Study guides › Level 9 · Trigonometry

The angle between clock hands

Level 9 · Trigonometry · for questions like “At 3:20, what is the smaller angle between the hands?”

💡 The idea

Measure both hands in degrees from 12 o'clock:

• minute hand = 6° × the minutes
• hour hand = 30° × the hour + ½° × the minutes (it creeps forward as the minutes pass)

🪜 The steps

1. Work out both positions.
2. Take the difference.
3. If it's more than 180°, take it away from 360°.

✏️ Worked examples

1. 3:20: minute hand 120°, hour hand 90 + 10 = 100°. The difference is 20°.

2. 9:30: minute hand 180°, hour hand 270 + 15 = 285°. The difference is 105°.

⚠️ Watch out

• Don't forget the hour hand moves. At 3:30 it's halfway between 3 and 4.

🎯 Try these

1. 2:00
2. 6:30
3. 4:40

Answers: 1) 60° 2) 15° 3) 100°

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