How this calculator works
The minute hand turns 360 degrees in an hour, so it moves 6 degrees each minute. The hour hand moves 30 degrees per hour plus 0.5 degrees per minute. Subtract their positions and reduce the result to one turn for the directed angle. The smaller angle is at most 180 degrees. Inverse modes use the directed hour-to-minute angle, which removes the ambiguity between the two directions. Seconds are represented by fractional minutes.
Formula
Hour position H = 30h + 0.5m degrees; minute position M = 6m degrees; clockwise difference D = (5.5m - 30h) mod 360 degrees. Smaller angle = min(D, 360 - D).
Worked example
At 3:30, H = 105 degrees and M = 180 degrees. The clockwise hour-to-minute angle and smaller separation are both 75 degrees; the reverse directed angle is 285 degrees. At 12:00 both directed separations are zero.
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Frequently asked questions
Why is the angle at 3:30 not 90 degrees?
The hour hand has moved halfway from 3 to 4 by then. Ignoring that motion would overestimate the separation by 15 degrees.
Can I enter a 24-hour time?
Yes. Hours 0 through 23 are accepted and reduced modulo 12. Enter minutes from 0 inclusive to 60 exclusive.
Why can an inverse calculation have no answer?
The requested directed angle may never occur inside the chosen hour, or may require a fractional hour label when the minutes are fixed. The table lists only valid times in the stated mode.