Angle Between Clock Hands
Also known as angle between the hands of a clock · clock angle problem · clock hands angle formula · angle between hour and minute hand
Worked example: 3:00 → hands exactly 90 degrees apart — press Try an example to run it live, then adjust anything.
Enter your known values, leave one input blank, and solves for the missing one. Tap a variable’s symbol to see what it means, with a typical value. Try different units for next level excitement!
Learning zone
The whole puzzle is that both hands move. The minute hand sweeps 6° per minute; the hour hand creeps 0.5° per minute — so the minute hand GAINS 5.5° on the hour hand every minute, and that 5.5 is the only number in the formula doing real work. At H o'clock the hour hand starts 30H degrees around; M minutes later the gap is |30H − 5.5M|, folded to the small side because an angle between hands is always reported as the one under 180°.
Where people go wrong: freezing the hour hand. At 3:30 the answer 'feels like' 90° minus something on the 6 — but the hour hand has spent half an hour drifting toward the 4, and the true gap is 75°. Any answer computed with a parked hour hand is off by up to 30°, which on this puzzle is the difference between right and embarrassingly wrong.
- = Angle between the hands (°)
- = Hour on the face
- = Minutes past the hour
- Angle between the hands — Law of Cosines, Dot Product from Magnitudes and Included Angle
- Hour on the face — Clock Hands Overlap Time, Recipe Scaling (Ingredient for a New Yield)
- Minutes past the hour — Clock Hands Overlap Time, Recipe Scaling (Ingredient for a New Yield)