Clock Angle Puzzles
A time on a clock face, and one question: how far apart are the hands? The catch that trips almost everyone is that the hour hand keeps moving — at twenty to four it is nowhere near the three. Every attempt here is worked through in full.
Also known as clock angle problems, hands of a clock and angle between clock hands.
The Angle Between The Hands
A clock angle puzzle asks for the angle between the hour and minute hands at a stated time. The minute hand moves 6° per minute; the hour hand moves 0.5° per minute — it never sits still, which is the fact most wrong answers forget.
How To Solve Clock Angles Puzzles
- Minute hand: minutes × 6°.
- Hour hand: hour × 30° + minutes × 0.5°.
- Take the difference; if it exceeds 180°, take 360° minus it.
- Sanity-check against a familiar time before answering.
The hour hand moves with the minutes. At 3:30 it is halfway between 3 and 4, not on the 3 — half of all wrong clock answers forget exactly this.
| Family | What it covers |
|---|---|
| On the hour | undefined |
| Half past | undefined |
| Quarter hours | undefined |
| Multiples of five | undefined |
| Awkward minutes | undefined |
| Hands close together | undefined |
| Hands far apart | undefined |
| Mixed times | undefined |
One Formula, Every Time
Both hands are just speeds: 6° a minute and 0.5° a minute from twelve. The explanations here compute both hands separately and then take the difference, so every answer can be checked line by line against your own arithmetic.
Clock Angle Puzzles Frequently Asked Questions
How do you calculate the angle between clock hands?
Work out each hand separately and subtract. The minute hand moves 6 degrees per minute, so multiply the minutes by six. The hour hand moves 30 degrees per hour plus half a degree per minute, so multiply the hour by 30 and add half the minutes. Take the difference between the two, and if it comes to more than 180 degrees subtract it from 360 to get the smaller angle.
Why does the hour hand move between the numbers?
Because it travels continuously rather than jumping. In the hour between three and four it has to cover the whole 30 degrees from one number to the next, which works out at half a degree per minute. At half past three it is exactly halfway between the three and the four, not sitting on the three. Forgetting this is the single commonest error in these puzzles.
What is the formula for clock angles?
The angle of the hour hand is 30H plus 0.5M, and the angle of the minute hand is 6M, both measured clockwise from twelve. The gap is the absolute difference of those two, and the answer is that value or 360 minus it, whichever is smaller. Written as one line it is often given as the absolute value of 30H minus 5.5M.
How often are the clock hands at right angles?
Twenty-two times in twelve hours, so forty-four times a day. The hands form a right angle slightly less often than you might expect because the minute hand has to gain a full 180 degrees on the hour hand between one right angle and the next but one, and the hour hand keeps moving while it does.
When do the hands overlap exactly?
Eleven times in twelve hours, roughly every 65 and a half minutes rather than every hour. They coincide at twelve, then at about 1:05, 2:11, 3:16 and so on. There is no overlap between eleven and twelve, which is why it is eleven times and not twelve.
Are the answers always whole numbers?
No, and that is worth expecting. Because the hour hand moves half a degree a minute, any odd number of minutes gives an answer ending in .5. The families here labelled as whole-number times stick to minutes that come out exactly; the rest do not, and the arithmetic is the same either way.
What is the angle between the hands at 3:15?
Seven and a half degrees, not zero — and this is the classic trap. The minute hand is exactly on the 3, but the hour hand has spent fifteen minutes drifting past it at half a degree a minute, so it sits at 97.5° while the minute hand sits at 90°. Using the formula, |30 × 3 − 5.5 × 15| = |90 − 82.5| = 7.5°.