Spherical wedge

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

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 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 $\tfrac{4}{3} \pi r^3$, the volume a spherical wedge of the same radius r is given by

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

This emerges also from a more general formula for less symmetric cases.

Extrapolating the same principle and considering that the surface area of a sphere is given by $4\pi r^2$, 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 $V_s$ is the volume of the sphere and $V_w$ is the volume of a given spherical wedge,

$\frac{V_w}{V_s} = \frac{\alpha}{2\pi}$

Also, if S_l is the area of a given wedge's lune, and S_s is the area of the wedge's sphere,

$\frac{S_l}{S_s} = \frac{\alpha}{2\pi}$

See also

• Spherical cap

Notes

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.