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.
Related Practice
The Angle Between The Hour And Minute Hands
A clock angle puzzle gives you a time and asks how many degrees apart the two hands are. It sounds like a reading exercise and it is really a small piece of arithmetic, because both hands are moving and you have to know how fast.
The reason it is worth practising is that it has one memorable trap, and once you have fallen into it you never do again.
The Hour Hand Does Not Sit Still
This is the whole puzzle. The hour hand travels continuously, so in the hour between three and four it must cover all 30 degrees separating them. That works out at half a degree per minute.
At 3:40, then, the hour hand is not on the three. It is 20 degrees past it — two thirds of the way to the four. Treating it as parked on the three is the mistake behind more wrong answers than everything else combined, and it is generated here as a named wrong option so you can see exactly what it costs.
The Two Speeds
| Hand | Per hour | Per minute | Why |
|---|---|---|---|
| Minute | 360° | 6° | A full turn every 60 minutes. |
| Hour | 30° | 0.5° | A full turn every 12 hours, and 30° spread across 60 minutes. |
So the hour hand sits at 30H + 0.5M degrees clockwise from twelve, the minute hand at 6M, and the gap between them is the difference. Some books compress that into a single expression, |30H − 5.5M|, which is the same thing with the subtraction done first.
Always Take The Smaller Angle
Two hands make two angles, and they add to 360°. Unless a question says otherwise it wants the smaller one, so a difference of 247.5° is really an answer of 112.5°. That is the second named wrong option on every puzzle here, because giving the reflex angle is a genuine reading of the picture — just not the one being asked for.
Two Facts Worth Knowing
| Arrangement | How often in 12 hours | Note |
|---|---|---|
| Hands overlap | 11 times | Roughly every 65½ minutes, not every hour — there is no overlap between 11 and 12. |
| Hands at a right angle | 22 times | Twice per overlap cycle, so 44 times in a full day. |
| Hands opposite | 11 times | Same cycle as the overlaps, offset by half of it. |
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°.