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 Incoming links: Hemisphere

Volume of a spherical cap Created 2026-10-18 Updated 2026-10-20

For a solid spherical cap of height

0 \leq h \leq 2 R

cut from a sphere of radius

R

,

V = π h^{2} (R - \frac{h}{3}) .

Centre the sphere at the origin with the cap on the side

z \geq R - h

. Its cross-section at height

z

is a circle of squared radius

R^{2} - z^{2}

. A volume integral in cylindrical coordinates gives

V = \int_{R - h}^{R} \int_{0}^{2 π} \int_{0}^{R^{2} - z^{2}} r d r d ϕ d z = π \int_{R - h}^{R} (R^{2} - z^{2}) d z = π h^{2} (R - h /3) .

The formula includes caps larger than a hemisphere, when

h > R

, and the whole solid sphere, when

h = 2 R

. For the retained region

z \geq - d

with

0 < d < R

h = R + d

to obtain

V = π (2 R^{3} /3 + R^{2} d - d^{3} /3)

. Multiplying by uniform mass density gives its mass.

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