# 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.

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 4/3π*r*^{3}, the volume a spherical wedge of the same radius *r* is given by^{[3]}

Extrapolating the same principle and considering that the surface area of a sphere is given by 4π*r*^{2}, it can be seen that the surface area of the lune corresponding to the same wedge is given by^{[A]}

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,

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]}

## See also

## 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.

## References

**^**Morton, P. (1830).*Geometry, Plane, Solid, and Spherical, in Six Books*. Baldwin & Cradock. p. 180.**^**Beman, D. W. (2008).*New Plane and Solid Geometry*. BiblioBazaar. p. 338. ISBN 0-554-44701-0.- ^
^{a}^{b}Hart, C. A. (2009).*Solid Geometry*. BiblioBazaar. p. 465. ISBN 1-103-11804-8. **^**Avallone, E. A.; Baumeister, T.; Sadegh, A.; Marks, L. S. (2006).*Marks' Standard Handbook for Mechanical Engineers*. McGraw-Hill Professional. p. 43. ISBN 0-07-142867-4.