The Moon's phase angle is simply that angle at the Moon that is subtended by the Earth and the Sun. This angle directly determines the percent phase of the Moon that is used in planetarium software.

The equation to determine the Moon's phase angle from its phase is:

f = cos^-1 [ 2P / 100 - 1]

where f = the Moon's phase angle; and
           P = the Moon's phase (in percent)

The Moon's phase at 9:22 p.m. May 23, 2010 was:

P = 82.346 %

Using this value, the phase angle of the Moon was determined to be:

f = 49^o.690216