Ratio of lateral areas

The lateral area of a frustum of a cone and a frustum of a pyramid is $\left(\dfrac{p+P}{2}\right)(L)$ where $p$ and $P$ are the base perimeters and $L$ is the slant height. From the diagram, the slant height $L=\sqrt{5^2+1^2}=\sqrt{26}$ .

For the frustum of a pyramid: $p=4(2)=8$ and $P=4(4)=16$ . So the lateral area is $\left(\dfrac{8+16}{2}\right)(\sqrt{26})=12\sqrt{26}$

For the frustum of a cone: $p=2\pi$ and $P=4\pi$ . So the lateral area is $\left(\dfrac{2\pi + 4\pi}{2}\right)(\sqrt{26})=3\pi \sqrt{26}$

Finally, the ratio is

$\dfrac{3\pi \sqrt{26}}{12\sqrt{26}}=$ $\boxed{\dfrac{\pi}{4}}$