Spherical Cap Volume Calculator
Find the volume of a spherical cap.
Result
How it is calculated
The tool takes the radius R of the full sphere and the height h of the cap, both in the same length unit, and computes V = (π·h²/3)·(3R − h) and curved area = 2·π·R·h. The height is measured along the axis from the flat cutting face to the top of the dome. It must be greater than 0 and no larger than 2R; a value above the diameter gives no result. At h = R the cap is a hemisphere and at h = 2R it is the whole sphere. The results are labelled generically as units³ and units², so read them in whatever unit you used. The curved area excludes the flat circular base, which you add separately as π·(2Rh − h²) if you need a closed surface.
Worked example
A dome is cut from a sphere of radius 10 and is 4 tall. Volume: π × 4² / 3 × (3 × 10 − 4) = π × 16 / 3 × 26 = 435.63 units³. Curved area: 2 × π × 10 × 4 = 251.33 units². The flat base radius is the square root of (2 × 10 × 4 − 16) = 8, so a base disc would add π × 64 = 201.06 units² if the shape were closed. As a check, with h = 20 the same formula returns the full sphere volume 4,188.79.
Common mistakes
- Entering the radius of the cap's flat base as R. The first field is the radius of the whole sphere the cap was cut from, which is larger than the base radius unless the cap is a hemisphere.
- Entering the diameter instead of the radius. This makes the volume about eight times too large for the same shape.
- Measuring h along the curved surface or from the sphere's centre. It is the straight height from the cutting plane to the top of the cap.
- Using the curved area as the total surface of a closed solid. The flat base is not included in the number.
- Mixing units, for example R in metres and h in centimetres. Both must use the same unit, and the volume is then in that unit cubed.
About this calculator
A spherical cap is the portion of a sphere cut off by a single plane, and this calculator finds its volume, V = (π·h²/3)·(3R−h), and curved surface area, 2·π·R·h, from the parent sphere's radius R and the cap's height h (measured from the cutting plane to the top of the cap). The formulas come from integrating circular cross-sections along the cap's axis.
Engineers use spherical-cap geometry to design dome roofs, tank heads (the rounded ends of pressure vessels), and lens surfaces, since a dome or a convex tank end is exactly a cap shape. Astronomers use the same geometry for polar ice caps and for the patch of a planet's surface visible from a given altitude, and manufacturers of domed containers, contact lenses, and architectural domes rely on this calculation during design. Engineering students and architects sizing a domed structure use this calculator to get exact volume and surface figures rather than approximations.
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