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 $a$, and the height of the cap is $h$, then the volume of the spherical cap is

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

and the curved surface area of the spherical cap is

$A=2\pi rh.$

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

The parameters $a$, $h$ and $r$ are not independent:

$r^{2}=(r-h)^{2}+a^{2}=r^{2}+h^{2}-2rh+a^{2},$

$r={\frac {a^{2}+h^{2}}{2h}}$.

Substituting this into the area formula gives:

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

Note also that in the upper hemisphere of the diagram, $\scriptstyle h=r-{\sqrt {r^{2}-a^{2}}}$, 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

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

Application

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

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

where

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

is the total of the two isolated spheres, and

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

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

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

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

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

where $\Gamma$ (the gamma function) is given by $\Gamma (z)=\int _{0}^{\infty }t^{z-1}\mathrm {e} ^{-t}\,\mathrm {d} t$.

The formula for $V$ can be expressed in terms of the volume of the unit n-ball $C_{n}={\scriptstyle \pi ^{n/2}/\Gamma [1+{\frac {n}{2}}]}$ and the hypergeometric function ${}_{2}F_{1}$ or the regularized incomplete beta function $I_{x}(a,b)$as

$V=C_{n}\,r^{n}\left({\frac {1}{2}}\,-\,{\frac {r-h}{r}}\,{\frac {\Gamma [1+{\frac {n}{2}}]}{{\sqrt {\pi }}\,\Gamma [{\frac {n+1}{2}}]}}{\,\,}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1-n}{2}};{\tfrac {3}{2}};\left({\tfrac {r-h}{r}}\right)^{2}\right)\right)={\frac {1}{2}}C_{n}\,r^{n}I_{(2rh-h^{2})/r^{2}}\left({\frac {n+1}{2}},{\frac {1}{2}}\right)$ ,

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

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

where $\scriptstyle 0\leq h\leq r$.

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

$G_{n}(q)=\int \limits _{0}^{q}(1-t^{2})^{(n-1)/2}dt.$

For odd $n=2k+1:$

$G_{n}(q)=\sum _{i=0}^{k}(-1)^{i}{\binom {k}{i}}{\frac {q^{2i+1}}{2i+1}}.$

It is shown in that if $n\to \infty$ then $p_{n}(q)\to 1-F(q{\sqrt {n}})$ where $F()$ is the integral of the standard normal distribution.

See also

• Circular segment — the analogous 2D object.
• Solid angle — contains formula for n-sphere caps
• Spherical segment
• Spherical sector
• Spherical wedge

References

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External links

• Weisstein, Eric W., "Spherical cap", MathWorld., derivation and some additional formulas
• Online calculator for spherical cap volume and area
• Summary of spherical formulas