Definitions

# Spherical cap

In geometry, a spherical cap is a portion of a sphere cut off by a plane. If the plane passes through the center of the sphere, so that the height of the cap is equal to the radius of the sphere, the spherical cap is called a hemisphere.

If the radius of the sphere is called $r$, the radius of the base of the cap called $a$, and the height of the cap called $h$, the volume of the spherical cap is then $scriptstyle pi h \left(3a^2 + h^2\right)/6$ and the curved surface area of the spherical cap is $scriptstyle 2 pi r h$.

Note also that in the upper hemisphere of the diagram, $scriptstyle h = r - sqrt\left\{r^2 - a^2\right\}$, and in the lower hemisphere $scriptstyle h = r + sqrt\left\{r^2 - a^2\right\}$; hence in either hemisphere $scriptstyle a = sqrt\left\{h\left(2r-h\right)\right\}$ and so an alternative expression for the volume is $scriptstyle pi h^2 \left(3r-h\right)/3$.

Generally, the volume of a hyperspherical cap of height $h$ and radius $r$ in $n$-dimensional Euclidean space is given by :$V = frac\left\{2 pi ^ \left\{frac\left\{n-1\right\}\left\{2\right\}\right\} r^\left\{n\right\}\right\}\left\{\left(n-1\right) Gamma left \left(frac\left\{n-1\right\}\left\{2\right\} right \right)\right\} intlimits_\left\{0\right\}^\left\{cos^\left\{-1\right\}left\left(frac\left\{r-h\right\}\left\{r\right\}right\right)\right\}sin^n \left(t\right) ,mathrm\left\{d\right\}t$
where $Gamma$ (the gamma function) is given by $Gamma\left(z\right) = int_0^infty t^\left\{z-1\right\} mathrm\left\{e\right\}^\left\{-t\right\},mathrm\left\{d\right\}t$.