Spherical cap

An example of a spherical cap in blue.

In geometry, a spherical cap, spherical dome, or spherical segment of one base 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^[1]

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

and the curved surface area of the spherical cap is^[1]

$A = 2 \pi r h$

or

$A=2 \pi r^2 (1-\cos \theta)$

The relationship between $h$ and $r$ is irrelevant as long as 0 ≤ $h$ ≤ $2r$. The red 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

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

The volume may also be found by integrating under a surface of rotation, using $\scriptstyle x = r \cos(\theta)$ and factorizing.

$V = \int_x^r \pi \left(r^2-x^2\right) dx = \pi \left(\frac{2}{3} r^3-r^2 x+\frac{1}{3} x^3\right) = \frac{\pi}{3} r^3 (\cos(\theta)+2) (\cos(\theta)-1)^2$.

Application

The volume of the union of two intersecting spheres of radii r₁ and r₂ is ^[2]

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

where

$V^{(1)} = \frac{4\pi}{3}r_1^3 +\frac{4\pi}{3}r_2^3$

is the sum of the volumes 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 volumes of the two spherical caps forming their intersection. If d < r₁ + r₂ is the distance between the two sphere centers, elimination of the variables h₁ and h₂ leads to^[3][4]

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

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 ^[5]

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

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

Earlier in ^[6] (1986, USSR Academ. Press) the formulas were received: $A=A_n p_ { n-2 } (q), V=V_n p_n (q)$, where $q= 1-h/r (0 \le q \le 1 ), p_n (q) =(1-G_n(q)/G_n(1))/2$,

$G _n(q)= \int \limits_{0}^{q} (1-t^2) ^ { (n-1) /2 } dt$.

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 ^[7] that, if $n \to \infty$ and $q\sqrt n = const.$, 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

1. Polyanin, Andrei D; Manzhirov, Alexander V. (2006), Handbook of Mathematics for Engineers and Scientists, CRC Press, p. 69, ISBN 9781584885023.
2. Connolly, Michael L. (1985). "Computation of molecular volume". J. Am. Chem. Soc 107: 1118–1124. doi:10.1021/ja00291a006.
3. Pavani, R.; Ranghino, G. (1982). "A method to compute the volume of a molecule". Comput. Chem. 6: 133–135. doi:10.1016/0097-8485(82)80006-5.
4. Bondi, A. (1964). "Van der Waals volumes and radii". J. Phys. Chem. 68: 441–451. doi:10.1021/j100785a001.
5. 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.
6. Chudnov, Alexander M. (1986). "On minimax signal generation and reception algorithms (rus.)". Problems of Information Transmission 22 (4): 49–54.
7. Chudnov, Alexander M (1991). "Game-theoretical problems of synthesis of signal generation and reception algorithms (rus.)". Problems of Information Transmission 27 (3): 57–65.

Additional reading

• Richmond, Timothy J. (1984). "Solvent accessible surface area and excluded volume in proteins: Analytical equation for overlapping spheres and implications for the hydrophobic effect". J. Mol. Biol. 178 (1): 63–89. doi:10.1016/0022-2836(84)90231-6.
• Lustig, Rolf (1986). "Geometry of four hard fused spheres in an arbitrary spatial configuration". Mol. Phys. 59 (2): 195–207. Bibcode:1986MolPh..59..195L. doi:10.1080/00268978600102011.
• Gibson, K. D.; Scheraga, Harold A. (1987). "Volume of the intersection of three spheres of unequal size: a simplified formula". J. Phys. Chem. 91 (15): 4121–4122. doi:10.1021/j100299a035.
• Gibson, K. D.; Scheraga, Harold A. (1987). "Exact calculation of the volume and surface area of fused hard-sphere molecules with unequal atomic radii". Mol. Phys. 62 (5): 1247–1265. Bibcode:1987MolPh..62.1247G. doi:10.1080/00268978700102951.
• Petitjean, Michel (1994). "On the analytical calculation of van der Waals surfaces and volumes: some numerical aspects". Int. J. Quant. Chem. 15 (5): 507–523. doi:10.1002/jcc.540150504.
• Grant, J. A.; Pickup, B. T. (1995). "A Gaussian description of molecular shape". J. Phys. Chem. 99 (11): 3503–3510. doi:10.1021/j100011a016.
• Busa, Jan; Dzurina, Jozef; Hayryan, Edik; Hayryan, Shura (2005). "ARVO: A fortran package for computing the solvent accessible surface area and the excluded volume of overlapping spheres via analytic equations". Comp. Phys. Commun. 165: 59–96. Bibcode:2005CoPhC.165...59B. doi:10.1016/j.cpc.2004.08.002.

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.