Moon Phases Almanac
You need one known new moon, a count of days and one division by 29.53059 — the remainder is the phase, and a cosine turns it into illumination.
Step one: fix an epoch. Take any new moon listed in an almanac or observatory ephemeris as day zero. The method does not care which new moon you choose, because the cycle repeats every 29.53059 days — 29 days, 12 hours, 44 minutes and 3 seconds.
Step two: count the days from your epoch to the target date, then divide by 29.53059. The whole part of the quotient is the number of complete cycles; discard it. The remainder is the number of days into the current cycle — the phase day.
On narrow screens, swipe or scroll the plate sideways.
Step three: read the phase day against the cycle marks: 0 is new, 7.38 is first quarter, 14.77 is full, 22.15 is last quarter, with the crescent and gibbous marks near 3.7, 11.1, 18.5 and 25.8. A remainder within about a day of a mark means that phase; each principal phase falls about 7.38 days after the previous one.
Step four: convert to illumination. Scale the phase day to an angle — elongation ≈ phase day ÷ 29.53059 × 360° — then lit fraction ≈ (1 − cos elongation) / 2. Worked example with round numbers: if your target date is 40.0 days after the epoch new moon, subtract one full cycle to get 10.47 days in; elongation ≈ 10.47 ÷ 29.53059 × 360° ≈ 128°, and illumination ≈ (1 − cos 128°) / 2 ≈ 0.81 — a waxing gibbous at about 81%, with full following in about four days.
Two checks keep the arithmetic honest. First, the Metonic cycle: 235 lunations equal 19 years, so your target phase should fall on nearly the same calendar date 19 years earlier or later — the same cycle some lunar calendars use to place leap months. Second, sanity-check against the sky: a remainder near 7.38 must show a half-lit disk visible from about noon to midnight, because first quarter stands 90° east of the Sun. For what each phase looks like, see The Eight Moon Phases Explained (/moon-phases-explained); for the rise times to expect, see Moonrise and Moonset (/moonrise-and-moonset).
Further reading