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.^[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 = 2 $π$ 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 4/3 $π$ r³, the volume a spherical wedge of the same radius r is given by^[3]

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

Extrapolating the same principle and considering that the surface area of a sphere is given by 4 $π$ r², it can be seen that the surface area of the lune corresponding to the same wedge is given by^[A]

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

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,

{\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,^[4]^[A]

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

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.

[1] Morton, P. (1830). Geometry, Plane, Solid, and Spherical, in Six Books. Baldwin & Cradock. p. 180.

[2] Beman, D. W. (2008). New Plane and Solid Geometry. BiblioBazaar. p. 338. ISBN 0-554-44701-0.

[Hart-3] Hart, C. A. (2009). Solid Geometry. BiblioBazaar. p. 465. ISBN 1-103-11804-8.

[4] 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.

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Spherical wedge

See also

Notes

References