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, so 6° per minute.
• The hour hand sweeps 360° in 12 hours, which is 720 minutes, so 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

Note the column heading. Two of those five are not acute angles at all, which is exactly the sloppiness the paragraph above warns about, and it is why this calculator says “smaller angle” rather than “acute angle” in its results.

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.