Clock Angle Calculator

How to Calculate the Angle Between Clock Hands

An analogue clock face is a full circle of 360 degrees. The clock has 12 hour markers, so each hour mark is 30 degrees apart (360 / 12 = 30). The minute hand sweeps the full 360 degrees in 60 minutes, so it moves 6 degrees per minute (360 / 60 = 6). The hour hand moves more slowly: it takes 12 hours to complete one rotation, which means it moves 0.5 degrees per minute (360 / 720 = 0.5).

The Formula

Given time H:M (where H is the hour in 12-hour format and M is the minutes):

• Hour hand angle = (H mod 12) × 30 + M × 0.5
• Minute hand angle = M × 6
• Difference = |hour angle − minute angle|
• Smaller angle = if the difference exceeds 180°, use 360° minus the difference

The key insight for the hour hand formula is that the hour hand does not jump sharply between hour marks. At 3:30, for example, the hour hand is halfway between 3 and 4, not sitting on the 3.

Worked Example: 3:30

At 3:30 (the default values in this calculator):

• Hour hand angle = (3 mod 12) × 30 + 30 × 0.5 = 90 + 15 = 105°
• Minute hand angle = 30 × 6 = 180°
• Difference = |105 − 180| = 75°
• 75° is less than 180°, so the smaller angle is 75 degrees
• The reflex angle = 360 − 75 = 285 degrees

Common Clock Angles

The Relative Speed of the Hands

The minute hand moves at 6 degrees per minute and the hour hand moves at 0.5 degrees per minute. The relative speed of the minute hand with respect to the hour hand is 5.5 degrees per minute. This means the minute hand gains a full lap (360 degrees) on the hour hand every 720/11 minutes, or approximately 65 minutes and 27 seconds. That is why the hands meet 11 times in 12 hours, not 12.

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Method: Standard clock angle formula. Hour hand angle = (H mod 12) × 30 + M × 0.5; Minute hand angle = M × 6; Smaller angle = min(|difference|, 360 − |difference|). Angles measured clockwise from the 12 o'clock position.

This calculator uses the standard mathematical formula for clock hand angles. Results are expressed in decimal degrees. For times including seconds, the second hand moves 6 degrees per second and the minute hand advances 0.1 degrees per second.