Direct Answer: The angle (\theta) between the hour hand and the minute hand of a clock at time (H) hours and (M) minutes is given by the master formula:
(\theta = \left| 30H - \frac{11}{2}M \right|)
If the calculated angle is greater than (180^\circ), the reflex angle is obtained by subtracting the result from (360^\circ) (i.e. (360^\circ - \theta)).
Step-by-Step Mathematical Derivation
A clock face is a complete circle consisting of (360^\circ) divided into 12 hours (60 minute divisions):
- Minute Hand Speed: Moves (360^\circ) in 60 minutes = (6^\circ) per minute.
- Hour Hand Speed: Moves (360^\circ) in 12 hours ((720) minutes) = (0.5^\circ) ((\frac{1}{2}^\circ)) per minute, or (30^\circ) per hour.
- Relative Speed: The minute hand gains over the hour hand at:
(6^\circ - 0.5^\circ = 5.5^\circ = \frac{11}{2}^\circ \text{ per minute}).
At time (H:M):
Position of hour hand from 12 = (30H + 0.5M)
Position of minute hand from 12 = (6M)
Angle difference (\theta = |(30H + 0.5M) - 6M| = |30H - 5.5M| = \mathbf{\left| 30H - \frac{11}{2}M \right|}).
Solved Practical Examples
Example 1: Find the angle between the hands at 4:20
Here, (H = 4), (M = 20):
(\theta = \left| 30(4) - \frac{11}{2}(20) \right| = |120 - 110| = \mathbf{10^\circ}).
Result: The angle between the hands at 4:20 is (10^\circ).
Example 2: Find the angle between the hands at 8:30
Here, (H = 8), (M = 30):
(\theta = \left| 30(8) - \frac{11}{2}(30) \right| = |240 - 165| = \mathbf{75^\circ}).
Result: The angle between the hands at 8:30 is (75^\circ).
Key Clock Facts for Competitive Exams
| Clock State / Event |
In 12 Hours |
In 24 Hours (1 Day) |
Angle (\theta) |
| Hands Coincide (Overlap) | 11 times (not 12; 11 to 1 overlap only once at 12:00) | 22 times | (0^\circ) |
| Hands in Opposite Directions (Straight line) | 11 times (5 to 7 opposite only once at 6:00) | 22 times | (180^\circ) |
| Hands at Right Angles (Perpendicular) | 22 times | 44 times | (90^\circ) |
| Hands in a Straight Line (Coincide or Opposite) | 22 times (11 + 11) | 44 times | (0^\circ \text{ or } 180^\circ) |
