What is the angle between the hour and minute hands at 3:15?

What is the angle between the hour and minute hands at 3:15?

• A. 22.5 degrees
• B. 15 degrees
• C. 45 degrees
• D. 7.5 degrees

Answer: (D)

The angle is found by translating the positions of both hands into degrees and comparing them, paying attention to the hour hand’s gradual movement. At 3:15 the minute hand is at the 15-minute mark, which is 15 × 6 = 90 degrees from 12 o’clock. The hour hand, starting at 3:00, sits at 3 × 30 = 90 degrees from 12. But it doesn’t stay fixed; it moves continuously as time passes. In 15 minutes, the hour hand advances 15 × 0.5 = 7.5 degrees (since it moves 30 degrees per hour, or 0.5 degrees per minute). So its position is 90 + 7.5 = 97.5 degrees from 12. The angle between the hands is the difference: 97.5 − 90 = 7.5 degrees. Therefore, the angle is 7.5 degrees. (The other, larger angle would be 360 − 7.5, which is not the acute angle typically asked for.)

The angle is found by translating the positions of both hands into degrees and comparing them, paying attention to the hour hand’s gradual movement.

At 3:15 the minute hand is at the 15-minute mark, which is 15 × 6 = 90 degrees from 12 o’clock. The hour hand, starting at 3:00, sits at 3 × 30 = 90 degrees from 12. But it doesn’t stay fixed; it moves continuously as time passes. In 15 minutes, the hour hand advances 15 × 0.5 = 7.5 degrees (since it moves 30 degrees per hour, or 0.5 degrees per minute). So its position is 90 + 7.5 = 97.5 degrees from 12.

The angle between the hands is the difference: 97.5 − 90 = 7.5 degrees. Therefore, the angle is 7.5 degrees. (The other, larger angle would be 360 − 7.5, which is not the acute angle typically asked for.)