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 $α$ . 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 $.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠4/3⁠πr3$ , 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 $4 π r 2$ , 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 }}\,.

^ 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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