Spherical wedge (Area of the lune)

Description

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. (Aspherical 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. Considering that the surface area of a sphere is given, ithe surface area of the lune corresponding to the same wedge can be calculated by the radius of the ball to which the wedge belongs and the dihedral angle of the wedge.

Related formulas

Variables

AThe Area of the lune (m2)
αThe dihedral angle of the wedge (radians)
rThe radius of the sphere (m)