Clock and Calendar: Telling Time by the Sky and by Numbers

Time from the sky: day, month and year

People measured time by watching the sky long before clocks.

• Day: one spin of the Earth. Measured from noon to noon (the Sun at its highest), it is the solar day of 24 hours.
• Month: the Moon shows the same phase again after about 29.53 days (synodic month).
• Year: the Earth goes round the Sun and the seasons repeat after about 365.2422 days (tropical year).

These numbers do not divide each other neatly. That is the whole problem of making a calendar.

Solar day and sidereal day

A sidereal day is one turn of the Earth measured against a far star: about 23 h 56 min 4 s. During that time the Earth has also moved about 1° along its orbit, so it must turn a little more (about 4 minutes) before the Sun is overhead again. So the solar day is about 24 h.

This is why a given star rises about 4 minutes earlier every night.

Time zones

The Earth turns 360° in 24 h, so 15° of longitude = 1 hour, and 1° = 4 minutes. Countries use time zones counted from UTC (Greenwich, 0°). India uses UTC+5:30 (82.5° E).

Types of calendars

• Lunar calendar: 12 Moon months ≈ 354 days. It drifts about 11 days a year against the seasons. Example: the Islamic Hijri calendar.
• Solar calendar: follows the year of seasons. Example: the Gregorian calendar used worldwide for civil life.
• Lunisolar calendar: Moon months, plus an extra month added every 2 to 3 years to stay with the seasons. Examples: Hindu, Hebrew and Chinese calendars.

Julian and Gregorian

The Julian calendar (46 BCE) added a leap day every 4 years: year = 365.25 days. That is 11 minutes too long, so by 1582 the calendar was 10 days off. The Gregorian reform dropped 10 days and changed the rule:

A year is a leap year if it divides by 4, except century years, which must divide by 400. So 2024 and 2000 are leap years; 1900 and 2100 are not. Average year = 365.2425 days, very close to 365.2422.

Clock problems: angle between the hands

The minute hand turns 360° in 60 min: 6° per minute. The hour hand turns 30° in 60 min: 0.5° per minute.

At H hours and M minutes:

Angle = |30H − 5.5M| (if it is more than 180°, take 360° minus it).

The minute hand gains 5.5° per minute on the hour hand. The hands overlap 11 times in 12 hours (every 65 5/11 minutes) and are opposite 11 times in 12 hours.

Calendar problems: odd days

Odd days are the days left over after full weeks.

• Common year: 365 = 52 × 7 + 1 → 1 odd day.
• Leap year: 366 → 2 odd days.
• 100 years (24 leap) → 5 odd days; 200 years → 3; 300 years → 1; 400 years → 0.

To move from one date to another, add up the odd days and move that many weekdays forward. Example: 1 Jan 2025 was a Wednesday. 2025 is common, so 1 Jan 2026 is Wednesday + 1 = Thursday.

Try it

Look at a wall calendar. Find your birthday's weekday this year. Predict next year's using odd days (+1, or +2 if 29 February comes in between). Check with the 3D year box.

Key formulas and definitions

• Solar day = 24 h; sidereal day ≈ 23 h 56 min
• Synodic month ≈ 29.53 days; tropical year ≈ 365.2422 days
• Leap year: divisible by 4, but century years only if divisible by 400
• Angle between hands = |30H − 5.5M|
• 15° longitude = 1 hour; 1° = 4 minutes
• Odd days: common year 1, leap year 2, 100 years 5, 400 years 0

Worked examples

Common mistakes

• Thinking every year divisible by 4 is a leap year; 1900 and 2100 are not.
• Forgetting that the hour hand also moves as minutes pass (at 3:30 it is halfway between 3 and 4).
• Using 6° per minute for the gain; the minute hand gains only 5.5° per minute on the hour hand.
• Mixing up the solar day (24 h) and the sidereal day (23 h 56 min).