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
- V = πh(3a2 + h2) / 6
and the curved surface area of the spherical cap is
- A = 2πrh.
Note also that in the upper hemisphere of the diagram, 
, and in the lower hemisphere 
; hence in either hemisphere 
and so an alternative expression for the volume is
- V = πh2(3r − h) / 3.
Hyperspherical cap
Generally, the n-dimensional volume of a hyperspherical cap of height h and radius r in n-dimensional Euclidean space is given by
where Γ (the gamma function) is given by 
.
The formula for V can be expressed in terms of the volume of the unit n-ball ![C_{n}={\scriptstyle \pi^{n/2}/\Gamma[1+\frac{n}{2}]}](http://wpcontent.answers.com/math/9/1/5/915fd60d8b00a6880264bbba9f90d9f3.png)
and the hypergeometric series 2F1 as
![V = C_{n} \, r^{n} \left( \frac{1}{2}\, - \,\frac{r-h}{r} \,\frac{\Gamma[1+\frac{n}{2}]}{\sqrt{\pi}\,\Gamma[\frac{n+1}{2}]}
{\,\,}_{2}F_{1}\left(\tfrac{1}{2},\tfrac{1-n}{2};\tfrac{3}{2};\left(\tfrac{r-h}{r}\right)^{2}\right)\right)](http://wpcontent.answers.com/math/d/7/7/d77c0390e295a3a2087b2fe9d47ce66a.png)
,
where 
.
See also
- Circular segment — the analogous 2D object.
- Dome (mathematics)
- Solid angle — contains formula for n-sphere caps
- Spherical wedge
External links
- Online calculator for spherical cap volume and area
- Summary of spherical formulae
- Derivation and some additional formulas, from MathWorld
 |
This geometry-related article is a stub. You can help Wikipedia by expanding it. |
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
