Find the measure of the smaller angle formed by the hands of a clock at the following times. 8:20

검증된 단계별 안내

Understand that the problem asks for the smaller angle between the hour and minute hands of a clock at 8:20.

Calculate the position of the minute hand: since each minute corresponds to 6 degrees (360 degrees / 60 minutes), multiply the number of minutes by 6. So, the minute hand angle is \(20 \times 6\) degrees.

Calculate the position of the hour hand: each hour corresponds to 30 degrees (360 degrees / 12 hours), and the hour hand also moves as the minutes pass. So, the hour hand angle is \(8 \times 30 + \frac{20}{60} \times 30\) degrees.

Find the difference between the two angles calculated: \(| \text{hour hand angle} - \text{minute hand angle} |\).

Since the clock is circular, the smaller angle between the hands is the minimum of the difference and \(360\) degrees minus the difference. So, calculate \(\min(\text{difference}, 360 - \text{difference})\).

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주요 개념

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Angle Calculation of Clock Hands

The angle between clock hands is found by calculating the positions of the hour and minute hands relative to 12 o'clock. Each minute, the minute hand moves 6 degrees, while the hour hand moves 0.5 degrees. Understanding these rates helps determine the exact angle at any given time.

Converting Time to Angles

To find the angle at a specific time, convert the hour and minute values into degrees. The minute hand’s angle is 6 times the minutes, and the hour hand’s angle is 30 times the hour plus 0.5 times the minutes. This conversion is essential for precise angle measurement.

Determining the Smaller Angle

Since the hands form two angles that sum to 360 degrees, the smaller angle is the minimum of the calculated angle and its supplement (360 degrees minus the angle). This ensures the answer reflects the smaller angle between the hands.