Calculate the smallest angle between clock hands for a given time or find the time when they form a target angle in degrees or radians.
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Clock Angle Formula
The calculator uses the standard clock hand equations. The hour hand moves 0.5° per minute. The minute hand moves 6° per minute.
theta = |30H - 5.5M|
If the result is greater than 180°, subtract it from 360° to get the smaller angle.
- theta = angle between the hour and minute hands, in degrees
- H = hour on a 12-hour clock (0 to 11)
- M = minutes past the hour (0 to 59)
The hour hand position from 12 is 30H + 0.5M. The minute hand position from 12 is 6M. The difference between these two values, taken as the smaller of the two arcs, is the clock angle.
For the reverse mode, the calculator solves for M given a target angle A and starting hour H:
M = (30H ± A) / 5.5
- A = target smaller angle in degrees (0 to 180)
- H = the starting hour on a 12-hour clock
- The ± gives up to two valid times within the hour
Assumptions: hands move continuously, not in discrete ticks. Radian inputs are converted to degrees internally. Hours from 12 to 23 are mapped to their 12-hour equivalent for the math, then displayed in both formats.
Mode 1 (Angle at a time): You enter H and M. The calculator returns the smaller angle, both clockwise sweeps, and each hand’s position from 12. Mode 2 (Time for an angle): You enter A and the starting hour. The calculator returns every minute value in that hour where the hands form A.
Reference Tables
Common clock angles at the top of each hour (minute = 0):
| Time | Angle | Type |
|---|---|---|
| 12:00 | 0° | Overlap |
| 1:00 | 30° | Acute |
| 2:00 | 60° | Acute |
| 3:00 | 90° | Right |
| 4:00 | 120° | Obtuse |
| 5:00 | 150° | Obtuse |
| 6:00 | 180° | Straight |
| 9:00 | 90° | Right |
Hand speed reference:
| Hand | Per minute | Per hour | Full revolution |
|---|---|---|---|
| Hour | 0.5° | 30° | 12 hours |
| Minute | 6° | 360° | 60 minutes |
| Relative (minute − hour) | 5.5° | 330° | ~65.45 minutes |
Worked Examples and FAQ
Example 1: angle at 3:15. H = 3, M = 15. Hour hand at 30(3) + 0.5(15) = 97.5°. Minute hand at 6(15) = 90°. Difference is 7.5°. The hands are not exactly at a right angle, even though it looks like they should be.
Example 2: angle at 10:14. Hour hand at 30(10) + 0.5(14) = 307°. Minute hand at 6(14) = 84°. Difference is |307 − 84| = 223°. Smaller angle is 360 − 223 = 137°. Obtuse.
Example 3: when do the hands form 90° between 4:00 and 5:00? Solve 30(4) − 5.5M = ±90. M = (120 − 90)/5.5 = 5.4545 min, or M = (120 + 90)/5.5 = 38.1818 min. So 4:05:27 and 4:38:11.
How many times a day do the hands overlap? 22 times. The overlap repeats every 12/11 hours, giving 11 overlaps per 12-hour cycle.
How many times do they form a right angle? 44 times in 24 hours. Two per hour, except the hours straddling 3:00 and 9:00 where one of the right angles falls exactly on the hour boundary.
Why is 3:15 not exactly 90°? The hour hand drifts forward as minutes pass. By minute 15 it has moved 7.5° past the 3, so the gap shrinks to 7.5°.
Can the angle exceed 180°? Geometrically no, since the smaller of the two arcs between the hands is always 180° or less. The reflex angle is 360° minus the smaller angle.
