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 \(7.5^\circ\).