Calculating the Angle Between Clock Hands at 2:13 p.m.

Understanding how the hour and minute hands move is key to solving clock angle problems. The hands of a clock move at constant speeds relative to the 12 o'clock position.

Hand Speeds Explained

The minute hand completes a full circle (360 degrees) in 60 minutes. The hour hand completes a full circle (360 degrees) in 12 hours.

Step-by-Step Angle Calculation at 2:13 p.m.

We need to find the angle of each hand relative to the 12 o'clock position at 2:13 p.m.

1. Angle of the Minute Hand

At 2:13 p.m., the minute hand is exactly at the 13-minute mark. Since the minute hand moves \(6^\circ\) per minute:

Angle of Minute Hand = \(13 \text{ minutes} \times 6^\circ/\text{minute}\)

Angle of Minute Hand = \(78^\circ\)

2. Angle of the Hour Hand

At 2:13 p.m., the hour hand is past the 2 o'clock mark. It has moved for 2 full hours and an additional 13 minutes from the 12 o'clock position.

The hour hand moves \(30^\circ\) per hour and \(0.5^\circ\) per minute.

Angle due to 2 hours = \(2 \text{ hours} \times 30^\circ/\text{hour} = 60^\circ\)

Angle due to 13 minutes = \(13 \text{ minutes} \times 0.5^\circ/\text{minute} = 6.5^\circ\)

Total Angle of Hour Hand = Angle due to hours + Angle due to minutes

Total Angle of Hour Hand = \(60^\circ + 6.5^\circ = 66.5^\circ\)

3. Finding the Difference in Angles

To find the angle between the hands, we calculate the absolute difference between their angles from the 12 o'clock mark:

Difference = \(|\text{Angle of Minute Hand} - \text{Angle of Hour Hand}|\)

Difference = \(|78^\circ - 66.5^\circ|\)

Difference = \(|11.5^\circ|\)

Difference = \(11.5^\circ\)

4. Determining the Acute Angle

The calculated difference is \(11.5^\circ\). Since this is less than \(180^\circ\), it is already the acute angle between the hands.

Thus, the acute angle between the hour-hand and the minute-hand at 2:13 p.m. is \(11.5^\circ\).

Revision Table: Clock Angle Formulas

Angle between hands \( = |(H \times 30 + M \times 0.5) - (M \times 6)|^\circ \)

Angle between hands \( = |30H + 0.5M - 6M|^\circ \)

Angle between hands \( = |30H - 5.5M|^\circ \)

Using the formula for H=2, M=13:

Angle \( = |30 \times 2 - 5.5 \times 13|^\circ \)

Angle \( = |60 - 71.5|^\circ \)

Angle \( = |-11.5|^\circ \)

Angle \( = 11.5^\circ \)

If the result is greater than \(180^\circ\), subtract it from \(360^\circ\) to get the acute angle.

Additional Information: Clock Angle Concepts

• The reference point for measuring angles is usually the 12 o'clock position, considered as \(0^\circ\).
• Angles are typically measured clockwise.
• The 'acute' angle is the smaller angle between the two hands (less than or equal to \(180^\circ\)). The other angle is the 'obtuse' or 'reflex' angle.
• Problems often ask for the angle at a specific time or the times when the hands are at a specific angle (like 0 degrees for coinciding, 180 degrees for opposite, or 90 degrees for perpendicular).
• The relative speed between the minute hand and the hour hand is \(6^\circ/\text{minute} - 0.5^\circ/\text{minute} = 5.5^\circ/\text{minute}\). This relative speed is useful for finding times when hands are at certain relative positions.