# Spherical cap

The spherical cap is the purple section.

In geometry, a spherical cap or spherical dome 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.

## Volume and surface area

If the radius of the base of the cap is ${\displaystyle a}$, and the height of the cap is ${\displaystyle h}$, then the volume of the spherical cap is

${\displaystyle V={\frac {\pi h}{6}}(3a^{2}+h^{2}),}$

and the curved surface area of the spherical cap is

${\displaystyle A=2\pi rh.}$

The relationship between ${\displaystyle h}$ and ${\displaystyle r}$ is irrelevant as long as 0 ≤ ${\displaystyle h}$${\displaystyle 2r}$. The blue section of the illustration is also a spherical cap.

${\displaystyle r^{2}=(r-h)^{2}+a^{2}=r^{2}+h^{2}-2rh+a^{2},}$
${\displaystyle r={\frac {a^{2}+h^{2}}{2h}}}$.

Substituting this into the area formula gives:

${\displaystyle A=2\pi {\frac {(a^{2}+h^{2})}{2h}}h=\pi (a^{2}+h^{2}).}$

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

${\displaystyle V={\frac {\pi h^{2}}{3}}(3r-h)}$.

## Application

The volume of all points which are in at least one of two intersecting spheres of radii r1 and r2 is [1]

${\displaystyle V=V^{(1)}-V^{(2)}}$,

where

${\displaystyle V^{(1)}={\frac {4\pi }{3}}r_{1}^{3}+{\frac {4\pi }{3}}r_{2}^{3}}$

is the total of the two isolated spheres, and

${\displaystyle V^{(2)}={\frac {\pi h_{1}^{2}}{3}}(3r_{1}-h_{1})+{\frac {\pi h_{2}^{2}}{3}}(3r_{2}-h_{2})}$

the sum of the two spherical caps of the intersection. If d <r1+r2 is the distance between the two sphere centers, elimination of the variables h1 and h2 leads to[2] [3]

${\displaystyle V^{(2)}={\frac {\pi }{12d}}(r_{1}+r_{2}-d)^{2}[d^{2}+2d(r_{1}+r_{2})-3(r_{1}-r_{2})^{2}].}$

## Generalizations

### Sections of other solids

The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid.

### Hyperspherical cap

Generally, the ${\displaystyle n}$-dimensional volume of a hyperspherical cap of height ${\displaystyle h}$ and radius ${\displaystyle r}$ in ${\displaystyle n}$-dimensional Euclidean space is given by [4]

${\displaystyle V={\frac {\pi ^{\frac {n-1}{2}}\,r^{n}}{\,\Gamma \left({\frac {n+1}{2}}\right)}}\int \limits _{0}^{\arccos \left({\frac {r-h}{r}}\right)}\sin ^{n}(t)\,\mathrm {d} t}$
${\displaystyle V=C_{n}\,r^{n}\left({\frac {1}{2}}\,-\,{\frac {r-h}{r}}\,{\frac {\Gamma [1+{\frac {n}{2}}]}{{\sqrt {\pi }}\,\Gamma [{\frac {n+1}{2}}]}}{\,\,}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1-n}{2}};{\tfrac {3}{2}};\left({\tfrac {r-h}{r}}\right)^{2}\right)\right)={\frac {1}{2}}C_{n}\,r^{n}I_{(2rh-h^{2})/r^{2}}\left({\frac {n+1}{2}},{\frac {1}{2}}\right)}$ ,

and the area formula ${\displaystyle A}$ can be expressed in terms of the area of the unit n-ball ${\displaystyle A_{n}={\scriptstyle 2\pi ^{n/2}/\Gamma [{\frac {n}{2}}]}}$ as

${\displaystyle A={\frac {1}{2}}A_{n}\,r^{n-1}I_{(2rh-h^{2})/r^{2}}\left({\frac {n-1}{2}},{\frac {1}{2}}\right)}$ ,

Earlier in [5] (1986, USSR Academ. Press) the formulas were received: ${\displaystyle A=A_{n}p_{n-2}(q),V=V_{n}p_{n}(q)}$, where ${\displaystyle q=1-h/r(0\leq q\leq 1),p_{n}(q)=(1-G_{n}(q)/G_{n}(1))/2,}$

## References

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4. Li, S. (2011). "Concise Formulas for the Area and Volume of a Hyperspherical Cap". Asian J. Math. Stat. 4 (1): 66–70. doi:10.3923/ajms.2011.66.70
5. Chudnov A.M. (1986). “On Minimax Signal Generation and Reception Algorithms”, Problems of Information Transmission, 22:4 (1986), 49–54, rus. (Mi ppi958, an: 0624.94005)
6. Chudnov A.M. (1991). “Game-Theoretical Problems of Synthesis of Signal Generation and Reception Algorithms”, Problems of Information Transmission, 27:3 (1991), 57–65, rus. (Mi ppi570, an:0778.94001)
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