Free · Angle calculators 🕑
Clock-Hand Directional Hour–Minute Angles
Calculate both directed angles between hour and minute hands for a twelve-hour analog clock.
- Formula & worked example
- Private in your browser
- No signup
Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
Inputs and results stay in this browser tab. Bookify does not upload or store the values you enter.
Formula and method
At 3:30, the hour hand is at 105° and the minute hand at 180°, giving directed angles of 75° and 285°.
Hour position = 30h + 0.5m; minute position = 6m; complementary directed angles sum to 360°.
Worked example
Enter these known values and leave the other values blank.
- Hour component (0 represents 12)
- 3
- Minute component
- 30
- Angle from hour to minute hand
- 75 deg
- Angle from minute to hour hand
- 285 deg
Assumptions and limitations
- Use an hour component from zero through eleven; represent twelve as zero. Minute component is at least zero and below sixty.
- The hour hand moves continuously with the minutes. Fractional minutes can represent seconds; the retained advisory note asks for integer components.
- Two directed angles are displayed. When the hands coincide, the captured convention gives 0° for one direction and 360° for the full-turn complement.
- Arithmetic uses 64 significant decimal digits, with separate rounding for displayed values.
Common questions
Does the hour hand stay exactly on the hour number?
The model advances it by 0.5° per minute, so at 3:30 it is halfway between three and four.
References
Bookify requires whole hour components while retaining fractional minutes and the captured continuous hand-position equations.