# 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?

## 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 $247^{\circ} - 84^{\circ} = 163^{\circ}$.