Spherical cap

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The spherical cap is the purple section.

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 $\scriptstyle \pi h (3a^2 + h^2)/6$ and the curved surface area of the spherical cap is $\scriptstyle 2 \pi r h$ . [1]

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

Generally, the $n$ -dimensional volume of a hyperspherical cap of height $h$ and radius $r$ in $n$ -dimensional Euclidean space is given by

$V = \frac{2 \pi ^ {\frac{n-1}{2}} r^{n}}{(n-1) \Gamma \left ( \frac{n-1}{2} \right )} \int\limits_{0}^{\cos^{-1}\left(\frac{r-h}{r}\right)}\sin^n (t) \,\mathrm{d}t$

where $Γ$ (the gamma function) is given by $\Gamma(z) = \int_0^\infty t^{z-1} \mathrm{e}^{-t}\,\mathrm{d}t$ .