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.
Related Calculators
- Angle Addition Calculator
- 12 Hour Time Calculator
- Pie Cut Calculator
- Line Array Angle Calculator
- Perpendicular Line Calculator
- All Math and Numbers Calculators
Clock Angle Formula
The calculator uses the standard clock hand equations. The hour hand moves 0.5ยฐ per minute. The minute hand moves 6ยฐ per minute.
If the result is greater than 180ยฐ, subtract it from 360ยฐ 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:
- A = target smaller angle in degrees (0 to 180)
- H = the starting hour on a 12-hour clock
- The ยฑ and full-turn terms give up to two distinct valid times from the starting hour, excluding the next 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 using the entered 24-hour notation. Reverse times are rounded to the nearest second; the unrounded minute offsets appear in Calculation details.
Mode 1 (Angle at a time): You enter H and M. The calculator returns the smaller angle and both clockwise sweeps; each hand's position from 12 appears in Calculation details. 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 | 0ยฐ | Overlap |
| 1:00 | 30ยฐ | Acute |
| 2:00 | 60ยฐ | Acute |
| 3:00 | 90ยฐ | Right |
| 4:00 | 120ยฐ | Obtuse |
| 5:00 | 150ยฐ | Obtuse |
| 6:00 | 180ยฐ | Straight |
| 9:00 | 90ยฐ | Right |
Hand speed reference:
| Hand | Per minute | Per hour | Full revolution |
|---|---|---|---|
| Hour | 0.5ยฐ | 30ยฐ | 12 hours |
| Minute | 6ยฐ | 360ยฐ | 60 minutes |
| Relative (minute โ hour) | 5.5ยฐ | 330ยฐ | ~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) = 97.5ยฐ. Minute hand at 6(15) = 90ยฐ. Difference is 7.5ยฐ. 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) = 307ยฐ. Minute hand at 6(14) = 84ยฐ. Difference is |307 โ 84| = 223ยฐ. Smaller angle is 360 โ 223 = 137ยฐ. Obtuse.
Example 3: when do the hands form 90ยฐ between 4:00 and 5:00? Solve 30(4) โ 5.5M = ยฑ90. M = (120 โ 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 90ยฐ? The hour hand drifts forward as minutes pass. By minute 15 it has moved 7.5ยฐ past the 3, so the gap shrinks to 7.5ยฐ.
Can the angle exceed 180ยฐ? Geometrically no, since the smaller of the two arcs between the hands is always 180ยฐ or less. The reflex angle is 360ยฐ minus the smaller angle.
