What is the angle between clock hands at 3:00?

At exactly 3:00, the hour hand points to 90 degrees (the 3 o'clock position) and the minute hand points to 0 degrees (the 12 o'clock position). The angle between them is exactly 90 degrees. This is a classic example often used in maths problems because the hands form a perfect right angle.

How do you calculate the angle between clock hands?

Use these two formulas. Hour hand angle (from 12 o'clock) = (H mod 12) x 30 + M x 0.5, where H is the hour and M is the minutes. This is because each hour mark is 30 degrees apart (360 / 12) and the hour hand also advances 0.5 degrees per minute. Minute hand angle = M x 6, because the minute hand moves 6 degrees per minute (360 / 60). The angle between the hands is the absolute difference between these two values. If the result is greater than 180 degrees, subtract it from 360 to get the smaller angle.

At what times are the clock hands exactly together (0 degrees)?

The hands coincide (0 degree angle) 11 times in every 12 hours, not 12. Starting from 12:00, the hands meet again at approximately 1:05:27, 2:10:54, 3:16:21, 4:21:49, 5:27:16, 6:32:43, 7:38:10, 8:43:38, 9:49:05, 10:54:32, and then back at 12:00. The minute hand gains on the hour hand at a rate of 5.5 degrees per minute, so each coincidence occurs every 720/11 minutes (approximately 65 minutes and 27 seconds).

Clock Angle Calculator

This calculator works out the exact angle between the hour and minute hands of an analogue clock at any time you choose, a common question in maths classes and clock-reading puzzles. Enter the hour (1 to 12) and the minutes (0 to 59), or click "Use current time" to load the time from your device. It instantly returns the smaller angle between the hands in degrees, along with the reflex angle, the larger arc on the other side, since both always add up to 360 degrees. It also shows the position of each hand measured clockwise from the 12 o'clock mark, and classifies the smaller angle as acute, right, obtuse, straight or coincident. A live clock face diagram redraws the hands to match your chosen time, and a step-by-step working panel shows how the hour hand angle, the minute hand angle and the final difference are worked out, using the standard formula of 30 degrees per hour plus 0.5 degrees per minute for the hour hand, and 6 degrees per minute for the minute hand. Use it to check homework, settle a classic 3 o'clock right-angle puzzle, or see why the hands line up exactly 11 times every 12 hours rather than 12. The results assume a perfectly accurate mechanical clock, so a real one may vary very slightly.

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Standard formula Hour hand: 0.5° per minute • Minute hand: 6° per minute

1. Enter the Time

Hours (1 to 12)

Minutes (0 to 59)

Or use current time

2. Clock Face

Angle Between the Hands

Smaller Angle
75°
Degrees

Reflex Angle

285°

Degrees (larger arc)

Hour Hand

105°

Degrees from 12

Minute Hand

180°

Degrees from 12

Step-by-Step Working

Enter a time above to see the working.

Angle Classification

Smaller angle75°

Angle typeAcute

Reflex angle285°

Both angles sum to360°

Hour hand positionBetween 3 and 4

Minute hand positionPointing at 30 min mark

Relative gain per minute5.5°

Result: Enter a time to calculate the angle between the clock hands.

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

Time	Hour Hand	Minute Hand	Smaller Angle	Type
12:00	0°	0°	0°	Coincident
3:00	90°	0°	90°	Right angle
6:00	180°	0°	180°	Straight line
9:00	270°	0°	90°	Right angle
3:30	105°	180°	75°	Acute
6:30	195°	180°	15°	Acute
12:30	15°	180°	165°	Obtuse

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.