Two hands, two speeds, one perennial puzzle: what angle do the clock’s hands make at a given time? The minute hand races at 6 degrees a minute while the hour hand creeps at half a degree — and that creep is where every wrong answer is born. This clock angle calculator takes any hour and minute and returns the smaller angle between the hands (and the larger reflex angle going the other way), with the hand positions shown so the geometry is checkable at a glance.
The Two Speeds That Run the Puzzle
Measure angles clockwise from 12. The minute hand completes 360° in 60 minutes: 6° per minute, position 6M. The hour hand completes 360° in 12 hours: 30° per hour plus a creep of 0.5° per minute, position 30H + 0.5M. The angle between them is the absolute difference, trimmed to at most 180°: angle = |30H − 5.5M|, folded into the smaller of the two arcs. Everything in this problem family — coincidences, oppositions, right angles — is bookkeeping on those two speeds and their difference of 5.5° per minute, the rate at which the minute hand gains on the hour hand.
Worked Example 1: 3:15 — the Famous Trap
At 3:15 the minute hand points exactly at the 3 — where intuition parks the hour hand too, answering 0°. Wrong: by 3:15 the hour hand has crept a quarter of the way toward 4, sitting at 30 × 3 + 0.5 × 15 = 97.5°, while the minute hand is at 90°. The angle is 7.5° — this page’s default calculation and the single most-missed clock answer anywhere. If a puzzle smells like a trick, the trick is always the hour hand’s creep.
Worked Examples 2 and 3: Straight Lines and Right Angles
At 6:00 the hands oppose exactly: hour at 180°, minute at 0°, angle 180°. At 12:30 the hour hand has crept to 15° while the minute hand points down at 180°, giving 165° — not the 180° a static reading suggests. Right angles occur 22 times a day, not 24 and not 48: the hands are perpendicular roughly every 32 minutes 44 seconds, and two of the expected slots are “absorbed” near 3:00 and 9:00 where the events merge across the hour boundary.
How Often Do the Hands Meet?
The minute hand gains 5.5° per minute, so it laps the hour hand every 360 ÷ 5.5 = 65.45 minutes — 65 minutes 27 seconds. Across 12 hours the hands coincide 11 times, not 12: the twelfth meeting is the starting position of the next cycle (12:00 itself). The full daily census of special configurations: 11 coincidences, 11 oppositions, 22 right angles per 12-hour cycle. Each figure is the same 5.5° gain rate wearing a different costume, and each is a favorite interview and puzzle-book question precisely because the naive answer is off by one.
The Reflex Angle
Two arcs always join the hands: the smaller angle (at most 180°) and the reflex angle completing the circle (360° minus the smaller). Convention reports the smaller — “the angle between the hands” means the tight one unless a puzzle specifies otherwise — but this calculator prints both, because oriented problems (bearings, rotations, clock-arithmetic riddles) sometimes want the long way round. At 3:15: 7.5° the short way, 352.5° the long way.
Common Mistakes
Freezing the hour hand on the hour numeral is the founding error — it moves 0.5° every minute, 30° every hour, and no correct answer ignores it. Using 6° per minute for the hour hand (that is the minute hand’s speed) scrambles the roles. Forgetting to fold answers above 180° into the smaller angle reports reflex values as the headline. And in 24-hour inputs, reduce the hour modulo 12 first — 15:15 is 3:15 on the dial, creep included.
Seconds Hands and Other Extensions
Add the second hand (6° per second, racing a hundred times faster than the minute hand’s creep) and triple coincidences become vanishingly rare: all three hands align only at 12:00:00 in each cycle — the gain rates never otherwise synchronize. Puzzle books also ask for the angle after a time swap (“when do the hands exchange places?”) — the answer, about every 65 minutes, is again the 5.5° gain rate in disguise.
Related Tools
More angle play lives in the Math Calculators hub — try the Heron’s Formula Calculator for angle-free triangle areas and the Dot Product Calculator for angles between vectors.
Frequently Asked Questions
What is the angle between the clock hands at 3:15?
7.5 degrees — not zero. The minute hand is at 90 degrees but the hour hand has crept to 97.5 degrees, a quarter of the way from 3 toward 4.
How do you calculate the clock angle for any time?
Hour hand position = 30 × hour + 0.5 × minutes; minute hand = 6 × minutes. Take the absolute difference and, if it exceeds 180, subtract it from 360. Compactly: angle = |30H − 5.5M|, folded to at most 180.
How many times a day do the clock hands overlap?
22 times a day — 11 per 12-hour cycle, roughly every 65 minutes 27 seconds. Not 24: each cycle’s twelfth meeting is the next cycle’s starting 12:00 position.
How many times are the hands at right angles in 12 hours?
22 times — about every 32 minutes 44 seconds. The naive guess of 24 double-counts the merged events near 3:00 and 9:00.
What is the angle at 6:00?
180 degrees — the hands point in exactly opposite directions, the maximum possible separation.
Why does the hour hand move between the numerals?
Because it traverses the full dial in 12 hours continuously: 30 degrees per hour is 0.5 degrees per minute. Analog hands never jump (except in novelty clocks); they creep, and the creep is the whole puzzle.
What is a reflex angle on a clock?
The larger of the two arcs between the hands — 360 degrees minus the smaller angle. Reports default to the smaller angle; the reflex figure matters for oriented and bearing-style problems.