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Clock Angle Puzzles — Questions and Answers

Clock questions look fiddly but reduce to two constants: the minute hand moves 6° per minute and the hour hand 0.5° per minute, so the minute hand gains on the hour hand at exactly 5.5° per minute. Every coincidence, right angle, straight line and gaining/losing-clock question follows from those numbers.

6 solved questionsClock PuzzlesFree · no signup

Clock Angle Puzzles— Concepts, Formulas & Shortcuts

  • Minute hand: 6° per minute. Hour hand: 0.5° per minute (30° per hour).
  • Relative gain of minute over hour hand = 5.5° per minute.
  • Angle at H hours M minutes = |30H − 5.5M| degrees (take 360 − θ if the result exceeds 180).
  • The hands coincide 11 times in 12 hours and 22 times in 24 hours.
  • The hands are at right angles 44 times in 24 hours and opposite 22 times in 24 hours.
  • Mirror-image time = 11:60 − given time.

Clock Angle Puzzles Practice Questions with Answers

Attempt each question first, then open the explanation. All 6 questions below are free to read and require no signup.

  1. Q1.The angle between the hour and minute hands at 3:00 is:

    Easy
    • A60°
    • B75°
    • C90°
    • D120°
    +Show Answer & Explanation

    Answer: C. 90°

    Explanation: |30 × 3 − 5.5 × 0| = 90°.

    Clock Angle Puzzles question 1 of 6

  2. Q2.The angle between the hands at 3:30 is:

    Moderate
    • A65°
    • B75°
    • C85°
    • D90°
    +Show Answer & Explanation

    Answer: B. 75°

    Explanation: |30 × 3 − 5.5 × 30| = |90 − 165| = 75°.

    Clock Angle Puzzles question 2 of 6

  3. Q3.At what time between 3 and 4 o'clock do the hands of a clock coincide?

    Difficult
    • A3:15
    • B3:16 4/11
    • C3:17
    • D3:18 2/11
    +Show Answer & Explanation

    Answer: B. 3:16 4/11

    Explanation: The minute hand must gain 90° at 5.5° per minute: 90/5.5 = 180/11 = 16 4/11 minutes past 3.

    Clock Angle Puzzles question 3 of 6

  4. Q4.The angle between the hands at 9:30 is:

    Moderate
    • A95°
    • B100°
    • C105°
    • D110°
    +Show Answer & Explanation

    Answer: C. 105°

    Explanation: |30 × 9 − 5.5 × 30| = |270 − 165| = 105°.

    Clock Angle Puzzles question 4 of 6

  5. Q5.How many times do the hands of a clock coincide in 12 hours?

    Moderate
    • A10
    • B11
    • C12
    • D24
    +Show Answer & Explanation

    Answer: B. 11

    Explanation: The minute hand laps the hour hand 11 times in 12 hours, so they coincide 11 times.

    Clock Angle Puzzles question 5 of 6

  6. Q6.How many times in a day are the hands of a clock at right angles?

    Difficult
    • A22
    • B24
    • C44
    • D48
    +Show Answer & Explanation

    Answer: C. 44

    Explanation: They form a right angle twice in most hours but only 44 times in 24 hours, because two occurrences are lost around 3 and 9 o'clock in each 12-hour cycle.

    Clock Angle Puzzles question 6 of 6

Clock Angle Puzzles — Frequently Asked Questions

What is the formula for the angle between clock hands?+

Angle = |30H − 5.5M|, where H is the hour and M the minutes. If the result exceeds 180°, subtract it from 360° to get the smaller angle. At 3:30 this gives |90 − 165| = 75°.

Why do the hands coincide only 11 times in 12 hours?+

The minute hand gains 360° on the hour hand every 65 5/11 minutes, so in 720 minutes it laps the hour hand 11 times — not 12, because the 12 o'clock coincidence is shared between cycles.

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