How this instrument works
A clock face is two independent pointers sharing one pivot. The minute hand sweeps a full 360° in 60 minutes — 6° every minute — while the hour hand creeps only 30° per hour, or 0.5° per minute, since it needs twelve hours to complete the same lap. The gap between them opens at a constant relative rate of 6 − 0.5 = 5.5° per minute, and that constant is exactly the 5.5M hiding inside 30H − 5.5M: the whole formula is just that steady drift, measured from wherever the hour hand started at H o'clock.
That drift can, in principle, wrap most of the way around the dial before the minute hand catches back up, so the sheet does not stop at |30H − 5.5M|. Two rays from a common center always split a circle into a pair of arcs that add to 360°, and 'the angle between the hands' by convention names the smaller arc — never more than a straight line's 180°. So the formula takes whichever is smaller, the raw gap or its complement to 360°, the same reflex-angle correction used anywhere two rays share a vertex, from clock puzzles to compass bearings.
That 5.5°-per-minute drift produces a count that surprises almost everyone the first time they meet it: across a 12-hour face the pointers coincide only 11 times, not 12, and sit exactly opposite one another 11 times too, because the minute hand needs 360° ÷ 5.5° ≈ 65.45 minutes to lap the hour hand — not the naive 60. Carry that same relative speed forward and the pointers square up at a right angle 22 times in 12 hours, twice for every lap. None of it needs an actual clock to check; it all falls out of one constant, 5.5° per minute.
- Type the hour on a 12-hour dial into Hour (0-11) — use 0 for both 12 midnight and 12 noon.
- Type the minutes past that hour into Minute (0-59).
- Read Angle between hands for the result; it defaults to degrees but switches to radians or turns without re-entering anything.
- Change either field and watch the angle swing between 0° (hands together) and 180° (hands opposite) as the minutes advance.
Worked example — three o'clock exactly
Set Hour (0-11) to 3 and Minute (0-59) to 0 — a plain three o'clock. The minute hand points straight up at the 12, so its position is 6 × 0 = 0° from noon. The hour hand sits exactly on the 3, its position 30 × 3 + 0.5 × 0 = 90° from noon. The raw gap between them is |90 − 0| = 90°, already under 180°, so the reflex correction never triggers and Angle between hands reports that figure directly.
That 90° is exact, not rounded: in the sheet's internal radians it comes out to 1.5707963267948966, precisely π/2, because 90° × π/180 collapses to a clean half of π with nothing left over. Switch the field's unit to radians and that is the digit string shown; switch to turns and it reads 0.25, a quarter of the way around the dial. Few clock times land on such a tidy figure — most give a jagged decimal, which is the ordinary case, not the exception.
Questions
What is the formula for the angle between clock hands?
θ = |30H − 5.5M|, then 360° minus that value if it exceeds 180°. 30H is the hour hand's position in degrees and 5.5M is how far ahead the minute hand has crept relative to it — 6°/minute for the minute hand minus the hour hand's own 0.5°/minute. At 3:00 this gives |90 − 0| = 90°, no correction needed.
Why does the formula subtract 5.5 times the minutes instead of 6?
Because the hour hand keeps moving too. The minute hand alone would open a 6° gap every minute, but the hour hand simultaneously creeps forward 0.5° every minute in the same direction, so the gap only widens by their difference, 6 − 0.5 = 5.5° per minute — the constant relative speed built into every clock-angle problem.
Why is the reported angle never more than 180°?
Two hands on one dial always split the circle into a pair of arcs that add up to 360°, and 'the angle between the hands' conventionally means the smaller of the two — never more than a straight line. That is why the sheet reports min(raw, 360° − raw); at 6:00 the raw gap and its complement are both exactly 180°, the largest the answer can ever be.
What is the most common mistake when working this out by hand?
Treating the hour hand as frozen on its hour mark. At 2:30 it has actually drifted halfway toward the 3, sitting at 30×2 + 0.5×30 = 75°, not 60°, while the minute hand is at 6×30 = 180°. Ignoring that drift gives a wrong answer of 120° instead of the correct 105°.
Does entering hour 0 handle both midnight and noon correctly?
Yes — the sheet works on a 12-hour dial where the 12 mark is hour 0, so 0 covers both 12 midnight and 12 noon. At 0:00 the hour hand and minute hand both sit at 0°, the raw gap is 0°, and the reported angle is 0°, meaning the hands fully overlap.
How many times a day do clock hands overlap or form a right angle?
Across each 12-hour lap the hands coincide 11 times and sit opposite each other 11 times, because the minute hand gains on the hour hand at a constant 5.5° per minute and needs 360° ÷ 5.5° ≈ 65.45 minutes to lap it — not the 60 minutes intuition suggests. The same drift puts the hands at a right angle 22 times every 12 hours.