What is "spherical wedge"

Wedge \Wedge\ (w[e^]j), n. [OE. wegge, AS. wecg; akin to D. wig, wigge, OHG. wecki, G. weck a (wedge-shaped) loaf, Icel. veggr, Dan. v[ae]gge, Sw. vigg, and probably to Lith. vagis a peg. Cf. Wigg.]

1. A piece of metal, or other hard material, thick at one end, and tapering to a thin edge at the other, used in splitting wood, rocks, etc., in raising heavy bodies, and the like. It is one of the six elementary machines called the mechanical powers. See Illust. of Mechanical powers, under Mechanical.

2. (Geom.) A solid of five sides, having a rectangular base, two rectangular or trapezoidal sides meeting in an edge, and two triangular ends.

3. A mass of metal, especially when of a wedgelike form. ``Wedges of gold.''
   --Shak.

4. Anything in the form of a wedge, as a body of troops drawn up in such a form.

   In warlike muster they appear, In rhombs, and wedges, and half-moons, and wings.
   --Milton.

5. The person whose name stands lowest on the list of the classical tripos; -- so called after a person (Wedgewood) who occupied this position on the first list of 1828. [Cant, Cambridge Univ., Eng.]
   --C. A. Bristed.

6. (Golf) A golf club having an iron head with the face nearly horizontal, used for lofting the golf ball at a high angle, as when hitting the ball out of a sand trap or the rough.

   Fox wedge. (Mach. & Carpentry) See under Fox.

   Spherical wedge (Geom.), the portion of a sphere included between two planes which intersect in a diameter.

n. (context geometry English) A volume of a sphere lying between two vertical planes through the centre of the sphere.

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. 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 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 $\tfrac{4}{3} \pi r^3$, the volume a spherical wedge of the same radius r is given by

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

Extrapolating the same principle and considering that the surface area of a sphere is given by 4πr, it can be seen that the surface area of the lune corresponding to the same wedge is given by

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

Hart (2009) states that the "volume of a spherical wedge is to the volume of the sphere as the number of degrees in the [angle of the wedge] is to 360". Hence, and through derivation of the spherical wedge volume formula, it can be concluded that, if V is the volume of the sphere and V is the volume of a given spherical wedge,

$$\frac{V_w}{V_s} = \frac{\alpha}{2\pi}$$

Also, if S is the area of a given wedge's lune, and S is the area of the wedge's sphere,

$$\frac{S_l}{S_s} = \frac{\alpha}{2\pi}$$