Once you know the time, you can also find the angle between the two hands
Learn how to calculate the angle formed by the hour and minute hands on a clock, and, in the other direction, how to find the exact time when the two hands overlap.
The minute hand makes a full 360° turn in 60 minutes, so it moves 6° each minute. The hour hand makes a full 360° turn in 12 hours (720 minutes), so it moves 0.5° each minute. The key point is that the hour hand keeps moving gradually even when it is not exactly on the hour.
To find the angle between the two hands, calculate each hand’s angle and take the difference. If that difference is greater than 180°, the angle going the other way around the clock (360° − the difference) is actually the smaller angle.
Enter a time to calculate the angle between the two hands, and then try finding the exact time when the two hands overlap.
Move the hour (H) and minute (M) sliders to see the angle formed by the hour and minute hands. The hour-hand angle is 30×H+0.5×M, and the minute-hand angle is 6×M.
Hour (H)3h
Minute (M)15min
Choose a whole-hour reference H. Between that time and the next whole hour, there is exactly one moment when the two hands overlap. Find the value of M that makes the hour-hand angle (30H+0.5M) equal to the minute-hand angle (6M).