Calculating the Angle Between Clock Hands at 2:50 AM: A Comprehensive Guide

Understanding how to calculate the angle between the hour and minute hands of a clock is a fascinating mathematical problem that can enhance your skills in basic arithmetic and proportions. This article will walk you through the process of finding the angle at a specific time, specifically 2:50 AM. We'll break down each step to ensure clarity and accuracy.

Step-by-Step Calculation of Clock Angles

Let's delve into the step-by-step process of calculating the angle between the clock's hands at 2:50 AM.

1. Calculating the Position of the Minute Hand

The first step is to determine the position of the minute hand. Each minute on a clock represents 6 degrees, as the full circle (360 degrees) is divided into 60 minutes.

At 2:50 AM, we need to calculate the position of the minute hand. Minute hand position 50 minutes * 6 degrees/minute 300 degrees

2. Calculating the Position of the Hour Hand

Next, we need to find the position of the hour hand. Here, we consider the basic facts about how the hour hand moves with time.

Each hour position represents 30 degrees (360 degrees / 12 hours). At 2:00, the hour hand is at 2 * 30 degrees 60 degrees. As the minutes pass, the hour hand also moves. For every minute, the hour hand moves 0.5 degrees (30 degrees / 60 minutes). For 50 minutes, the additional movement is 50 * 0.5 degrees 25 degrees. Therefore, the position of the hour hand at 2:50 is 60 degrees 25 degrees 85 degrees.

3. Calculating the Angle Between the Hands

The angle between the two hands is the absolute difference between their positions.

Angle between hands |300 degrees - 85 degrees| 215 degrees.

4. Determining the Smaller Angle

On a clock, the angle can be measured in two directions. To find the smaller angle, subtract the calculated angle from 360 degrees.

Smaller angle 360 degrees - 215 degrees 145 degrees.

Thus, the angle between the two hands of the clock at 2:50 AM is 145 degrees.

Alternative Method: Verifying the Calculation

To verify our calculation, let's use another approach:

The minute-hand is at the 10th minute mark, representing 60 degrees (since 5 minute increments 30 degrees, and 10 minutes 2 * 30 degrees). The hour-hand is between 2 and 3, which are 30 degrees apart. In 50 minutes, it has moved 5/6 of the way from 2 to 3, which is 25 degrees (5/6 * 30 degrees). Adding these values gives 60 degrees 25 degrees 85 degrees, and the total angle between the hands is 360 degrees - 85 degrees 275 degrees. The smaller angle is still 360 degrees - 275 degrees 145 degrees.

Do you agree with this result? Yes, the angle between the hands is indeed 145 degrees.

Understanding the Circle's Degrees

Let's further break down the circle into degrees to understand the concept better:

A full circle has 360 degrees, and a clock shows 12 hours, so each hour has 360/12 30 degrees. The minute hand pointing directly at 10 is at the 10th hour, which is 2 hours away from 12, and thus 2*30 degrees 60 degrees. The hour hand is between 2 and 3, which are 30 degrees apart. In 50 minutes, it has moved 5/6 of the way from 2 to 3, which is 25 degrees (5/6 * 30 degrees). Adding these values gives 60 degrees 25 degrees 85 degrees. The angle between the hands is therefore 360 degrees - 85 degrees 275 degrees. The smaller angle is 360 degrees - 275 degrees 145 degrees.

The final answer is 145 degrees.

Conclusion

The angle between the hour and minute hands of a clock at 2:50 AM is 145 degrees. Understanding the basics of clock angle calculation can be both a useful skill and an engaging mathematical exercise. By using these steps, you can calculate the angle for any given time with precision.

For further practice and exploration, try calculating angles for different times and share your results with peers or teachers to enhance your problem-solving skills.