Calculating the Angle Between the Hour and Minute Hands on a Clock
Understanding how to calculate the angle between the hour and minute hands of a clock is a fundamental concept in timekeeping and has practical applications in various fields such as mathematics, physics, and engineering. This article focuses on the calculation of the angle between the hands of a clock at 3:50 PM, providing a step-by-step method and an explanation of the different possible angles.
The Step-by-Step Calculation
Let's calculate the angle between the hour and minute hands of a clock at 3:50 PM.
1. Calculate the Position of the Minute Hand
Each minute, the minute hand moves 6 degrees (360 degrees / 60 minutes).
At 50 minutes, the angle from the 12 o'clock position is:
Minute hand angle 50 times; 6 300 degrees
2. Calculate the Position of the Hour Hand
Each hour, the hour hand moves 30 degrees (360 degrees / 12 hours).
At 3 o'clock, the hour hand is at:
Hour hand angle at 3:00 3 times; 30 90 degrees
Additionally, the hour hand also moves as the minutes pass. In 50 minutes, it moves:
Hour hand movement (50 / 60) times; 30 25 degrees
Therefore, the total angle of the hour hand at 3:50 is:
Hour hand angle 90 25 115 degrees
3. Calculate the Angle Between the Hour Hand and the Minute Hand
The angle between the two hands is the absolute difference between their angles:
Angle between hands 300 - 115 185 degrees
4. Determine the Smaller Angle
Since the maximum angle between the hands can be 360 degrees, the smaller angle is:
Smaller angle 360 - 185 175 degrees
Alternative Calculation Methods
Hour Hand Speed: 0.5°/min
The speed of the hour hand is 0.5deg;/min.
In 53 minutes, it will move:
26.5° from the 4 o'clock mark, i.e. at 146.5° from 12 position.
The minute hand is at:
318° from 12 position.
The difference:
318 - 146.5 171.5° (clockwise direction)
188.5° (anticlockwise direction)
Using Ratios and Proportions
The hour hand moves 30deg; in 60 minutes.
The hour hand moves in 53 minutes:
53 times; 30 / 60 26.5°
The minute hand moves 360deg; in 60 minutes.
The minute hand moves in 53 minutes:
53 times; 360 / 60 318°
The angle between the hands in the clockwise direction:
318 - 26.5 291.5°
The angle between the hands in the anticlockwise direction:
42 - 26.5 68.5° or 360 - 291.5 68.5°
Conclusion
The method of calculation involves understanding the degrees each hand moves per unit of time and then finding the difference between these angles. By mastering these calculations, one can easily determine the angle between the hour and minute hands at any given time.