Directions: List the measurements of three different rectangular prisms that each have a surface area of 72 square units?

### Hint

How does the total surface area change when one dimension of the rectangular prism changes?

### Answer

One solution is 2 units by 3 units by 6 units. Julie Wright found another integer solution of 2 units by 2 units by 8 units while Kate Fractal generalized the solution in the comments. Another would be a cube where each side has an area that is 12 square units (one sixth of 72 square units). So, the side lengths are all root 12. If you find another, post it in the comments.

Source: Robert Kaplinsky

Another solution is 1 unit by 4 units by 6.4 units.

Yet another solution a cube with an edge length of 2*sqrt(3) units.

In general, this problem can be solved algebraically as follows:

Let x, y and z be the lengths of the edges. Then the given is

2xy+2yz+2xz=72

xy+yz+xz=36

Let’s isolate x!

xy+xz=36-yz

x(y+z)=36-yz (Yay, distributive law!)

x=(36-yz)/(y+z)

Choose your favorite positive values for y and z, and as long as their product is less than 36, you’ll get a valid solution.

Kate, I love your strategy. Who would think that this kind of thinking could come from a problem that appears so simple at first glance.

I’m looking for integer dimensions using Kate’s formula. Besides the 2,3,6 trio listed, I see 2,2,8.

Remember sixth graders don’t yet know about square roots, at least not by CC standards.

Great point Julie. I hadn’t considered that. Thanks for highlighting the conflict. I’ll add your integer dimension solution to the problem.

Verified by brute force (i.e., Excel): 2, 2, 8 and 2, 3, 6 are indeed the only integer solutions. (However, there are 3 integer solutions when the surface area is 96: 2, 2, 11; 1, 6, 6; and 4, 4, 4.)

Very cool Ilona. Thanks for sharing this!

The solution is 2 by 3 by 6.

1.5 x 11.25 x 1.5

1 x 1 x 17.5