Hands Apart: 6:30
Read the puzzle
An analogue clock face with the numbers 12, 3, 6 and 9 and tick marks for every hour shows half past six; a digital display underneath reads 6:30. The minute hand points straight down at the 6, and the hour hand sits partway between the 6 and the 7. What is the smaller angle between the two hands?
What is the SMALLER angle between the hands?
The hour hand is not on the 6 at 6:30.
15°
At 6:30 the hour hand is halfway from 6 to 7: 15° from the minute hand.
How to solve it
Measure both hands from the 12, going clockwise. The full circle is 360°, so each hour gap is 360 ÷ 12 = 30°, and each minute moves the minute hand 360 ÷ 60 = 6°.
The minute hand at 30 minutes is at 30 × 6 = 180°, pointing at the 6.
The hour hand moves too: 30° per hour, which is 0.5° per minute. At 6:30 it has gone 6 × 30 = 180° for the six full hours plus 30 × 0.5 = 15° for the extra half hour, so it is at 195°, halfway between the 6 and the 7.
The gap is 195 − 180 = 15°. The other way round the clock is 360 − 15 = 345°, so the smaller angle is 15°.
The usual wrong answer is 0°, from picturing both hands on the 6. That only happens if the hour hand stands still for the whole hour, which a real clock never does. A quick formula check: |30 × 6 − 5.5 × 30| = |180 − 165| = 15°.
