How to Calculate the Angle Between Clock Hands at 6:50

Have you ever wondered how to determine the angle between the hour and minute hands of a clock at a specific time? Let's explore the process of calculating the angle between the hour and minute hands at 6:50. This article will provide a step-by-step guide and detailed explanation.

Understanding the Basic Principles

Before diving into the calculation, it's essential to understand the basic principles behind the movement of the clock hands. Each hour represents 30 degrees (360/12), and each minute represents 6 degrees (360/60).

Calculating the Position of the Minute Hand

At 6:50, the minute hand is at 50 minutes. To calculate its position:

Each minute represents 6 degrees. Therefore, at 50 minutes, the minute hand is at: 50 * 6 300 degrees.

The minute hand is at 300 degrees from the 12 o'clock position.

Calculating the Position of the Hour Hand

At 6:00, the hour hand is at:

6 * 30 180 degrees. As the minutes pass, the hour hand also moves. For every minute, the hour hand moves 0.5 degrees (30/60). At 50 minutes past 6:00, the hour hand has moved: 50 * 0.5 25 degrees. Therefore, the position of the hour hand at 6:50 is: 180 25 205 degrees.

Calculating the Angle Between the Hands

To find the angle between the hour and minute hands:

Subtract the smaller position from the larger position: 205 - 300 -95 degrees. Since we are dealing with positive angles, we take the absolute value: | -95 | 95 degrees.

The angle between the hour and minute hands of the clock at 6:50 is 95 degrees.

Additional Examples for Clarity

Angle at 6:45

At 6:45, the calculation is a bit easier:

The minute hand is at: 45 * 6 270 degrees. The hour hand is at: 6 * 30 45 * 0.5 180 22.5 202.5 degrees. The angle between the hands is: 270 - 202.5 67.5 degrees.

Misconceptions and Alternative Approaches

It's worth addressing a common misconception. At 6:45, the angle is not 67.5 degrees as some might initially assume, but 95 degrees. This is because the hour hand has moved an additional 2.5 degrees from its position at 6:40. Therefore, the correct calculation is:

Minute hand at: 45 * 6 270 degrees. Hour hand at: 6 * 30 45 * 0.5 202.5 degrees. Angle between hands: 270 - 202.5 67.5 degrees. However, due to additional movement, the correct angle is: 360 - 67.5 292.5 degrees, but since we are looking for the smaller angle, it is 360 - 292.5 95 degrees.

Another approach might be mistakenly calculating to 120 degrees at 6:50, which is incorrect. The correct position and calculation are as described above.

Additional Examples: 7:50

Let's calculate the angle for 7:50 to further illustrate the process:

Minute hand at: 10 * 30 300 degrees. Hour hand at: 7 * 30 50 * 0.5 210 25 235 degrees. Angle between hands: 300 - 235 65 degrees.

Conclusion

Calculating the angle between the hour and minute hands of a clock can be a fun and educational exercise. By understanding the basic principles and following the detailed steps, you can easily determine the angle for any given time. Whether it's for a homework problem, a clock conundrum, or simply a curious mind, the process remains straightforward.