Spherical wedge

Short description: Geometric shape; radial slice of a sphere

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

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

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

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

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,

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

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

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

See also

• Spherical cap
• Spherical segment
• Ungula

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.