Clock Angle Calculator
Calculate the angle between the hour and minute hands of an analog clock at any time.
Find acute and reflex angles for any HH:MM combination.
Clock Angle Calculator
This calculator finds the angle between the hour hand and minute hand of an analog clock at any given time. The problem is a classic in math puzzles, programming interviews, and elementary geometry.
Hand Speeds
A full circle is 360 degrees and is divided across the clock face.
- The minute hand sweeps 360° in 60 minutes — that is 6° per minute.
- The hour hand sweeps 360° in 12 hours (720 minutes) — that is 0.5° per minute.
The hour hand moves continuously, not in jumps. At 3:30 it is halfway between 3 and 4, not pointing exactly at 3.
Formulas
Position of minute hand from 12 o’clock:
- M_angle = 6 × M (where M is the minute, 0–59)
Position of hour hand from 12 o’clock:
- H_angle = 30 × (H mod 12) + 0.5 × M (where H is the hour)
Angle between hands:
- raw = |H_angle − M_angle|
- angle = min(raw, 360 − raw)
The minimum gives the acute (smaller) angle. The reflex angle is 360° minus that value.
Worked Example — 3:15
- Minute angle: 6 × 15 = 90°
- Hour angle: 30 × 3 + 0.5 × 15 = 97.5°
- Difference: |97.5 − 90| = 7.5° (acute)
- Reflex: 352.5°
Special Times
| Time | Acute Angle |
|---|---|
| 12:00 | 0° (overlap) |
| 6:00 | 180° (straight line) |
| 3:00 | 90° (right angle) |
| 9:00 | 90° (right angle) |
Hands Overlap
The hands overlap exactly 11 times every 12 hours, every 65.4545 minutes (or 1h 5m 27.27s). The hands form a straight line (180°) 11 times every 12 hours as well.
Why It Matters
Beyond puzzle value, clock-angle problems appear in robotics (rotation tracking), gaming (clockwise indicators), and as a teaching tool for modular arithmetic and continuous motion.