Finding the Clock Hand Angle at 6:10

This explanation details how to find the angle between the hour hand and the minute hand of a standard analog clock when the time is exactly 6:10.

Understanding Clock Hand Movement

To calculate the angle, we need to know how the clock hands move:

• The minute hand completes a full circle (360 degrees) in 60 minutes. Therefore, its speed is calculated as: $$ \text{Minute Hand Speed} = \frac{360^\circ}{60 \text{ minutes}} = 6^\circ \text{ per minute} $$
• The hour hand completes a full circle (360 degrees) in 12 hours. Its base speed is: $$ \text{Hour Hand Speed (per hour)} = \frac{360^\circ}{12 \text{ hours}} = 30^\circ \text{ per hour} $$
• Additionally, the hour hand moves continuously as the minutes pass. In 60 minutes, the hour hand moves 30 degrees (the distance between two hour marks). So, its speed per minute is: $$ \text{Hour Hand Speed (per minute)} = \frac{30^\circ}{60 \text{ minutes}} = 0.5^\circ \text{ per minute} $$

Calculating Hand Positions at 6:10

We need to determine the exact position (in degrees) of both hands relative to the '12' mark at 6:10.

Minute Hand Position

At 6:10, the minute hand is pointing at the '2' (10 minutes past the hour). We calculate its position using its speed per minute:

• Position = Minutes past the hour $ \times $ Speed per minute
• $ \text{Minute Hand Angle} = 10 \text{ min} \times 6^\circ/\text{min} $
• $ \text{Minute Hand Angle} = 60^\circ $

So, the minute hand is at 60 degrees from the 12 o'clock position.

Hour Hand Position

At 6:10, the hour hand has moved past the '6'. We calculate its position based on the hours and the additional minutes:

• The position at the exact hour (6) is $ 6 \text{ hours} \times 30^\circ/\text{hour} = 180^\circ $.
• In addition, we account for the movement during the 10 minutes past 6: $ 10 \text{ min} \times 0.5^\circ/\text{min} = 5^\circ $.
• Total Position = Position at 6 o'clock + Movement due to minutes
• $ \text{Hour Hand Angle} = 180^\circ + 5^\circ $
• $ \text{Hour Hand Angle} = 185^\circ $

Therefore, the hour hand is at 185 degrees from the 12 o'clock position.

Determining the Angle Between Hands

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

• $ \text{Angle} = |\text{Hour Hand Angle} - \text{Minute Hand Angle}| $
• $ \text{Angle} = |185^\circ - 60^\circ| $
• $ \text{Angle} = |125^\circ| $
• $ \text{Angle} = 125^\circ $

Since this angle ($125^\circ$) is less than 180 degrees, it is the smaller angle between the hands.

Conclusion

The angle between the hour hand and the minute hand at 6:10 is 125 degrees.