Clock Angle Calculator: Angle Between Hour and Minute Hands
The angle between clock hands depends on the position of both hands at a chosen time. The minute hand moves around the dial once an hour, while the hour hand moves gradually between hour marks. This calculator finds the smaller angle, the other angle around the circle, and each hand’s position measured clockwise from 12.
How to calculate the angle between clock hands
A clock face is a full circle of 360°. Twelve equally spaced hour marks are 30° apart, and 60 minute marks are 6° apart. Let H be the hour on a 0-to-11 scale, treating 12 as 0, and M the minutes.
Hour-hand angle = 30H + 0.5M
Minute-hand angle = 6M
Find the absolute difference D between the two hand angles. The smaller angle is min(D, 360 − D). It is always between 0° and 180°. All hand positions are measured clockwise from the top of the clock.
Worked examples: 3:00, 3:30, and 6:00
At 3:00, the hour hand is at 90° and the minute hand is at 0°. Their smaller angle is 90°, a right angle.
At 3:30, the hour hand has moved halfway from 3 to 4. Its angle is 30 × 3 + 0.5 × 30 = 105°. The minute hand is at 6 × 30 = 180°. The difference is 75°, rather than 90°.
At 6:00, the hands point in opposite directions. The smaller angle is 180°, a straight angle. At 12:00, the hands overlap and the smaller angle is 0°.
At 11:59, the hour hand is at 359.5° and the minute hand at 354°. Their smaller angle is 5.5°, showing why near-noon calculations must include the hour hand’s gradual movement.
How seconds affect the clock angle
For an ideal clock with continuously moving hands, seconds add a small amount to both hand positions. With S representing seconds, use:
Hour-hand angle = 30H + 0.5M + S ÷ 120
Minute-hand angle = 6M + 0.1S
At 3:00:30, the hour hand is at 90.25° and the minute hand is at 3°. The smaller angle is 87.25°. Leaving seconds at 00 gives the standard minute-only calculation. Displayed results are rounded to at most three decimal places.
What is the other angle around the clock?
Two hands divide a circle into two angles. Subtract the smaller angle from 360° to find the other one. At 3:30, the angles are 75° and 285°. At 6:00, both are 180°. When the hands overlap, the smaller angle is 0° and the other is a full turn of 360°.
A smaller angle below 90° is acute, exactly 90° is right, between 90° and 180° is obtuse, and exactly 180° is straight. Overlapping hands are labeled separately.
Why does the hour hand move between hour marks?
The hour hand completes 360° in 12 hours. It therefore travels 30° per hour, 0.5° per minute, and 1/120° per second. The minute hand travels 6° per minute and 0.1° per second.
Ignoring the hour hand’s movement is a common source of mistakes. At 9:30, it is halfway between 9 and 10, at 285°, rather than exactly at 270°. With the minute hand at 180°, the smaller angle is 105°.
Frequently asked questions
Does AM or PM change the angle?
No. The same 12-hour clock positions repeat in the morning and evening. For example, 3:30 AM and 3:30 PM both give 75°.
How do I enter a 24-hour time?
Convert it to the equivalent 12-hour hour. For 15:30, choose hour 3 and minute 30. Midnight and noon both use hour 12.
Can the smaller angle exceed 180°?
No. The smaller of the two angles around a circle cannot exceed half a turn. The other angle can be greater than 180°.
Do the hands overlap once every hour?
The minute hand gains on the hour hand at 5.5° per minute. Overlaps occur approximately every 65.455 minutes, so there are 11 overlaps in a 12-hour cycle.
Does the diagram show the second hand?
No. The calculation compares only the hour and minute hands. The seconds input adjusts their positions for the selected time.