Clock Angle Calculator
Enter any time in hours and minutes and this calculator instantly finds the angle between the hour and minute hands of an analog clock. You get the smaller angle, the reflex angle (the larger sweep going the other way), the position of each hand from 12 o'clock, and a step-by-step breakdown of the math. Switch to reverse mode to find all the exact minute marks within any hour where the hands form a specific angle.
Formula
Worked example
At 3:20 - Hour hand: 30 x 3 + 0.5 x 20 = 90 + 10 = 100 deg. Minute hand: 6 x 20 = 120 deg. Difference: |100 - 120| = 20 deg. Since 20 < 180 the smaller angle is 20 deg and the reflex is 340 deg.
How the clock angle formula works
A clock face is a circle of 360 degrees, divided into 12 hours, so each hour mark is 30 degrees apart (360 / 12 = 30). The minute hand sweeps the full circle in 60 minutes, so it moves 6 degrees per minute (360 / 60 = 6). The hour hand is slower: it advances one hour-segment (30 degrees) in 60 minutes, which works out to 0.5 degrees per minute. At any time H hours and M minutes, the hour hand sits at 30H + 0.5M degrees from 12 o'clock, and the minute hand sits at 6M degrees. The angle between them is the absolute difference of those two positions, capped at 180 degrees (if the raw difference exceeds 180, subtract it from 360 to get the smaller of the two arcs).
Why 3:15 is not exactly 90 degrees
This is the most common clock angle misconception. At 3:00 the hour hand points exactly at the 3 (90 degrees from 12). But by 3:15 the hour hand has already moved an additional 15 x 0.5 = 7.5 degrees, so it now sits at 97.5 degrees, while the minute hand is at 6 x 15 = 90 degrees. The angle between them is |97.5 - 90| = 7.5 degrees, nowhere near 90. The same logic applies at every quarter hour: the hour hand is never exactly on the hour mark once any minutes have passed.
Reverse solving: finding the time for a given angle
Sometimes you want to work backwards: at what minute mark within a specific hour do the hands form exactly 90 degrees (or any other target angle)? Setting |30H - 5.5M| = A and solving for M gives two equations: M = (30H - A) / 5.5 and M = (30H + A) / 5.5. Both solutions are valid if they fall in the range 0-60. For a target of 90 degrees in the 3 o'clock hour, the two solutions are about 0 min (3:00) and 32 min 43 sec (3:32:43). Try this yourself in the reverse mode above.
How many times do the hands overlap in 12 hours?
The minute hand gains exactly 5.5 degrees on the hour hand each minute. Starting from 12:00 (overlap), the next overlap occurs after 360 / 5.5 = 65.45 minutes, or 65 min 27 sec. Over 12 hours the hands overlap exactly 11 times (not 12), because the last overlap of the cycle is back at 12:00, which is also the first overlap of the next cycle. Similarly, the hands are exactly opposite (180 degrees apart) 11 times per 12-hour period.
Common clock angles
| Time | Hour hand (deg) | Minute hand (deg) | Angle between hands |
|---|---|---|---|
| 12:00 | 0.0 | 0.0 | 0.0 deg (overlap) |
| 12:15 | 7.5 | 90.0 | 82.5 deg |
| 3:00 | 90.0 | 0.0 | 90.0 deg |
| 3:15 | 97.5 | 90.0 | 7.5 deg |
| 3:30 | 105.0 | 180.0 | 75.0 deg |
| 4:00 | 120.0 | 0.0 | 120.0 deg |
| 6:00 | 180.0 | 0.0 | 180.0 deg (straight) |
| 6:30 | 195.0 | 180.0 | 15.0 deg |
| 9:00 | 270.0 | 0.0 | 90.0 deg |
| 11:59 | 359.5 | 354.0 | 5.5 deg |
Exact angles between the hour and minute hands at notable times (12-hour clock).
Frequently asked questions
What is the angle between the hands at 3:00?
At 3:00 the hour hand is at 90 degrees (three hour-marks of 30 degrees each) and the minute hand is at 0 degrees (at 12). The angle between them is exactly 90 degrees. This is one of the few whole-degree values that also has a clean intuitive explanation.
How do I find the angle at any time without a calculator?
Use the formula: hour hand position = 30H + 0.5M, minute hand position = 6M, where H is the hour (0-11) and M is the minutes. Subtract one from the other, take the absolute value, and if the result is more than 180 subtract it from 360. For example, at 7:40: hour hand = 30 x 7 + 0.5 x 40 = 230 deg; minute hand = 6 x 40 = 240 deg; difference = |230 - 240| = 10 deg.
How many times do clock hands overlap in 24 hours?
Exactly 22 times. In a 12-hour period the hands overlap 11 times (roughly every 65 minutes 27 seconds), and that pattern repeats twice in 24 hours.
What angle do the clock hands form at 6:00?
At 6:00 exactly the hour hand is at 180 degrees and the minute hand is at 0 degrees, so they form a straight angle of 180 degrees. The hands point in exactly opposite directions.
Does the angle depend on whether the clock shows AM or PM?
No. An analog clock runs on a 12-hour cycle, so 3:00 AM and 3:00 PM produce identical hand positions and the same angle. This calculator accepts 24-hour input for convenience but automatically maps it to the 12-hour face (any value above 11 is treated as hour mod 12).