Find the height of the cylinder. The height of the diagonal in a square is 12 ft. What is the area and perimeter of the square? What is the surface area of a rectangular prism with the dimensions 15 cm x 8 cm x 9 cm?

The diameter $d = 10 \text{ cm}$.
The radius $r$ is half of the diameter.

The volume of a cylinder is given by $V = \pi r^2 h$. We are given $V = 800 \text{ cm}^3$ and we found $r = 5 \text{ cm}$.

The height of the cylinder is approximately:

The exact height is $\boxed{\frac{32}{\pi} \text{ cm}}$.

12. The height of the diagonal in a square is 12 ft. What is the area and perimeter of the square?

For a square with side length $s$, the diagonal $L$ is given by $L = s\sqrt{2}$. We are given $L = 12 \text{ ft}$.

The area of a square is $A = s^2$.

The perimeter of a square is $P = 4s$.

The area of the square is $\boxed{72 \text{ ft}^2}$ and the perimeter is $\boxed{24\sqrt{2} \text{ ft}}$.

13. What is the surface area of a rectangular prism with the dimensions 15 cm x 8 cm x 9 cm?

Length $l = 15 \text{ cm}$
Width $w = 8 \text{ cm}$
Height $h = 9 \text{ cm}$

The surface area $SA$ of a rectangular prism is given by the formula $SA = 2(lw + lh + wh)$.

The surface area of the rectangular prism is $\boxed{654 \text{ cm}^2}$.