How to Calculate the Angle Between Clock Hands
Calculate the angle between clock hands using |30H - 5.5M|, with the 180-degree correction rule and a worked example for interviews.
Expected Interview Answer
The angle between the hour and minute hands at H hours and M minutes is |30×H − 5.5×M| degrees, taking the smaller of that value and 360 minus that value if it exceeds 180.
The minute hand moves 360 degrees in 60 minutes, so it sweeps 6 degrees per minute. The hour hand moves 360 degrees in 12 hours (720 minutes), so it sweeps 0.5 degrees per minute, which also means it drifts 30 degrees per hour plus an extra 0.5 degrees for every minute past the hour. Computing each hand’s position from 12 o’clock separately — 30×H + 0.5×M for the hour hand and 6×M for the minute hand — and subtracting gives the same |30H − 5.5M| formula after simplification, since 6M − (30H + 0.5M) = 5.5M − 30H. Because a clock face is circular, always take the absolute value and, if it is over 180 degrees, subtract it from 360 to get the smaller angle between the hands.
- One formula, |30H − 5.5M|, replaces separate hand-position tracking
- The 180-degree correction always yields the smaller, more natural angle
- Generalizes directly to “when are the hands at angle X” reverse problems
AI Mentor Explanation
Two fielders running between wickets at different fixed speeds — one covering a set distance per second, the other covering it faster — separate over time exactly like clock hands, whose gap opens at a fixed rate because the minute hand moves 6 degrees per minute while the hour hand moves only 0.5 degrees per minute. Subtracting the slower fielder’s progress from the faster one’s at any instant gives their separation, precisely how |30H − 5.5M| gives the hands’ angular separation at any time.
Worked example
Formula
- |30H − 5.5M|
Substitute
- |30×3 − 5.5×40| = |90 − 220|
Result
- = 130 degrees
Step-by-Step Explanation
Step 1
Compute hour-hand angle
Hour hand position = 30×H + 0.5×M degrees from 12.
Step 2
Compute minute-hand angle
Minute hand position = 6×M degrees from 12.
Step 3
Take the absolute difference
This simplifies directly to |30H − 5.5M|.
Step 4
Correct for the reflex angle
If the result exceeds 180, subtract it from 360 for the smaller angle.
What Interviewer Expects
- Correct derivation of the 5.5 degrees-per-minute relative speed
- Correct application of |30H − 5.5M| rather than tracking hands separately from scratch
- Correct 360-minus-angle correction when the raw result exceeds 180
- Ability to solve the reverse problem: given an angle, find the time(s)
Common Mistakes
- Forgetting the hour hand also moves within the hour (using only 30×H)
- Not taking the absolute value, leading to a negative angle
- Forgetting to subtract from 360 when the computed angle exceeds 180
- Confusing degrees-per-minute rates between the two hands
Best Answer (HR Friendly)
“I compute where each hand actually points in degrees from 12: the hour hand at 30 times the hour plus half a degree per minute, and the minute hand at 6 degrees per minute. Subtracting those two positions and taking the absolute value gives |30H − 5.5M|, and if that number is more than 180 degrees, I subtract it from 360 to report the smaller, standard angle between the hands.”
Follow-up Questions
- How would you find all times in a 12-hour period when the hands are exactly opposite (180 degrees)?
- How would you find all times when the hands overlap exactly?
- How does the formula change on a 24-hour analog clock face?
- How would you compute the angle between the minute and second hands instead?
MCQ Practice
1. What is the angle between the hands at 6:00?
|30×6 − 5.5×0| = |180 − 0| = 180 degrees, the hands point in exactly opposite directions.
2. What is the angle between the hands at 9:15?
|30×9 − 5.5×15| = |270 − 82.5| = 187.5 degrees, which exceeds 180, so the smaller angle is 360 − 187.5 = 172.5 degrees.
3. At what approximate rate does the angle between the hands change per minute (away from a coincidence point)?
The minute hand gains on the hour hand at 6 − 0.5 = 5.5 degrees per minute, which is the relative angular speed.
Flash Cards
Core clock-angle formula? — |30H − 5.5M| degrees, corrected to at most 180.
Minute hand speed? — 6 degrees per minute.
Hour hand speed? — 0.5 degrees per minute.
Relative speed between the hands? — 5.5 degrees per minute (6 − 0.5).