Clocks

A dial is 360° split into 12 hour marks, so one hour mark = 30° and one minute mark = 6°.

The two speeds are everything:

• The minute hand moves 360° in 60 minutes = 6° per minute.
• The hour hand moves 30° in 60 minutes = 0.5° per minute.
• So the minute hand gains 5.5° every minute on the hour hand.

From that come the standard results:

1. Angle at H:M = |30H − 5.5M|. If the answer exceeds 180°, subtract it from 360°.
2. Hands coincide when the gap is 0: 5.5M = 30H, i.e. M = 30H/5.5 minutes past H.
3. Right angle when the gap is 90°: 5.5M = 30H ± 90 — twice in most hours.
4. In 12 hours the hands coincide 11 times, are opposite 11 times and are at right angles 22 times. In 24 hours, double each. It is 11 and not 12 because between 11 and 12 o'clock they meet only at 12 itself.

The single most common mistake is treating the hour hand as parked on its number. At 4:20 the hour hand is not at the 4; it has crept a third of the way to the 5, i.e. 10° past. Always subtract 5.5M, never 6M, from 30H.

minute hand 6°/min · hour hand 0.5°/min · gain 5.5°/min · angle = |30H − 5.5M| · in 12 h: 11 coincidences, 11 oppositions, 22 right angles

Find the angle between the hands at 4:20. → Angle = |30 × 4 − 5.5 × 20| = |120 − 110| = 10°. Sanity check: the minute hand at 20 minutes sits exactly on the 4, and in those 20 minutes the hour hand has moved 20 × 0.5 = 10° past the 4 — so the gap really is 10°, and the careless |120 − 120| = 0° would mean the hands coincide, which they clearly do not.