In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the center of the sphere (forming a great circle), so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.

The volume of the spherical cap and the area of the curved surface may be calculated using combinations of

These variables are inter-related through the formulas ${\displaystyle a=r\sin \theta }$, ${\displaystyle h=r(1-\cos \theta )}$, ${\displaystyle 2hr=a^{2}+h^{2}}$, and ${\displaystyle 2ha=(a^{2}+h^{2})\sin \theta }$.

If ${\displaystyle \phi }$ denotes the latitude in geographic coordinates, then ${\displaystyle \theta +\phi =\pi /2=90^{\circ }\,}$, and ${\displaystyle \cos \theta =\sin \phi }$.

Note that aside from the calculus based argument below, the area of the spherical cap may be derived from the volume ${\displaystyle V_{sec}}$ of the spherical sector, by an intuitive argument, as

The intuitive argument is based upon summing the total sector volume from that of infinitesimal triangular pyramids. Utilizing the pyramid (or cone) volume formula of ${\displaystyle V={\frac {1}{3}}bh'}$, where ${\displaystyle b}$ is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and ${\displaystyle h'}$ is the height of each pyramid from its base to its apex (at the center of the sphere). Since each ${\displaystyle h'}$, in the limit, is constant and equivalent to the radius ${\displaystyle r}$ of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and:

The volume and area formulas may be derived by examining the rotation of the function

for ${\displaystyle x\in [0,h]}$, using the formulas the surface of the rotation for the area and the solid of the revolution for the volume. The area is