A rectangle with length 8 cm and width $\pi$ cm has area $8\pi \mathrm{cm}^{2}$. Suppose that the radius of the semi-circle is $r \mathrm{~cm}$. The area of a circle with radius $r \mathrm{~cm}$ is $\pi r^{2} \mathrm{~cm}^{2}$ and so the area of the semi-circle is $\frac{1}{2} \pi r^{2} \mathrm{~cm}^{2}$. Since the rectangle and the semi-circle have the same area, then $\frac{1}{2} \pi r^{2}=8\pi$ and so $\pi r^{2}=16\pi$ or $r^{2}=16$. Since $r>0$, then $r=4$ and so the radius of the semi-circle is 4 cm.