Clocks — Angle Between Hands, Overlaps and Wrong Clocks

A clock face has 360°, 12 hours and 60 minutes. Every clock question uses these speeds:

• Minute hand: 360° in 60 min → 6° per minute
• Hour hand: 30° in 60 min → ½° per minute
• The minute hand gains 5.5° per minute on the hour hand.

1. Angle between the hands

Angle = | 30H − 5.5M | (H = hour, M = minutes) If the answer is more than 180°, subtract it from 360° to get the smaller angle.

Example 1

What is the angle between the hands at 3:40?

|30 × 3 − 5.5 × 40| = |90 − 220| = 130°.

Example 2

At 7:20?

|210 − 110| = 100°.

Example 3

At 9:00? |270 − 0| = 270 → 360 − 270 = 90°.

Why the hour hand matters: at 3:40 the hour hand is not on 3 — it has moved ⅔ of the way to 4. Students who forget this get 120° instead of 130°.

2. When do the hands overlap?

The hands are together when 30H = 5.5M → M = 60H/11.

Example 4

At what time between 4 and 5 are the hands together?

M = 240/11 = 21 9/11 minutes past 4.

Useful facts in 12 hours:

• Hands overlap 11 times (not 12 — between 11 and 1 they overlap only once, at 12).
• Hands are at a right angle (90°) 22 times.
• Hands are opposite (180°) 11 times.

In 24 hours: 22 overlaps, 44 right angles, 22 opposite.

3. Hands opposite each other

Example 5

Between 2 and 3, when are the hands opposite?

|30 × 2 − 5.5M| = 180 → 5.5M = 240 → M = 480/11 = 43 7/11 minutes past 2.

4. Mirror images

The time seen in a mirror = 11:60 − actual time (for times between 1:00 and 11:00).

Example 6

A clock shows 3:25. What does its mirror image show? 11:60 − 3:25 = 8:35.

5. Fast and slow clocks

Example 7

A clock gains 5 minutes every hour. It is set right at 8 am. What will it show at 2 pm (actual time)?

6 hours × 5 min = 30 min gained → it shows 2:30 pm.

Common mistakes

• Treating the hour hand as fixed on the hour number.
• Saying the hands overlap 12 times in 12 hours (it is 11).
• Forgetting to take 360° minus the angle when it is over 180°.

Quick recap

• Minute hand 6°/min, hour hand ½°/min, gap closes at 5.5°/min.
• Angle = |30H − 5.5M|.
• Overlap at M = 60H/11. 11 overlaps in 12 hours.
• Mirror time = 11:60 − time.