What this calculator does
The trap in this classic problem is forgetting that the hour hand moves continuously. At 3:15 the minute hand points exactly at 3, but the hour hand has already crept a quarter of the way to 4.
That creep is why the answer is 7.50 degrees rather than zero. The hour hand advances half a degree per minute, so in fifteen minutes it moves 7.5 degrees past the 3.
The formula
The hour hand sits at 30 degrees per hour plus half a degree per minute. The minute hand sits at 6 degrees per minute. The smaller of the difference and its complement to 360 is the answer.
| Term | Meaning |
|---|---|
| Hour hand rate | 30 degrees per hour, or half a degree per minute. |
| Minute hand rate | 6 degrees per minute, twelve times faster. |
| Reflex angle | The larger angle going the other way round, summing to 360. |
The inputs explained
| Field | What to enter |
|---|---|
| Hour (0–12) | The hour, from 0 to 12. |
| Minutes (0–59) | The minutes past the hour, from 0 to 59. |
When to use it
Solving the classic puzzle
A standard interview and exam question that catches people out.
Understanding continuous motion
The hour hand never jumps, which is the whole point.
Finding when the hands align
Working backwards from an angle of zero gives the overlap times.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the angle change through the hour?
The same hour at four points through it.
| Minutes past | Angle between the hands | Hour hand position |
|---|---|---|
| 3:00 | 90.00° | 90.00° |
| 3:15 | 7.50° | 97.50° |
| 3:30 | 75.00° | 105.00° |
| 3:45 | 157.50° | 112.50° |
Questions
Why is 3:15 not zero degrees?
Because the hour hand moves continuously rather than jumping between hours. By quarter past, it has travelled a quarter of the way from 3 to 4, which is 7.5 degrees, and the minute hand is still at the 3.
How often do the hands overlap?
Every 65 minutes and 27 seconds, which works out to 11 times in 12 hours rather than 12. The minute hand has to make up a full lap on the hour hand each time, and it gains at only 5.5 degrees per minute.
Why does the minute hand move at 6 degrees a minute?
Because it covers 360 degrees in 60 minutes. The hour hand covers 360 degrees in 12 hours, which works out to 0.5 degrees per minute, exactly one twelfth of the speed.
What is the reflex angle for?
The two hands divide the clock face into two angles that sum to 360. The convention is to report the smaller one, but the reflex angle is the other valid answer to the same question.
For angle reduction generally, see the reference angle calculator. For trigonometric work, see the trigonometry calculator.