In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune (termed the wedge's base). The angle between the radii lying within the bounding semidisks is the dihedral angle of the wedge α. If AB is a semidisk that forms a ball when completely revolved about the z-axis, revolving AB only through a given α produces a spherical wedge of the same angle α. Beman (2008) remarks that "a spherical wedge is to the sphere of which it is a part as the angle of the wedge is to a perigon."[A] A spherical wedge of α = π radians (180°) is called a hemisphere, while a spherical wedge of α = 2π radians (360°) constitutes a complete ball.
The volume of a spherical wedge can be intuitively related to the AB definition in that while the volume of a ball of radius r is given by , the volume a spherical wedge of the same radius r is given by
Extrapolating the same principle and considering that the surface area of a sphere is given by , it can be seen that the surface area of the lune corresponding to the same wedge is given by[A]
Hart (2009) states that the "volume of a spherical wedge is to the volume of the sphere as the number of degrees in the [angle of the wedge] is to 360".[A] Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if is the volume of the sphere and is the volume of a given spherical wedge,
- A. ^ A distinction is sometimes drawn between the terms "sphere" and "ball", where a sphere is regarded as being merely the outer surface of a solid ball. It is common to use the terms interchangeably, as the commentaries of both Beman (2008) and Hart (2008) do.
- P. Morton (1830). Geometry, plane, solid, and spherical, in six books. Baldwin and Cradock. p. 180.
- D. W. Beman (2008). New Plane and Solid Geometry. BiblioBazaar, LLC. p. 338. ISBN 0-554-44701-0.
- C. A. Hart (2009). Solid Geometry. BiblioBazaar, LLC. p. 465. ISBN 1-103-11804-8.
- E. A. Avallone, T. Baumeister, A. Sadegh, L. S. Marks (2006). Marks' standard handbook for mechanical engineers. McGraw-Hill Professional. p. 43. ISBN 0-07-142867-4.