Moon Illumination Fraction Calculator
The fraction of the Moon's visible disk that is lit by the Sun depends on the phase angle between the Sun, Moon, and Earth. Over one synodic month, about 29.53 days, that angle sweeps through a full circle, taking the Moon from new through first quarter, full, last quarter, and back. Using a uniform-phase approximation, this calculator estimates the illuminated fraction from the number of days since the last new moon. It reports the fraction, the percentage lit, the phase angle, and the named phase. The synodic month length is user-editable so you can refine the estimate.
Moon illumination formula
age = days mod synodic month
phase angle = 360 * age / synodic month (degrees)
fraction = (1 - cos(phase angle)) / 2
percent = fraction * 100
At new moon the phase angle is 0 and the fraction is 0; at full moon the angle is 180 degrees and the fraction is 1. The cosine term smoothly traces the crescent and gibbous phases between.
Lunar phase notes
- The default 7.38 days is about first quarter (roughly half lit).
- Fraction 0 is new moon, 0.5 is a quarter phase, 1.0 is full moon.
- The synodic month averages about 29.53 days.
- This uniform-phase model is accurate to a few percent of fraction.
- For exact figures use the U.S. Naval Observatory phase data.
Moon illumination: frequently asked questions
How is the illuminated fraction of the Moon estimated?
The fraction of the Moon's disk that is lit depends on the phase angle between the Sun, Moon, and Earth. Over one synodic month (about 29.53 days) the phase angle sweeps through 360 degrees. The illuminated fraction equals one minus the cosine of the phase angle, all divided by two. This calculator drives the phase angle from the days since the last new moon.
What is the synodic month?
The synodic month is the average time from one new moon to the next, about 29.53 days. It is longer than the Moon's orbital period because the Earth-Moon system also moves around the Sun. The synodic month length is a user-editable input so you can use a more precise local value.
Why is this an estimate rather than an exact ephemeris?
The Moon's orbit is elliptical and inclined, so the true phase angle varies slightly from a uniform sweep. This calculator uses a uniform-phase approximation that is accurate to a few percent for the illuminated fraction. For exact values use a published ephemeris from the U.S. Naval Observatory.
What do the fraction values mean?
Zero means new moon (no lit disk visible), 0.5 means a quarter phase (half the disk lit), and 1.0 means full moon (fully lit). Values between describe crescent and gibbous phases. The percentage output is just the fraction times 100.
How do I find days since the last new moon?
Look up the date of the most recent new moon from a calendar or the U.S. Naval Observatory phase tables, then count the days to your date of interest. Enter that count here. The calculator handles values beyond one month by wrapping through the cycle.
Official sources
- U.S. Naval Observatory: moon phase data and fraction illuminated.
- NASA Science: Moon phases explained.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.