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
The relationship between and is irrelevant as long as 0 ≤ ≤ . The blue section of the illustration is also a spherical cap.
The parameters , and are not independent:
Substituting this into the area formula gives:
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
The volume of all points which are in at least one of two intersecting spheres of radii r1 and r2 is 
is the total of the two isolated spheres, and
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.
Generally, the -dimensional volume of a hyperspherical cap of height and radius in -dimensional Euclidean space is given by 
where (the gamma function) is given by .
and the area formula can be expressed in terms of the area of the unit n-ball as
Earlier in  (1986, USSR Academ. Press) the formulas were received: , where ,
- Circular segment — the analogous 2D object
- Solid angle — contains formula for n-sphere caps
- Spherical segment
- Spherical sector
- Spherical wedge
- Polyanin, Andrei D; Manzhirov, Alexander V. (2006), Handbook of Mathematics for Engineers and Scientists, CRC Press, p. 69, ISBN 9781584885023.
- Connolly, Michael L. (1985). "Computation of molecular volume". J. Am. Chem. Soc: 1118–1124. doi:10.1021/ja00291a006.
- Pavani, R.; Ranghino, G. (1982). "A method to compute the volume of a molecule". Comput. Chem. doi:10.1016/0097-8485(82)80006-5.
- Bondi, A. (1964). "Van der Waals volumes and radii". J. Phys. Chem. (68): 441–451. doi:10.1021/j100785a001.
- 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
- 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)
- 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)
- 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. Molec. 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.
|Wikimedia Commons has media related to Spherical caps.|
- Weisstein, Eric W., "Spherical cap", MathWorld. Derivation and some additional formulas.
- Online calculator for spherical cap volume and area.
- Summary of spherical formulas.