Calculate the angle of the minute hand from the 12 o'clock position.

The minute hand moves $360^\circ$ in 60 minutes, which is $6^\circ$ per minute.
At 15 minutes past the hour, the minute hand's angle is:

$$\text{Minute hand angle} = 15 \text{ minutes} \times 6^\circ/\text{minute} = 90^\circ$$

The hour hand moves $360^\circ$ in 12 hours, which is $30^\circ$ per hour. It also moves $0.5^\circ$ per minute ($30^\circ / 60 \text{ minutes}$).
At 3:15, the hour hand's angle is based on 3 full hours and 15 minutes:

$$\text{Hour hand angle} = (3 \text{ hours} \times 30^\circ/\text{hour}) + (15 \text{ minutes} \times 0.5^\circ/\text{minute})$$ $$\text{Hour hand angle} = 90^\circ + 7.5^\circ = 97.5^\circ$$

The angle between the hour and minute hands is the absolute difference of their positions:

$$\text{Angle} = |\text{Hour hand angle} - \text{Minute hand angle}|$$ $$\text{Angle} = |97.5^\circ - 90^\circ| = 7.5^\circ$$

The angle between the hour and minute hands at 3:15 is $\boxed{\text{7.5}^\circ}$.