jmc

The probability of winning on one spin is equal to the ratio of the area of the WIN sector to the area of the entire circle. The area of the entire circle is $\pi \cdot 5^2=25\pi$. In math terms, our ratio is: $\frac{2}{5}=\frac{\text{area of the win sector}}{25\pi }$. Solving for the area of the win sector, we find it equal to $\boxed{10\pi }$ square centimeters.