Problem 243

A rectangle with length 8 cm and width $\pi$ cm has area $8\pi\text{ cm}^2$. Suppose that the radius of the semi-circle is $r\text{ cm}$. The area of a circle with radius $r$ cm is $\pi r^2\text{ cm}^2$ and so the area of the semi-circle is $\frac{1}{2}\pi r^2\text{ cm}^2$. 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.