How to find the angle between clock hands
The classic mistake in clock angle problems is treating the hour hand as if it parks on the hour mark. It does not: the hour hand covers 30 degrees per hour, which means it keeps sliding 0.5 degrees every minute. Measuring clockwise from the 12, the hour hand angle is therefore 30° × hour + 0.5° × minute, while the minute hand angle is simply 6° × minute.
At 3:15, for example, the minute hand sits at 90° but the hour hand has moved 7.5° past the 3 and sits at 97.5°. The angle between them is 7.5°, not 0°. This calculator shows both hand angles separately so you can see exactly where the difference comes from.
Two hands always form two angles whose sum is 360°, so the smaller angle of 180° or less appears first in large type and the larger angle is listed right below it. When the hands overlap, line up straight or form a right angle, an extra row reports that position. Minutes accept decimals, so seven minutes thirty seconds can be entered as 7.5, and a 24-hour value is read back in a 12-hour format with AM or PM. The second hand is not supported.
Frequently Asked Questions
It is 7.5 degrees. The minute hand sits at 90 degrees, but the hour hand has drifted 7.5 degrees past the 3 to 97.5 degrees. Assuming the hour hand stays at 90 degrees and answering zero is the most common mistake.
The hour hand covers 30 degrees per hour, which is 0.5 degrees per minute. The minute hand covers 6 degrees per minute, so the hands close on each other at 5.5 degrees per minute.
Two hands always form two angles that add up to 360 degrees. Textbook problems ask for the one that is 180 degrees or less, so the smaller angle leads and the larger angle is listed next to it.