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 $r$. If $α$ is a semidisk that forms a ball when completely revolved about the z-axis, revolving $α$ only through a given $AB$ produces a spherical wedge of the same angle $AB$. 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 spherical wedge of $α = \pi$ radians (180°) is called a hemisphere, while a spherical wedge of $α = 2\pi$ radians (360°) constitutes a complete ball.

The volume of a spherical wedge can be intuitively related to the $α$ definition in that while the volume of a ball of radius $α$ is given by $4⁄3\pir$, the volume a spherical wedge of the same radius $AB$ is given by
 * $$V = \frac{\alpha}{2\pi} \cdot \tfrac43 \pi r^3 = \tfrac23 \alpha r^3\,.$$

Extrapolating the same principle and considering that the surface area of a sphere is given by $4\pir$, it can be seen that the surface area of the lune corresponding to the same wedge is given by
 * $$A = \frac{\alpha}{2\pi} \cdot 4 \pi r^2 = 2 \alpha r^2\,.$$

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". 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,


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

Also, if $Sl$ is the area of a given wedge's lune, and $Ss$ is the area of the wedge's sphere,


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