It is quarter past three in your watch. The angle between the hour hand and the minute hand is

Clock Hand Angle Calculation at 3:15

Understanding how to calculate the angle between the hands of a clock is a common question in many examinations. Let's break down the process for "quarter past three," which translates to 3:15 on a clock.

Clock Hand Movement Basics

Before we calculate the angle for 3:15, it's essential to understand how the hour hand and the minute hand move on a clock face:

- A full circle on a clock is $360^{\circ }$.
- There are 60 minutes in an hour, so the minute hand completes a full $360^{\circ }$ in 60 minutes.
- This means the minute hand moves at a rate of $\frac{360^{\circ }}{60\text{ minutes}}=6^{\circ }$ per minute.
- There are 12 hours marked on a clock, so the hour hand completes a full $360^{\circ }$ in 12 hours.
- This means the hour hand moves at a rate of $\frac{360^{\circ }}{12\text{ hours}}=30^{\circ }$ per hour.
- Since the hour hand also moves as minutes pass, its movement per minute is $\frac{30^{\circ }}{60\text{ minutes}}=0.5^{\circ }$ per minute.

Minute Hand Position at 3:15

At 3:15, the minute hand is exactly at the '3' mark on the clock. We can calculate its angle from the '12' (which is our reference point for $0^{\circ }$) as follows:

- The minute hand moves $6^{\circ }$ for every minute.
- At 15 minutes, its angle from the 12 will be:
- Angle of minute hand $=15\text{ minutes}\times 6^{\circ }/\text{minute}=90^{\circ }$

So, the minute hand is at $90^{\circ }$ from the 12.

Hour Hand Position at 3:15

The hour hand moves based on both the hour and the additional minutes past that hour. At 3:15:

- The hour hand has moved past the '3' mark.
- Its position is based on 3 full hours plus 15 minutes.
- Angle due to hours $=3\text{ hours}\times 30^{\circ }/\text{hour}=90^{\circ }$
- Angle due to minutes $=15\text{ minutes}\times 0.5^{\circ }/\text{minute}=7.5^{\circ }$
- Total angle of hour hand $=90^{\circ }+7.5^{\circ }=97.5^{\circ }$

So, the hour hand is at $97.5^{\circ }$ from the 12.

Angle Calculation between the Hands

To find the angle between the hour hand and the minute hand, we take the absolute difference between their positions:

- Angle between hands $=|\text{Angle of Hour Hand}-\text{Angle of Minute Hand}|$
- Angle between hands $=|97.5^{\circ }-90^{\circ }|$
- Angle between hands $=7.5^{\circ }$

This is the smaller angle between the two hands. If the calculated angle were greater than $180^{\circ }$, we would subtract it from $360^{\circ }$ to get the smaller, conventional angle. In this case, $7.5^{\circ }$ is already the smaller angle.

Therefore, the angle between the hour hand and the minute hand at "quarter past three" (3:15) is $7.5^{\circ }$.