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.)