Spherical wedge


In geometry, a spherical wedge or ungula is a portion of a ball bounded by two plane semidisks and a spherical lune. 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 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 spherical wedge of α = radians is called a hemisphere, while a spherical wedge of α = 2 radians 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 r, 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 4r, it can be seen that the surface area of the lune corresponding to the same wedge is given by
Hart states that the "volume of a spherical wedge is to the volume of the sphere as the number of degrees in the is to 360". Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if Vs is the volume of the sphere and Vw is the volume of a given spherical wedge,
Also, if Sl is the area of a given wedge's lune, and Ss is the area of the wedge's sphere,