Angles on a clock

A clock face is a circle, and its two hands always make an angle. At 3 o'clock they make a perfect right angle, like the corner of a page. At 6 o'clock they make a straight line. Finding the angle at any time is a favorite puzzle in school contests and job interviews, because it looks hard but needs only one fact: a full turn is 360°. Once you know how fast each hand moves, you can find the angle at any time, and even count how often the hands meet.

Thirty degrees between two numbers

An angle measures how far something turns. A full turn, all the way around, is 360°. The 12 numbers on a clock are spread evenly around the circle, so the gap from one number to the next is 360° ÷ 12 = 30°. There are also 60 small minute marks around the circle, so the gap from one minute mark to the next is 360° ÷ 60 = 6°.

Two hands always make two angles, one on each side, and together they add up to 360°. The questions ask for the smaller angle. It is never more than 180°.

Times on the hour

At a time like 4:00, the minute hand points straight up at 12 and the hour hand points exactly at the hour. Count the gaps between the two hands, the short way round, and multiply by 30°.

When the minute hand is not on 12

This is the step that many people miss. The hour hand does not jump from one number to the next. It moves slowly all the time. At 3:30 it is not on the 3: it is halfway between 3 and 4.

In one hour, the hour hand moves from one number to the next: 30°. One hour is 60 minutes, so in one minute it moves 30° ÷ 60 = 0.5°. The minute hand moves 6° every minute. Now measure where each hand is, starting from 12 and going clockwise (the way the hands turn).

1. Minute hand: 6° times the minutes.
2. Hour hand: 30° times the hour, plus 0.5° times the minutes.
3. Take the smaller position away from the bigger one.
4. If the result is more than 180°, take it away from 360°.

How often do the hands meet?

The hands meet when they point in exactly the same direction. You might think this happens once every hour, so 24 times a day. It does not.

In 12 hours, the minute hand goes all the way round 12 times, but the hour hand goes round only once. The minute hand is always chasing the hour hand, and it passes it once for every extra lap it makes. It makes 12 − 1 = 11 extra laps, so the hands meet 11 times in 12 hours.

You can see where the missing meeting went. The hands meet at 12:00, then a little after each hour: at about 1:05, 2:11, 3:16, 4:22, 5:27, 6:33, 7:38, 8:44, 9:49 and 10:55. Between 11:00 and 1:00 they meet only once, at 12:00. There is no meeting "a little after 11", because that one would be at 12:00.