How to Find the Angle Between Clock Hands

The angle between the hands at a given time is found by measuring each hand clockwise from twelve o'clock and taking the difference.

$H$ is the hour, $M$ the minutes, $h$ the angle of the hour hand and $m$ that of the minute hand. The hour hand goes round in 12 hours, so it moves 0.5 degrees a minute; the minute hand goes round in 60 minutes, so it moves 6.

The hour hand moves with the minutes

At twenty past three the hour hand is not at 90 degrees. Twenty minutes have passed since three, so it has crept $0.5 \times 20 = 10$ degrees further on, to $90 + 10 = 100$ degrees. Leaving it at 90 is the most common mistake in this calculation.

Example

At twenty past three the hour hand sits at $30 \times 3 + 0.5 \times 20 = 100$ degrees and the minute hand at $6 \times 20 = 120$ degrees. The difference is 20 degrees, which is the angle between them.

Notes

The two hands form two angles that add to 360 degrees. The smaller one is given here, so the answer falls between 0 and 180 degrees.

Hours on a 24 hour clock, such as 13, are accepted and folded back onto the face.