Understanding Clock Angles at 6:44 p.m.

To find the angle between the hour hand and the minute hand of a clock at a specific time, we need to calculate the position of each hand relative to the 12 o'clock mark.

Clock Hand Movement Rates

• The minute hand completes a full circle (360 degrees) in 60 minutes. So, its speed is $360^\circ / 60 \text{ minutes} = 6^\circ \text{ per minute}$.
• The hour hand completes a full circle (360 degrees) in 12 hours. So, its speed is $360^\circ / 12 \text{ hours} = 30^\circ \text{ per hour}$.
• Since the hour hand moves continuously, we also need to consider its movement within the hour. In 60 minutes, the hour hand moves $30^\circ$. So, its speed is $30^\circ / 60 \text{ minutes} = 0.5^\circ \text{ per minute}$.

Calculating Hand Positions at 6:44 p.m.

We will calculate the angle of each hand clockwise from the 12 o'clock position.

Minute Hand Position

At 44 minutes past the hour, the minute hand's angle is:

$\text{Minute hand angle} = \text{Number of minutes} \times \text{Degrees per minute}$

$\text{Minute hand angle} = 44 \times 6^\circ = 264^\circ$

Hour Hand Position

At 6:44 p.m., the hour hand is past the 6. We need to account for the 6 full hours and the 44 minutes past the hour.

Angle due to full hours: $6 \text{ hours} \times 30^\circ/\text{hour} = 180^\circ$

Angle due to minutes past the hour: $44 \text{ minutes} \times 0.5^\circ/\text{minute} = 22^\circ$

Total hour hand angle from 12 o'clock:

$\text{Hour hand angle} = 180^\circ + 22^\circ = 202^\circ$

Finding the Angle Between the Hands

To find the angle between the hands, we find the absolute difference between their positions:

$\text{Difference} = |\text{Minute hand angle} - \text{Hour hand angle}|$

$\text{Difference} = |264^\circ - 202^\circ| = |62^\circ| = 62^\circ$

Determining the Smaller Angle

A clock has two angles between the hands: a smaller one and a larger one. If the absolute difference is greater than 180 degrees, the smaller angle is $360^\circ$ minus the difference. If the difference is 180 degrees or less, that difference is the smaller angle.

In this case, the difference is $62^\circ$, which is less than $180^\circ$. Therefore, the smaller angle between the hour hand and the minute hand at 6:44 p.m. is $62^\circ$.

Absolute difference: $|264^\circ - 202^\circ| = 62^\circ$.

Since $62^\circ \le 180^\circ$, the smaller angle is $62^\circ$.

Revision Table: Key Concepts for Clock Angle Problems

Additional Information on Clock Angles

Calculating the angle between clock hands is a common problem in time and angle calculations. The key is understanding the relative speeds of the hour and minute hands. The 12 o'clock position is typically used as the reference point (0 degrees) for measuring the angle clockwise. Remember that the hour hand's position depends on both the hour and the minutes past the hour.

For any given time in H hours and M minutes (where H is 0-11), the angles from 12 o'clock are:

• Minute hand angle: $M \times 6^\circ$
• Hour hand angle: $(H \times 30^\circ) + (M \times 0.5^\circ)$

The difference is then the absolute value of the difference between these two angles. If this difference is greater than 180 degrees, subtract it from 360 degrees to get the smaller angle.