The Earth rotates \( 360^\circ \) in 24 hours. To calculate the angle turned in 3 hours 20 minutes, convert 3 hours 20 minutes to a fraction of 24 hours:
\[
3 \, \text{hours} 20 \, \text{minutes} = 3 + \frac{20}{60} = 3 + \frac{1}{3} = \frac{10}{3} \, \text{hours}
\]
The angle turned in \( \frac{10}{3} \) hours is:
\[
\text{Angle} = \frac{10}{3} \times \frac{360^\circ}{24} = \frac{10}{3} \times 15^\circ = 50^\circ
\]
Thus, the answer is \( 120^\circ \).