The angle between the hands, with the hour hand where it really is.
The angle between the hands
With the hour hand where it really is.
at 03:30
75°
The smaller of the two angles. Going the other way round is 285°, and the two add to 360°.
Angle between
75°
the smaller one
Reflex angle
285°
the other way round
Hour hand at
105°
30° per hour plus 0.5° per minute
Minute hand at
180°
6° per minute
The thick hand is the hour hand and the coloured one is the minute hand. At 03:30 the hour hand is 105° round from twelve — past the 3, not on it.
The hour hand is not fixed between the numbers. It moves half a degree per minute, so at 3:30 it sits halfway between 3 and 4 — which is why the answer at 3:30 is 75°, not 90°.
The minute hand moves 6° per minute and the hour hand 0.5°, so they close on each other at 5.5° per minute.
The hands overlap 22 times in 24 hours, not 24 — they line up every 65 5/11 minutes rather than every hour.
Hand positions are exact fractions of a degree, so half-degree-per-minute drift is carried rather than rounded.
What this tool shows
At 3:30 the hands are 75° apart, not 90°. The hour hand has moved halfway towards 4 by then — it does not wait on the 3 until the hour is up. That half-degree per minute is the whole difficulty of the question.
The angle between the hands at any time
The reflex angle going the other way
Where each hand actually points, in degrees
Whether the hands overlap, are opposite, or are square
Why the hour hand moves between the numbers
How often the hands line up
The face drawn Both angles given Each hand’s position Exact, not rounded
Exact fractions of a degree; hands move continuously.
Updated 7 September 2026 · Works in any browser, no installation
Work out where each hand really is, then subtract. The minute hand is at 6° per minute. The hour hand is at 30° per hour plus half a degree per minute. At 3:30 that puts them at 180° and 105°, which is 75° apart — not the 90° you get by leaving the hour hand on the 3.
At a glance
Formula shown
Hour hand at 30H + 0.5M degrees, minute hand at 6M degrees, so the difference is |30H \u2212 5.5M|. If that exceeds 180\u00b0, the angle usually wanted is 360\u00b0 minus it.
Scenario support
A puzzle or interview question about clock hands; a maths exercise on angles; checking when the hands will next be square or opposite.
Educational estimate
Planning support from the values you enter — not professional advice.
The 3:30 trap
Ask most people for the angle at 3:30 and they say 90°. The minute hand is on the 6, the hour hand is on the 3, and those are three hours apart at 30° each.
The hour hand is not on the 3. It is half an hour into its journey towards the 4, so it sits 15° past the 3, at 105°. The minute hand is at 180°. The gap is 75°.
This is the whole content of the problem. The minute hand is easy — it points at the number. The hour hand drifts continuously, and forgetting that drift is the single error that makes clock angle questions worth asking.
You can see it on any real clock: at half past, the hour hand is visibly between two numbers.
The formula, derived
The minute hand. A full turn in 60 minutes is 360/60 = 6° per minute. At M minutes it is at 6M degrees from twelve.
The hour hand. A full turn in 12 hours is 30° per hour. But it also creeps through each hour: 30° over 60 minutes is 0.5° per minute. At H hours and M minutes it is at 30H + 0.5M.
The difference. |30H + 0.5M − 6M| = |30H − 5.5M|. That 5.5 is the rate at which the hands close on each other — 6 minus 0.5.
At 3:30: |90 − 165| = 75. At 9:45: |270 − 247.5| = 22.5. The half-degrees are why the answers so often end in .5, and why this page keeps them as exact fractions.
Which angle do you mean
Two hands at a point always make two angles, and they add to 360°.
At 3:30 they are 75° apart one way and 285° the other. Almost every question wants the smaller one, which is the convention this page reports first — but the reflex angle is a real answer to a real reading of the question, so it is given too.
The rule: if |30H − 5.5M| comes out above 180°, subtract it from 360° to get the smaller one. At 1:00 the raw difference is 30° and that is already the smaller. At 11:00 it is 330°, and the answer wanted is 30°.
How often the hands overlap
Twenty-two times in twenty-four hours, not twenty-four. This surprises people, and the reason is worth following.
The hands coincide when 30H = 5.5M. Solving for consecutive overlaps gives a gap of 720/11 = 65 5/11 minutes — just over an hour and five minutes. In twelve hours that fits eleven times, not twelve.
So the overlaps are at 12:00, about 1:05:27, about 2:10:55, and so on. There is no overlap in the eleven o’clock hour at all: the hands meet at 12:00 instead.
The same argument gives the other alignments. The hands are opposite eleven times in twelve hours, and at right angles twenty-two times.
The classic interview question
“What is the angle between the hands at 3:15?” is a standing favourite, and the expected wrong answer is 0°.
At 3:15 the minute hand is at 90°, on the 3. The hour hand is at 90 + 7.5 = 97.5°. So the angle is 7.5°, not zero — they are close but not together.
The follow-up is usually “when ARE they together, between 3 and 4?” Setting 90 + 0.5M = 6M gives M = 180/11 ≈ 16.36 minutes, so at about 3:16:22.
What is really being tested is whether you noticed the hour hand moves. Everything after that is arithmetic.
Real clocks that tick
This page models an ideal analogue clock whose hands sweep continuously. Most real clocks do not.
A standard quartz movement steps the second hand once a second and advances the minute hand in jumps. Between jumps the minute hand is stationary, so the true angle differs from the model for most of every minute.
Mechanical movements and “sweep” quartz movements do move continuously, and match this model closely. So do the analogue clock faces drawn by software, which is why puzzle answers agree with the formula.
It matters only at the level of seconds. For any question phrased in whole minutes, the continuous model is the one intended.
Sources and methodology
The geometry is elementary; these are the references for the conventions and the timekeeping.
Method. Hand positions are exact fractions of a degree, so the hour hand’s half-degree-per-minute movement is carried rather than rounded — which is what makes 9:45 come out as exactly 22.5°. Both the smaller angle and its reflex are reported, since the question is ambiguous without saying which. The suite checks the result against the closed form |30H − 5.5M| at every one of the 720 distinct times on a twelve-hour face, which is exhaustive rather than sampled. That engine is verified on every change against 82 hand-written assertions, including agreement with the closed form at all 720 times on the face, and that the reported angle and its reflex always add to exactly 360°. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Coterminal AngleThe principal angle between 0° and 360°, the nearest negative one, and the family θ + 360n they all belong to — reduced by exact division, not by looping.
Reference AngleThe acute angle to the x-axis — never the y-axis — with the quadrant rule written out using your own numbers and the sign of each trig function beside it.
Complementary and Supplementary AnglesBoth at once, plus the explement — and an honest answer when there is none, since an obtuse angle has no complement and a negative number is not one.
Central AngleLeave one of radius, arc and angle blank and the page solves for it — with the sector drawn, plus chord, sector area and segment area from the same angle.
ModuloAll three conventions at once, because −7 mod 3 is −1 in JavaScript and 2 in Python and a page that gives only one of those is wrong for half its readers.
Time PercentageWork out what percentage one span of time is of another, or the percentage increase or decrease between an old and a new duration.
A clock face is arithmetic modulo 12 and modulo 360 at once — the Modulo Calculator covers the wrap-around that both of those depend on.
Educational use disclaimer
This is an educational tool. It models an ideal analogue clock whose hands move continuously; a stepping quartz movement jumps the minute hand instead, which changes the answer between ticks.
Published the clock angle page with the hour hand at 30° per hour plus half a degree per minute, which is the whole content of the problem and the reason 3:30 is 75° rather than 90°.
The face is drawn because a number alone does not say which of the two angles at a point is being reported; both the smaller and the reflex are given.
Checked against the closed form |30H − 5.5M| at every one of the 720 distinct times on a twelve-hour face, which is exhaustive rather than sampled.
Add this calculator to your site
Responsive embed — and private: nothing your visitors type leaves their browser.