Spherical wedge

A spherical wedge with radius Template:Mvar and angle of the wedge Template:Mvar

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 Template:Mvar. If Template:Mvar is a semidisk that forms a ball when completely revolved about the z-axis, revolving Template:Mvar only through a given Template:Mvar produces a spherical wedge of the same angle Template:Mvar. 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." Template:Ref A spherical wedge of Template:Math radians (180°) is called a hemisphere, while a spherical wedge of Template:Math radians (360°) constitutes a complete ball.

The volume of a spherical wedge can be intuitively related to the Template:Mvar definition in that while the volume of a ball of radius Template:Mvar is given by Template:Math, the volume a spherical wedge of the same radius Template:Mvar 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 Template:Math, it can be seen that the surface area of the lune corresponding to the same wedge is given byTemplate:Ref

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

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".Template:Ref Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if Template:Math is the volume of the sphere and Template:Math is the volume of a given spherical wedge,

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

Also, if Template:Math is the area of a given wedge's lune, and Template:Math is the area of the wedge's sphere,Template:Ref

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

See also

• Spherical cap
• Spherical segment
• Ungula

Notes

A. Template:Note 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.