Spherical Cap Calculator
Compute the volume, curved (lateral) surface area, base radius and total surface area of a spherical cap from the sphere radius and cap height.
units
units
Volume enclosed by the spherical cap
- 1
Cap height squared
3² = 9 - 2
3R − h
3 × 5 − 3 = 12 - 3
Volume = (π × h² ÷ 3) × (3R − h)
π × 9 ÷ 3 × 12 = 113.10Pappus-derived formula for the solid dome above the cutting plane.
How does this calculator work?
A spherical cap (dome) of sphere radius R and height h has volume (π·h²/3)(3R − h) and curved surface 2πRh. Its flat base radius is √(h(2R − h)), and the total surface area adds the base disk πa² to the curved surface, valid whenever 0 < h ≤ 2R.
Formula
How this is calculated
A spherical cap is the portion of a sphere cut off by a plane (think of a dome). It is defined by the radius R of the full sphere and the height h of the cap measured perpendicular from the cutting plane to the top of the dome. A valid cap satisfies 0 < h ≤ 2R; when h = 2R the cap is the entire sphere.
The volume is V = (π·h²/3)(3R − h). The curved (lateral) surface area — the dome surface only — is 2πRh, which interestingly depends only on R and h, not on where the cut is taken. The flat circular base has radius a = √(h(2R − h)), giving a base area of πa². The total surface area adds the curved surface and the base disk: 2πRh + πa².
All inputs share the same length unit; areas come out in square units and volume in cubic units. Edge cases: if h equals R the cap is a hemisphere; if h exceeds 2R the geometry is impossible and no result is shown. The calculator also rejects non-positive radius or height.
Frequently asked questions
The curved surface (2πRh) is just the dome-shaped outer skin of the cap. The total surface also includes the flat circular base (πa²), so total = 2πRh + πa².
The cap height is measured along the sphere, whose maximum extent is its diameter 2R. A height greater than 2R cannot exist on a sphere of radius R, so the calculator treats it as invalid.
When the cap height equals the sphere radius, the cap is exactly a hemisphere: volume = (2/3)πR³ and curved surface = 2πR².
Also known as
TG we-Calculate Editorial Team. (2026). Spherical Cap Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/spherical-cap-calculator
TG we-Calculate Editorial Team. "Spherical Cap Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/spherical-cap-calculator.
TG we-Calculate Editorial Team, "Spherical Cap Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/spherical-cap-calculator
@misc{wecalculate_spherical_cap_calculator, title = {Spherical Cap Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/spherical-cap-calculator}}, year = {2026}, note = {TG we-Calculate} }
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