Spherical lune

Common examples of great circles are lines of longitude (meridians) on a sphere, which meet at the north and south poles.

The surface area of a spherical lune is 2θ R2, where R is the radius of the sphere and θ is the dihedral angle in radians between the two half great circles.

When this angle equals 2π radians (360°) — i.e., when the second half great circle has moved a full circle, and the lune in between covers the sphere as a spherical monogon — the area formula for the spherical lune gives 4πR2, the surface area of the sphere.

For example, a regular hosotope {2,p,q} has digon faces, {2}2π/p,2π/q, where its vertex figure is a spherical platonic solid, {p,q}.

Or more specifically, the regular hosotope {2,4,3}, has 2 vertices, 8 180° arc edges in a cube, {4,3}, vertex figure between the two vertices, 12 lune faces, {2}π/4,π/3, between pairs of adjacent edges, and 6 hosohedral cells, {2,p}π/3.

The two great circles are shown as thin black lines, whereas the spherical lune (shown in green) is outlined in thick black lines. This geometry also defines lunes of greater angles: {2} π-θ , and {2} 2π-θ .
A full circle lune, {2} 2π
The phases of the moon make spherical lunes perceived as the intersection of a semicircle and semi-ellipse.
Stereographic projection of the 3-sphere 's parallels (red), meridians (blue) and hypermeridians (green). Lunes exist between pairs of blue meridian arcs.