geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a spherical wedge or ungula is a portion of a

ball A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...

bounded by two plane

semidisk In geometry, a disk (also spelled disc). is the region in a plane bounded by a circle. A disk is said to be ''closed'' if it contains the circle that constitutes its boundary, and ''open'' if it does not. For a radius, r, an open disk is usua ...

s and a

spherical lune In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an example of a digon, θ, with dihedral angle θ. The word "lune" derives from ''luna'', the ...

(termed the wedge's ''base''). The angle between the

radii In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...

lying within the bounding semidisks is the dihedral . If is a semidisk that forms a ball when completely revolved about the ''z''-axis, revolving only through a given produces a spherical wedge of the same angle . 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." A spherical wedge of

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

s (180°) is called a ''

hemisphere Hemisphere refers to: * A half of a sphere As half of the Earth * A hemisphere of Earth ** Northern Hemisphere ** Southern Hemisphere ** Eastern Hemisphere ** Western Hemisphere ** Land and water hemispheres * A half of the (geocentric) celes ...

'', while a spherical wedge of radians (360°) constitutes a complete ball. The

volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...

of a spherical wedge can be intuitively related to the definition in that while the volume of a ball of radius is given by , the volume a spherical wedge of the same radius is given by :

V = \frac \cdot \tfrac43 \pi r^3 = \tfrac23 \alpha r^3\,.

Extrapolating the same principle and considering that the surface area of a sphere is given by , it can be seen that the surface area of the lune corresponding to the same wedge is given by :

A = \frac \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

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

s in the ngle of the wedgeis to 360". Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if is the volume of the sphere and is the volume of a given spherical wedge, :

\frac = \frac\,.

Also, if is the

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

of a given wedge's lune, and is the area of the wedge's sphere, :

\frac = \frac\,.

Notes

:A. A distinction is sometimes drawn between the terms "

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

" and "

", 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

{{reflist Spherical geometry

See also

Notes

References