spherical wedge

{{short description|Geometric shape; radial slice of a sphere}}

File:Spherical Wedge.svg

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 {{mvar|α}}. If {{mvar|AB}} is a semidisk that forms a ball when completely revolved about the z-axis, revolving {{mvar|AB}} only through a given {{mvar|α}} produces a spherical wedge of the same angle {{mvar|α}}.{{cite book|title=Geometry, Plane, Solid, and Spherical, in Six Books|url=https://archive.org/details/geometryplaneso00goog|first=P.|last=Morton|publisher=Baldwin & Cradock|year=1830|page=[https://archive.org/details/geometryplaneso00goog/page/n192 180]}} Beman (2008){{cite book|title=New Plane and Solid Geometry|page=338|first=D. W.|last=Beman|publisher=BiblioBazaar|year=2008|isbn=0-554-44701-0}} 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." {{ref|a|[A]}} A spherical wedge of {{math|1=α = {{pi}}}} radians (180°) is called a hemisphere, while a spherical wedge of {{math|1=α = 2{{pi}}}} radians (360°) constitutes a complete ball.

The volume of a spherical wedge can be intuitively related to the {{mvar|AB}} definition in that while the volume of a ball of radius {{mvar|r}} is given by {{math|{{sfrac|4|3}}{{pi}}r{{isup|3}}}}, the volume a spherical wedge of the same radius {{mvar|r}} is given by

:$V\; =\; \backslash frac\{\backslash alpha\}\{2\backslash pi\}\; \backslash cdot\; \backslash tfrac43\; \backslash pi\; r^3\; =\; \backslash tfrac23\; \backslash alpha\; r^3\backslash ,.$

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

:$A\; =\; \backslash frac\{\backslash alpha\}\{2\backslash pi\}\; \backslash cdot\; 4\; \backslash pi\; r^2\; =\; 2\; \backslash alpha\; r^2\backslash ,.$

Hart (2009){{cite book|title=Solid Geometry|first=C. A.|last=Hart|page=465|publisher=BiblioBazaar|year=2009|isbn=1-103-11804-8}} 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".{{ref|a|[A]}} Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if {{math|V{{sub|s}}}} is the volume of the sphere and {{math|V{{sub|w}}}} is the volume of a given spherical wedge,

:$\backslash frac\{V\_\backslash mathrm\{w\}\}\{V\_\backslash mathrm\{s\}\}\; =\; \backslash frac\{\backslash alpha\}\{2\backslash pi\}\backslash ,.$

Also, if {{math|S{{sub|l}}}} is the area of a given wedge's lune, and {{math|S{{sub|s}}}} is the area of the wedge's sphere,{{cite book|title=Marks' Standard Handbook for Mechanical Engineers|page=43|first1=E. A.|last1=Avallone|first2=T.|last2=Baumeister|first3=A.|last3=Sadegh|first4=L. S.|last4=Marks|publisher=McGraw-Hill Professional|year=2006|isbn=0-07-142867-4}}{{Ref|a|[A]}}

:$\backslash frac\{S\_\backslash mathrm\{l\}\}\{S\_\backslash mathrm\{s\}\}\; =\; \backslash frac\{\backslash alpha\}\{2\backslash pi\}\backslash ,.$