Spherical wedge

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.[1] Beman (2008)[2] 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.

A spherical wedge with radius r and angle of the wedge α

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 4/3πr³, the volume a spherical wedge of the same radius r is given by[3]

${\displaystyle V={\frac {\alpha }{2\pi }}\cdot {\tfrac {4}{3}}\pi r^{3}={\tfrac {2}{3}}\alpha r^{3}\,.}$

Extrapolating the same principle and considering that the surface area of a sphere is given by 4πr², it can be seen that the surface area of the lune corresponding to the same wedge is given by^[A]

${\displaystyle A={\frac {\alpha }{2\pi }}\cdot 4\pi r^{2}=2\alpha r^{2}\,.}$

Hart (2009)[3] 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 V_s is the volume of the sphere and V_w is the volume of a given spherical wedge,

${\displaystyle {\frac {V_{\mathrm {w} }}{V_{\mathrm {s} }}}={\frac {\alpha }{2\pi }}\,.}$

Also, if S_l is the area of a given wedge's lune, and S_s is the area of the wedge's sphere,[4]^[A]

${\displaystyle {\frac {S_{\mathrm {l} }}{S_{\mathrm {s} }}}={\frac {\alpha }{2\pi }}\,.}$