For a unit radius sphere, the dividing equatorial line between the light and dark hemispheres projects to

when

, and to

for

. Therefore when

, the dark area is given by the area integral from the bottom of the sphere,

, to the dividing line

. Similarly, when

the dark area is given by the area integral from the dividing line to the top of the sphere,

, and is

. Because the total area of the sphere is

, dividing these area integrals by

gives us the fraction that is dark,

. The fraction of the moon that is illuminated is (1 – the fraction that is dark), and is given by

.

[1] D. B. Taylor, S. A. Bell, J. L. Hilton, and A. T. Sinclair,

*Computation of the Quantities Describing the Lunar Librations in the Astronomical Almanac, *Ft. Belvoir: Defense Technical Information Center, 2010.