Find the measure of the smaller angle formed by the hour hand and the minute hand of a clock at 8 O'clock

Options

A. `160^(@)`
B. `240^(@)`
C. `90^(@)`
D. `120^(@)`

Answer

To find the measure of the smaller angle formed by the hour hand and the minute hand of a clock at 8 o'clock, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Position of the Hands:**
- At 8 o'clock, the hour hand points directly at 8.
- The minute hand points directly at 12.

2. **Calculate the Angle for Each Hour:**
- A clock is a circle, which is 360 degrees.
- The hour hand completes a full rotation (360 degrees) in 12 hours.
- Therefore, the angle covered by the hour hand in one hour is:
\[ \text{Angle per hour} = \frac{360 \text{ degrees}}{12 \text{ hours}} = 30 \text{ degrees} \]

3. **Determine the Number of Hour Sections Between 12 and 8:**
- The hour hand is at 8 and the minute hand is at 12.
- The number of hour sections between 12 and 8 is:
\[ 12 - 8 = 4 \text{ sections} \]

4. **Calculate the Total Angle Between the Hands:**
- Each section (1 hour) corresponds to 30 degrees.
- Therefore, the total angle between the hour hand and the minute hand is:
\[ \text{Total angle} = 30 \text{ degrees/section} \times 4 \text{ sections} = 120 \text{ degrees} \]

5. **Identify the Smaller Angle:**
- Since the total angle formed is 120 degrees, and it is less than 180 degrees, it is already the smaller angle.

### Final Answer:
The measure of the smaller angle formed by the hour hand and the minute hand of a clock at 8 o'clock is **120 degrees**.

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