Clock Angle Calculator (angle between the hour and minute hands)

Understanding the clock angle calculator — concept and principle

This calculator finds the angle between the hour and minute hands of an analogue clock at any time, both the smaller angle and the reflex angle, plus the position of each hand and the next times the hands overlap or form a straight line. Use it for aptitude-test puzzles, geometry homework or curiosity.

The hour hand also creeps 0.5° per minute, so it sits past the 10.

How it is calculated

Minute hand = 6° × m (from 12 o'clock)

Hour hand = 30° × h + 0.5° × m (h on a 12-hour clock)

Angle = |30h − 5.5m|; if it is over 180°, the smaller angle is 360° minus it

The minute hand gains 5.5° per minute on the hour hand, so the hands overlap every 720/11 ≈ 65.45 minutes and are opposite halfway between overlaps.
Times are shown as h:mm:ss.

The principle

Both hands turn at constant speeds. The minute hand makes a full 360° turn in 60 minutes, 6° per minute. The hour hand makes 360° in 12 hours, 30° per hour or 0.5° per minute, so it does not jump from number to number but creeps forward as the minutes pass. The angle between them is the difference of their positions. Since the minute hand catches up 5.5° every minute and must gain a full 360° to lap the hour hand, overlaps come every 360 ÷ 5.5 = 720/11 minutes, 11 times in 12 hours.

Worked example

Default input: 10:10.

Minute hand = 6 × 10 = 60° from 12.
Hour hand = 30 × 10 + 0.5 × 10 = 305° from 12.
Difference = |305 − 60| = 245°, which is over 180°, so the smaller angle = 360 − 245 = 115°; reflex angle 245°.
Next overlap: 10 × 720/11 = 654.55 minutes after 12:00, which is 10:54:33.
Next straight line: 9.5 × 720/11 = 621.82 minutes after 12:00, which is 10:21:49.

How to read the result

0° means the hands overlap and 180° means they point in opposite directions. Many puzzle answers assume a "smooth" hour hand, as here; if a question treats the hour hand as fixed on the hour, the answer will differ by 0.5° per minute. Watch advertisements often show about 10:10; a common explanation is that the hands then frame the maker's name.

Common mistakes

Forgetting that the hour hand moves during the hour.
Using 24-hour values without converting to 12-hour.
Giving the reflex angle when the smaller angle is asked.
Assuming the hands overlap exactly every hour.

Keep in mind: The calculation assumes an ideal clock whose hands move continuously. Clocks with jumping second or minute hands may differ slightly.

Frequently asked questions

What is the angle between the hands at 3:15?

7.5°, because the hour hand has moved a quarter of the way from 3 to 4.

How many times do the hands overlap in a day?

22 times, 11 in each 12 hours.

Why do overlaps not happen every hour?

The hour hand moves too, so the minute hand needs about 65.45 minutes to catch it again.

Sources: Standard clock-hand rates: 6° per minute for the minute hand, 0.5° per minute for the hour hand

Checked 9 Nov 2026. Standard clock-hand rates: the minute hand turns 360° per 60 minutes (6° per minute) and the hour hand 360° per 12 hours (0.5° per minute); the overlap period 720/11 minutes follows from their difference of 5.5° per minute.

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