Clock Angle
The calculator finds the angle between the hour and minute hands for any time, draws the clock face and shows a step-by-step solution.
The clock angle calculator finds the angle between the hour hand and the minute hand for any time, draws the clock face and explains the solution. It is a popular puzzle in math contests and aptitude tests.
How fast the hands move
| Hand | Per hour | Per minute |
|---|---|---|
| Minute | 360° | 360° ÷ 60 = 6° |
| Hour | 360° ÷ 12 = 30° | 30° ÷ 60 = 0.5° |
The main trap: the hour hand does not stay still while the minutes pass. At 3:30 it points not at 3 but halfway between 3 and 4.
Clock angle formula
The angle of each hand is measured clockwise from the 12 mark. For a time of H hours and M minutes:
minute hand: 6° · M
hour hand: 30° · H + 0.5° · M
angle = |30° · H − 5.5° · M|
If the angle is greater than 180°, the smaller angle between the hands is 360° minus that angle. For times after noon (13:00–23:59), subtract 12 from the hours.
Example. Find the angle at 3:40.
Minute hand: 6° · 40 = 240°
Hour hand: 30° · 3 + 0.5° · 40 = 90° + 20° = 110°
Angle: |110° − 240°| = 130°
Fun facts
- The hands overlap 11 times every 12 hours — about every 65 minutes and 27 seconds.
- At 3:00 and 9:00 the angle is a right angle (90°); at 6:00 the hands point in opposite directions (180°).
- At 12:15 the angle is not 90° but 82.5°, because the hour hand has already moved 7.5°.