Moon Phase Calculator
The Moon's appearance changes predictably over a 29.53-day cycle called the synodic period. Starting from new moon (invisible disk), the Moon waxes through crescent, quarter, and gibbous phases to full moon, then wanes back through the same sequence. The illuminated fraction at any point in the cycle follows a cosine curve. Enter the number of days elapsed since the last new moon to find the current illuminated fraction, phase percentage, and the recognized phase name.
Moon phase formula
f = (1 - cos(2π * D / T)) / 2
T = 29.53059 days (synodic period)
Where D is days since the last new moon (0 to 29.53) and f is the illuminated fraction (0 = new moon, 1 = full moon). The phase cycles smoothly and symmetrically; the same fraction occurs twice per cycle, once while waxing and once while waning.
Moon phase schedule
New Moon: D = 0 (0% lit). Waxing Crescent: D = 1.85 to 7.38. First Quarter: D = approximately 7.38 (50% lit). Waxing Gibbous: D = 7.38 to 14.77. Full Moon: D = 14.77 (100% lit). Waning Gibbous: D = 14.77 to 22.15. Last Quarter: D = approximately 22.15 (50% lit). Waning Crescent: D = 22.15 to 29.53.
Moon phase: frequently asked questions
What is the synodic period of the Moon?
The synodic period is the time between successive identical Moon phases (e.g., full Moon to full Moon). It is approximately 29.53059 days. This is slightly longer than the sidereal period (27.32 days) because Earth also moves around the Sun.
How is the illuminated fraction calculated?
The illuminated fraction f = (1 - cos(2 pi D / T)) / 2, where D is the number of days since the last new moon and T = 29.53059 days. At new moon (D=0), f = 0. At full moon (D = 14.77), f = 1. At first quarter (D = 7.38), f = 0.5.
What are the eight recognized Moon phases?
New Moon (0-1.85 days), Waxing Crescent (1.85-7.38 days), First Quarter (7.38-9.22 days), Waxing Gibbous (9.22-14.77 days), Full Moon (14.77-16.61 days), Waning Gibbous (16.61-22.15 days), Last Quarter (22.15-24.0 days), Waning Crescent (24.0-29.53 days).
How does the Moon's phase affect tides?
Tides are strongest (spring tides) at new moon and full moon when the Sun, Moon, and Earth align. Tides are weakest (neap tides) at first and last quarter when the Moon is 90 degrees from the Sun-Earth line. The Moon's gravitational influence is the primary driver of ocean tides.
Is the illuminated fraction the same as percentage lit?
Yes. A fraction of 0.75 means 75% of the visible lunar disk is lit. Note that the illuminated fraction refers only to the visible face; the far side is always unlit (facing away from observers on Earth) regardless of phase.
Official sources
- USNO Moon phase data and synodic period: aa.usno.navy.mil/data/MoonPhases.
- NASA Moon phases overview: moon.nasa.gov.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.