Calculating the Angle Between Clock Hands at 9:15 AM

Understanding the angle between the hands of a clock at specific times is a common problem in both mathematics and everyday life. In this article, we will break down the process of determining the angle between the hands of a clock at 9:15 AM, providing detailed calculations and explanations.

Understanding the Basics of Clock Angles

When considering clock problems, it's essential to understand the basic positions and movements of the clock hands. Each hour mark on a clock represents 30 degrees, and each minute mark (or minute) represents 6 degrees. This understanding is key to solving clock problems accurately.

Position of the Hour Hand at 9:15 AM

At 9:15 AM, we need to determine the exact position of the hour hand. Here’s how we can calculate it:

1. Starting Position

At 9:00 AM, the hour hand is at the 9 o'clock mark, which is 270 degrees from the 12 o'clock mark.

2. Calculation of Movement

The hour hand moves 0.5 degrees per minute. From 9:00 to 9:15, the hour hand moves 15 minutes.

Therefore, the movement of the hour hand from 9:00 to 9:15 is:

15 minutes * 0.5 degrees/minute 7.5 degrees.

The position of the hour hand at 9:15 AM is:

270 degrees (initial position) 7.5 degrees (movement) 277.5 degrees.

Position of the Minute Hand at 9:15 AM

The minute hand at 15 minutes past the hour points to the 3 o'clock mark. Here’s the calculation:

Each minute mark represents 6 degrees. At 15 minutes, the minute hand is at 15 * 6 90 degrees from the 12 o'clock mark.

Calculating the Angle Between the Hands

To find the angle between the hour and minute hands, subtract the angle of the minute hand from the angle of the hour hand:

277.5 degrees (hour hand) - 90 degrees (minute hand) 187.5 degrees.

However, for practical purposes, we often calculate the smaller angle, which is 360 degrees minus the calculated angle:

360 degrees - 187.5 degrees 172.5 degrees.

Conclusion

In summary, the angle between the hands of the clock at 9:15 AM is 187.5 degrees clockwise, but for practical purposes, it is usually considered as 172.5 degrees.

Understanding these calculations can help in solving various clock angle problems. Keep these basic principles in mind to tackle other clock angle questions with ease.