What angle is formed between the minute and hour hands of a clock showing the time 3:15?

Let's find the positions of the hour and minute hands in degrees relative to 12 o'clock. A full circle is 360 degrees.

• Minute hand: At 3:15, the minute hand is exactly on the 15-minute mark. Since a full circle (60 minutes) is 360 degrees, 1 minute equals 360/60 = 6 degrees. The position of the minute hand: 15 minutes * 6 degrees/minute = 90 degrees.

• Hour hand: At 3 o'clock, the hour hand is at 90 degrees. Each hour, the hour hand moves 360/12 = 30 degrees. In 15 minutes (1/4 hour), it moves 1/4 of 30 degrees: 30 degrees / 4 = 7.5 degrees. Total position of the hour hand: 90 degrees (exactly at 3 o'clock) + 7.5 degrees (shift in 15 minutes) = 97.5 degrees.

The angle between the hands is the absolute difference of their positions: |97.5° - 90°| = 7.5°.

# Calculating the position of the minute hand
minutes = 15
minutes_angle = minutes * 6 # 6 degrees per minute

# Calculating the position of the hour hand
hours = 3
# Exact position at the hour
hours_angle_at_hour = hours * 30 # 30 degrees per hour
# Additional shift due to minutes
hours_angle_from_minutes = (minutes / 60.0) * 30 # Shift of the hour hand due to minutes
total_hours_angle = hours_angle_at_hour + hours_angle_from_minutes

# The angle between the hands
angle = (total_hours_angle - minutes_angle).abs

# Reduce to an angle <= 180 degrees
angle = [angle, 360 - angle].min