Clock face angles
The time is 20:14. What is the smaller angle between the hour hand and the minute hand on an accurate analogue clock?
Age
11 to 14
Challenge level
Problem
The time is 20:14.
What is the smaller angle between the hour hand and the minute hand on an accurate analogue clock?
This problem is taken from the UKMT Mathematical Challenges.
Student Solutions
Each minute, the minute hand moves $\frac{360}{60} = 6^{\circ}$, so the minute hand makes an angle of $14 \times 6 = 84^{\circ}$ with the upwards vertical.
Each hour, the hour hand moves $\frac{360}{12} = 30^{\circ}$ so the hour hand makes an angle of $(8 + \frac{14}{60}) \times 30 = 247^{\circ}$ with the same vertical.
Therefore, the angle between the two hands is $247^{\circ} - 84^{\circ} = 163^{\circ}$.
Each hour, the hour hand moves $\frac{360}{12} = 30^{\circ}$ so the hour hand makes an angle of $(8 + \frac{14}{60}) \times 30 = 247^{\circ}$ with the same vertical.
Therefore, the angle between the two hands is $247^{\circ} - 84^{\circ} = 163^{\circ}$.