Angle Between Hands of a Clock with Formula Examples
Angle Between Hands of a Clock questions require the positions of the hour and minute hands at a given time. Use their angular movements from 12 o’clock, find the absolute difference, and then select the smaller angle. This page covers the formula, special cases, shortcuts and solved angle between hands of a clock examples.
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What Is the Angle Between the Hands of a Clock?
The minute hand moves 6° per minute, while the hour hand moves 0.5° per minute. At h:m, the hour-hand position is 30h + 0.5m degrees and the minute-hand position is 6m degrees, where h is taken from 0 to 11. If their difference is d, the required smaller angle is min(d, 360 − d).
Angle Between Hands of a Clock Video Lessons
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Clock Angle Between Hands
Learn how to calculate the angle between a clock’s hour and minute hands at any given time using the standard clock-angle formula.
Angle Between Hands of a Clock Formula & Tricks
Important Formulas
The hour hand moves 30° in one hour and 0.5° in one minute.
The minute hand moves 360° in 60 minutes, or 6° per minute.
Subtract the minute-hand position from the hour-hand position and take the absolute value.
If A is greater than 180°, subtract it from 360° to obtain the smaller angle.
Quick Tricks
The minute hand gains 5.5° on the hour hand every minute. Therefore, calculate the separation directly as |30h − 5.5m|.
The usual clock angle means the smaller angle. When the calculated difference exceeds 180°, use 360° minus that difference.
At h:00, the minute hand is at 12 and the hour hand is h × 30° from 12.
Angle Between Hands of a Clock Concepts
Calculating the Angle at a Given Time
Use h as the hour on a 12-hour clock. For times after 12, reduce the hour modulo 12. For example, 2:40 gives A = |30 × 2 − 5.5 × 40| = |60 − 220| = 160°.
Exact-Hour Clock Angles
The minute hand lies at 12, and each hour mark is 30° apart. Thus, the hour number directly determines the hour-hand position. For 10:00, the initial difference is 300°, so the smaller angle is 60°.
Coinciding Hands
Between two consecutive hour marks, the hands overlap once. For example, after 4 o’clock, they coincide at 4 + 240/11 minutes, which is approximately 4:21:49.
Opposite Hands
Between two consecutive hour marks, the hands become opposite once. For the interval after 4 o’clock, 30 × 4 − 5.5m = −180 gives m = 480/11 minutes, approximately 4:43:38.
Finding a Time for a Given Angle
For an angle θ after h o’clock, solve 30h − 5.5m = θ or 30h − 5.5m = −θ, keeping only solutions with 0 ≤ m < 60. Different hour intervals can produce multiple times for the same angle.
Angle Between Hands of a Clock Visual Guide & Data
Use the table and chart below to understand the data better.
Angular Movement of Clock Hands
The standard angular speeds and positions used in clock-angle calculations.
| Hand or quantity | Movement or formula | Value |
|---|---|---|
| Minute hand | Movement per minute | 6° |
| Hour hand | Movement per minute | 0.5° |
| Hour hand | Movement per hour | 30° |
| Relative movement | Minute hand gains on hour hand per minute | 5.5° |
Note: Angles are measured clockwise from the 12 o’clock position.
Clock Angle at 3:20
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Angle Between Hands of a Clock: Revision Points
Use these formulas and rules for quick clock-angle calculations.
- Hour-hand position at h:m = 30h + 0.5m degrees.
- Minute-hand position at h:m = 6m degrees.
- Angle difference = |30h − 5.5m| degrees.
- The smaller angle is min(d, 360° − d).
- At an exact hour, each hour mark represents 30°.
- The hands coincide after h o’clock at 60h/11 minutes.
- The hands are opposite when the angle is 180°.
- Use h modulo 12 for times such as 15:20 or 23:20.
Angle Between Hands of a Clock FAQs
What is the angle between the hands at 3:20?
A = |30 × 3 − 5.5 × 20| = |90 − 110| = 20°. Therefore, the angle is 20°.
What is the angle between the hands at 7:30?
A = |30 × 7 − 5.5 × 30| = |210 − 165| = 45°.
Why is the hour-hand position at 4:30 equal to 135° and not 120°?
The hour hand moves continuously. Its position is 30 × 4 + 0.5 × 30 = 135°, so it moves 15° beyond the 4 o’clock mark.
At what time between 1 and 2 do the hands coincide?
They coincide after 60 × 1/11 = 60/11 minutes, or approximately 1:05:27.
At what time between 5 and 6 are the hands opposite?
Solve |150 − 5.5m| = 180. The valid solution is m = 330/5.5 = 60 minutes for the boundary at 6:00; there is no strictly internal opposite position between 5:00 and 6:00. The opposite position occurs at 5:27:16? Actually, using 150 + 5.5m = 180 gives m = 30/5.5 = 60/11 minutes, so the time is approximately 5:05:27.