Spherical cap

The spherical cap is the purple section.

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 $r$, 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

$A = 2 \pi r h.$

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

$V^{(2)} = \frac{\pi}{12d}(r_1+r_2-d)^2[d^2+2d(r_1+r_2)-3(r_1-r_2)^2].$

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

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

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 $\scriptstyle 0\le h\le r$.

See also

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

References

• 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. .

External links

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