3)πh²(3R − h), where R is the full sphere's radius and h is the cap's height. If you only measured the flat base, use V = (1/6)πh(3a² + h²) instead — the two are algebraically identical.
The spherical cap formula, step by step
or from base radius a: V = (1/6) π h (3a² + h²)
where R = sphere radius · h = cap height · a = base radius
Worked example — a cap of height 2 cut from a sphere of radius 5:
- Square the height: 2² = 4
- Work out (3R − h): (3 × 5) − 2 = 13
- Multiply and divide by 3: π × 4 × 13 ÷ 3 = 54.45 cubic units
The two forms are linked by a² = h(2R − h). In this example a² = 2 × 8 = 16, so the base radius is 4 — and feeding a = 4 into the second formula returns 54.45 as well.
Two checks that prove the formula
This formula is easy to mistrust, so it is worth seeing it collapse into shapes you already know. Both cases fall straight out of it:
| Set the height to… | Formula becomes | Which is |
|---|---|---|
| h = R | (1/3)πR²(3R − R) = (2/3)πR³ | A hemisphere — exactly half a sphere |
| h = 2R | (1/3)π4R²(3R − 2R) = (4/3)πR³ | The whole sphere |
| h → 0 | → 0 | A flat disc with no thickness |
If a cap calculation ever looks wrong, set h = R by hand and confirm you get two-thirds of the sphere. That single check catches most input errors.
Cap volume as a fraction of its sphere
For a sphere of radius R, how much of the total volume sits below a cut at depth h:
| Cap height h | Base radius a | Cap volume | Share of sphere |
|---|---|---|---|
| 0.1 R | 0.436 R | 0.030 R³ | 0.7% |
| 0.25 R | 0.661 R | 0.180 R³ | 4.3% |
| 0.5 R | 0.866 R | 0.654 R³ | 15.6% |
| R (hemisphere) | R | 2.094 R³ | 50% |
| 1.5 R | 0.866 R | 3.534 R³ | 84.4% |
| 2R (full sphere) | 0 | 4.189 R³ | 100% |
Note how slowly volume accumulates at first. Filling a spherical tank to a quarter of its radius gives barely 4% of its capacity — which is why sight glasses on spherical vessels are so misleading near the bottom.
Where spherical caps show up
Dished tank heads
Pressure vessels are rarely flat-ended. Torispherical and hemispherical heads are close to spherical caps, and their volume has to be added to the cylindrical shell to get true capacity. The tank volume calculator handles the common combinations directly.
Liquid in a spherical tank
Partly filled spherical and dished-bottom vessels hold a cap of liquid. Because the relationship between depth and volume is strongly non-linear — as the table above shows — a dipstick reading on a spherical tank cannot be read proportionally the way it can on a vertical cylinder.
Domes and architecture
Shallow domes are spherical caps. Knowing the enclosed volume matters for heating and ventilation loads long before it matters for anything structural.
Optics and lens blanks
The sagitta — the cap height of a curved lens surface — determines how much glass is removed when grinding. The same a² = h(2R − h) relation converts between a lens's curvature radius and its sag.
Common mistakes & pro tips
- Using the curved distance as the height. h is the straight, perpendicular rise from the base plane to the top — not the arc along the surface.
- Mixing up R and a. R belongs to the whole sphere; a is only the radius of the flat cut face. They are equal only for a hemisphere.
- Assuming a half-depth tank is half full. A spherical tank filled to half its radius holds under 16% of capacity, not 50%.
- Pro tip — recover R from the cap. If the sphere is gone and you only have the dome, R = (a² + h²) ÷ 2h rebuilds it.
- Pro tip — check with the hemisphere case. Setting h = R must return two-thirds of the sphere volume. If it doesn't, an input is wrong.
How to use this calculator
- Pick what you measured. Choose Sphere radius + cap height if you know the parent sphere, or Base radius + cap height if you can only measure the cap itself.
- Enter the cap height h — the perpendicular distance from the flat circular base up to the top of the dome, not the curved distance along the surface.
- Enter the second measurement — either the full sphere radius R, or the radius a of the flat circular base where the cap was cut.
- Read the volume. The calculator substitutes into V = (1/3)πh²(3R − h) and shows the working, plus the curved surface area and the base radius.
Frequently asked questions
What is the formula for the volume of a spherical cap?
V = (1/3)πh²(3R − h), where R is the radius of the full sphere and h is the height of the cap. If you know the base radius a instead of R, the equivalent form is V = (1/6)πh(3a² + h²). Both give the same answer.
What is a spherical cap?
A spherical cap is the piece of a sphere cut off by a flat plane — the dome above the cut. It has one flat circular face and one curved face. A hemisphere is the special case where the plane passes through the centre.
How is a spherical cap different from a hemisphere?
A hemisphere is exactly half a sphere, cut through the centre. A spherical cap is any slice, usually shallower. Set the cap height equal to the sphere radius and the formula collapses to (2/3)πR³ — the hemisphere volume — which is a useful way to check your numbers.
How do I find the volume of liquid in a spherical tank?
The liquid sitting in the bottom of a spherical tank is a spherical cap, so use the cap formula with h as the liquid depth and R as the tank's internal radius. A 3 m radius tank filled to 1 m deep holds (1/3)π × 1² × (9 − 1) = 8.38 m³, or about 8,378 litres.
How do I find the sphere radius if I only measured the cap?
From the base radius a and the height h, the parent sphere radius is R = (a² + h²) ÷ 2h. A dome 4 m across the base (a = 2) and 1 m high sits on a sphere of radius (4 + 1) ÷ 2 = 2.5 m.