- Spherical cap
In
geometry , a spherical cap is a portion of asphere 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 (thegamma 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] , fromMathWorld
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