Angle Between the Hands

🎯 Learning Objective

Compute the angle between the hour and minute hands at any time using |30H − 5.5M|.

💡 Concept

• Minute hand speed = 360° in 60 min = 6° per minute
• Hour hand speed = 360° in 12 hours = 0.5° per minute
• At H:M — hour hand at 30H + 0.5M degrees, minute hand at 6M degrees
• Angle = |30H − 5.5M|; if it exceeds 180°, take 360° − angle
• The 5.5 is simply 6 − 0.5 — the relative speed of the two hands

🧮 Key Formulas

Angle = |30H − 5.5M|

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Minute hand: 6°/min; Hour hand: 0.5°/min

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If angle > 180°, take 360° − angle

✏️ Easy Example

Q. Find the angle between the hands of a clock at 3:30.

1. Angle = |30×3 − 5.5×30|
2. = |90 − 165| = 75

Answer: 75°

🇮🇳 Real-Life Example

The big station clock at 9:30 — the hands spread |270 − 165| = 105°. Next time your train is late, verify the formula on the platform clock!

📝 Exam-Level Example

Q. What is the angle between the hands at 4:20?

1. |30×4 − 5.5×20|
2. = |120 − 110| = 10

Answer: 10°

📝 Exam-Level Example

Q. Find the angle between the hands at 8 o'clock.

1. |30×8 − 5.5×0| = 240
2. 240 > 180 → angle = 360 − 240

Answer: 120°

🪄 Memory Trick

The hour hand drifts 0.5° every minute — at half past, it sits midway between two numbers. Never park it exactly on the hour mark unless M = 0.

⚠️ Common Mistakes

• ❌ Keeping the hour hand fixed at the hour mark
• ❌ Reporting the reflex angle (more than 180°) instead of the smaller one
• ❌ Mixing the terms — 30 goes with H and 5.5 goes with M, never the reverse

🏆 Exam Tips

• ✅ Sanity anchors: right angle at 3:00 and 9:00, straight line at 6:00
• ✅ Angles always come in 0.5° steps — an option like 17.2° must be wrong

📌 Summary

• Minute 6°/min, hour 0.5°/min
• Angle = |30H − 5.5M|
• Above 180° → subtract from 360°
• The hour hand always drifts with the minutes