Angle Between the Hands of a Clock: The 30H − 5.5M Formula

Clock questions become arithmetic once you know the speed of each hand in degrees per minute. A full turn is 360 degrees.

• The minute hand makes a full turn in 60 minutes: 360 ÷ 60 = 6° per minute.
• The hour hand moves from one number to the next, 30°, in 60 minutes: 0.5° per minute.
• So the minute hand gains 5.5° per minute on the hour hand.

The most common mistake is forgetting that the hour hand keeps moving between the numbers. At 3:30 the hour hand is halfway between 3 and 4, not on the 3, so the angle is not 90°.

At H hours and M minutes, the hour hand has turned 30H + 0.5M degrees from 12, and the minute hand has turned 6M degrees. The angle between them is the difference:

Use H as the hour on a 12-hour dial. At 12 o'clock, H is 0 (or 12, since 360° is the same as 0°).

To find when the hands make an angle θ between H and H + 1 o'clock, solve 30H − 5.5M = ±θ for M. Each sign can give one valid time; keep only the values of M between 0 and 60.

Because the minute hand gains 5.5° a minute, it catches up a full 360° every 360 ÷ 5.5 = 720/11 minutes, about 65 minutes 27 seconds. In 12 hours (720 minutes) that happens 11 times.

For a full day, double the 12-hour figure: the hands coincide 24 × 11 ÷ 12 = 22 times.

Right angles occur twice in each gap between meetings, once as the gap widens through 90° and once as it closes through 270°, so there are 2 × 22 = 44 a day.

A related question shows a clock in a mirror and asks for the real time, or the reverse. A vertical mirror swaps left and right, so a hand pointing to 3 appears to point to 9. The real time and the mirror time always add up to 12:00. To find one from the other, subtract from 11:60, which is the same as 12:00 but avoids borrowing.

For a time after 11:00 and before 12:00, subtract from 23:60 instead so that the answer stays on the 12-hour dial.

• ✘ Keeping the hour hand fixed on the hour number.
• ✘ Giving the reflex angle when the question wants the smaller one, or the reverse.
• ✘ Counting 12 or 24 meetings instead of 11 or 22.
• ✘ Keeping a value of M that is negative or 60 and above when solving for times.

Clocks that run fast or slow are a separate question type; see fast and slow clocks. Day-of-the-week questions are covered in calendar problems by odd days.

Why does the minute hand gain 5.5 degrees a minute on the hour hand?

The minute hand turns 360 degrees in 60 minutes, which is 6 degrees a minute. The hour hand turns 30 degrees in 60 minutes, which is half a degree a minute. The difference, 5.5 degrees a minute, is how fast the gap between them changes.

Why do the hands meet 11 times in 12 hours, not 12?

The minute hand laps the hour hand once every 720/11 minutes, about 65.45 minutes, not every 60. In 720 minutes that gives 11 meetings. The meeting between 11 and 1 happens exactly at 12 o'clock, so no separate meeting falls between 11 and 12.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.