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 $scriptstyle\; pi\; h\; (3a^2\; +\; h^2)/6$ and the curved surface area of the spherical cap is $scriptstyle\; 2\; pi\; r\; h$. [http://mathforum.org/library/drmath/view/52133.html]

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 volume of a hyperspherical cap of height $h$ and radius $r$ in $n$-dimensional Euclidean space is given by :$V\; =\; frac\{2\; pi\; ^\; \{frac\{n-1\}\{2\; r^\{n\{(n-1)\; Gamma\; left\; (\; frac\{n-1\}\{2\}\; ight\; )\}\; intlimits\_\{0\}^\{cos^\{-1\}left(frac\{r-h\}\{r\}\; ight)\}sin^n\; (t)\; ,mathrm\{d\}t$
where $Gamma$ (the gamma function) is given by $Gamma(z)\; =\; int\_0^infty\; t^\{z-1\}\; mathrm\{e\}^\{-t\},mathrm\{d\}t$.

ee also

* Circular segment - the analogous 2D object.
* Dome (mathematics)

External links

* [http://formularium.org/?go=81 Online calculator for spherical cap volume and area]
* [http://mathforum.org/dr.math/faq/formulas/faq.sphere.html#spherecap Summary of spherical formulae]
* [http://mathworld.wolfram.com/SphericalCap.html Derivation and some additional formulas] , from MathWorld