Clock Angle Calculator

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)

Taking the minimum gives the smaller of the two angles the hands cut the face into; 360° minus it gives the larger one. Watch the wording here, because a lot of textbooks are sloppy about it. The smaller angle is usually acute, but not always: at 6:00 it is 180°, and at 4:00 it is 120°, which is obtuse. Only when that smaller angle comes out below 180° is the larger one genuinely a reflex angle.

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

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.