Calculating the Angle Between Clock Hands at Quarter to Five

Understanding how to calculate the angle between the hour and minute hands of a clock is a common type of question. We need to find the angle at a specific time: quarter to five. Quarter to five means the time is 4:45.

To solve this, we need to figure out the position of both the hour hand and the minute hand relative to the 12 o'clock mark (which we consider 0 degrees) at 4:45.

Movement of Clock Hands

• The minute hand completes a full circle (360 degrees) in 60 minutes.
• This means the minute hand moves at a speed of \(360^\circ / 60 \text{ minutes} = 6^\circ\) per minute.
• The hour hand completes a full circle (360 degrees) in 12 hours.
• This means the hour hand moves at a speed of \(360^\circ / 12 \text{ hours} = 30^\circ\) per hour.
• Alternatively, the hour hand moves \(30^\circ\) in 60 minutes, so its speed is \(30^\circ / 60 \text{ minutes} = 0.5^\circ\) per minute.

Step-by-Step Calculation for 4:45

1. Position of the Minute Hand at 4:45

At 4:45, the minute hand is exactly on the 9. The angle is calculated based on the minutes past 12:

Angle of minute hand = Minutes × \(6^\circ\)/minute

Angle of minute hand = \(45 \times 6^\circ = 270^\circ\) from the 12 o'clock position.

2. Position of the Hour Hand at 4:45

At 4:45, the hour hand is not exactly on the 4. It has moved past the 4 because of the 45 minutes that have passed since 4:00. We calculate its position based on the total time past 12 o'clock in hours.

4 hours and 45 minutes is equivalent to 4 + \(45/60\) hours = 4 + 0.75 hours = 4.75 hours.

Angle of hour hand = (Hours past 12) × \(30^\circ\)/hour

Angle of hour hand = \(4.75 \times 30^\circ = 142.5^\circ\) from the 12 o'clock position.

3. Angle Between the Hands

The angle between the hands is the absolute difference between their positions:

Angle = \(|\)Angle of minute hand - Angle of hour hand\(|\)

Angle = \(|270^\circ - 142.5^\circ|\)

Angle = \(|127.5^\circ| = 127.5^\circ\)

Sometimes, the angle calculated this way might be greater than 180 degrees. In such cases, the smaller angle is \(360^\circ\) minus the calculated angle. However, \(127.5^\circ\) is less than 180 degrees, so this is the angle we are looking for.

Summary Table

Difference in angles = \(|270^\circ - 142.5^\circ| = 127.5^\circ\).

Conclusion

The angle between the hour hand and the minute hand at quarter to five (4:45) is \(127.5^\circ\).

Revision Table: Clock Angle Formulas

Angle between hands = \(|\)Hour Hand Angle - Minute Hand Angle\(|\). If the result is > 180°, subtract from 360°.

Additional Information: Understanding Clock Angles

Clock angle problems involve calculating the angle between the hands of an analog clock. The key is understanding the relative speeds of the hour and minute hands. The minute hand moves much faster than the hour hand.

• At exactly 4:00, the hour hand is on the 4 and the minute hand is on the 12. The angle is \(4 \times 30^\circ = 120^\circ\).
• As time passes, the minute hand moves clockwise, and the hour hand also moves clockwise, but slower.
• At 4:45, the minute hand is pointing directly at the 9 (\(270^\circ\) from 12).
• The hour hand has moved 45 minutes past the 4 o'clock position. Since it moves \(0.5^\circ\) per minute, in 45 minutes it moves \(45 \times 0.5^\circ = 22.5^\circ\) past the 4.
• The 4 is at \(4 \times 30^\circ = 120^\circ\) from the 12. So the hour hand is at \(120^\circ + 22.5^\circ = 142.5^\circ\).
• The angle between the hands is the difference between their positions: \(|270^\circ - 142.5^\circ| = 127.5^\circ\). This confirms the previous calculation method.