Rectangular Prism Problem

Given a rectangular prism, the surface area is 94, and the sum of the lengths of the edges is 48. Find the length of an interior diagonal of the prism.

Solution

This problem is one of a very large class of interesting math problems in which it seems as though there isn't enough information to solve it. You have three unknowns (length, width, and height), but only two equations. How can you possibly find the lengths of the sides?

The answer to that question is: you don't need to find the lengths of the sides, because that's not what the problem is asking for! It's possible to find the length of the interior diagonal without knowing the lengths of the sides.

Let the dimensions of the prism be x, y, and z.

4(x + y + z) = 48, so (x + y + z) = 12 (Eq. #1).

2xy + 2yz + 2xz = 94 (Eq. #2).

Now square (x + y + z) and you get:

x² + y² + z² + 2xy + 2yz + 2xz = 144 (Eq. #3).

Subtract Eq. #2 from this and you get:

x² + y² + z² = 50.

This is the square of the length of an interior diagonal, so the answer is: the square root of 50.

(By the way, in creating this problem, I started with a prism with dimensions 3, 4, and 5. You can verify that it works, but you don't ever actually use those numbers in solving the problem.)

Isn't that slick?

Questions

1. If the surface area is 64, and the sum of the edges is 40, what is the length of the diagonal?
2. Create your own rectangular prism problem; start by picking the side lengths of the prism.