Every time in a 12-hour cycle is listed on the right. Click one to load it.
A clock face is 360°, so each of the 12 hour marks is 30° apart and each minute mark is 6° apart. The minute hand moves 6° per minute. The hour hand moves 0.5° per minute — thirty degrees an hour, spread across sixty minutes. Almost every mistake in these problems comes from forgetting that second number and leaving the hour hand parked on its mark.
At 3:30 the minute hand is at 30 × 6 = 180°. The hour hand is not at 90° — it has spent half an hour creeping toward 4, so it sits at (3 × 30) + (30 × 0.5) = 105°. The difference is 75°, not the 90° most people answer. The shortcut |30H − 5.5M| gives |90 − 165| = 75 in one step.
Take the absolute difference between the hour hand at 30H + 0.5M and the minute hand at 6M, which simplifies to |30H − 5.5M|. If that exceeds 180°, subtract from 360° for the smaller angle between the hands.
It covers 360° in 12 hours — 30° per hour — and 30 ÷ 60 = 0.5° per minute. Leaving the hour hand on its mark is the single most common error in these problems.
75°, not 90°. The minute hand is at 180° and the hour hand has moved halfway from 3 to 4, reaching 105°.
Every 720/11 minutes — 65 minutes and 27.27 seconds. That is 11 overlaps in 12 hours and 22 in a day, not 12 and 24, because the minute hand must make up a whole lap on a target that is itself moving.
44. The hands pass through 90° twice in each overlap cycle, and there are 22 cycles a day.