Directions: What is least amount of surface area possible on a rectangular prism with a volume of 64 cubic inches?

### Hint

What kind of rectangular prism maximizes volume and minimizes surface area?

### Answer

Understanding that the optimal shape is a cube is critical to solving this problem. With that in mind, each side of the cube has a length of 4 inches. So the area of one face is 16 square units and the total surface area is 96 square units.

Source: Robert Kaplinsky

That is the wrong answer the right answer is length to be 4 height to be 8 and with to be 2

Hi Holly. I believe that there might be a misunderstanding. A 4 x 8 x 2 rectangular prism has a surface area of 112 square units from these calculations: 2(4 x 8) + 2(4 x 2) + 2(8 x 2).

A cube has a surface area of 96 square units. The question asks about the least amount of surface area possible on a rectangular prism with a volume of 64 cubic inches. So, it doesn’t appear that the answer is wrong.

What am I missing?

I like the idea behind the problem, but I’m not crazy about the solution method. If a kid does know that the cube is optimal, it’s not very hard to figure out the dimensions that give 64 cubic inches. It’s a more interesting problem if they DON’T know the cube is optimal (which my kids don’t), but then I think it will be very difficult for them to prove it is.

What about:

What are the possible combinations for 3 whole-number dimensions for a rectangular prism with volume 64 cm^3?

Which of these combinations gives the greatest surface area? the least? the median?

(If the median is what I think it is, that’s kind of cool and makes me wonder about other cube numbers for volume…)

I’ll post what I think are the answers in a different comment in case they’re spoilers.

OK, here are my answers to my questions. Hope somebody will check my work! Maybe I’ll just give it to my brightest problem-solvers tomorrow!

Possible dimensions:

1 x 1 x 64 SA 258 greatest = long skinny

1 x 2 x 32 SA 196

1 x 4 x 16 SA 168

1 x 8 x 8 SA 160 median = flat square [interesting, no?]

2 x 2 x 16 SA 136

2 x 4 x 8 SA 112

4 x 4 x 4 SA 96 least = cube

(Yeah, I know, units, units. Also, these are literally back of an envelope calculations, so use with caution.)

I love all your contributions Julie, especially how you take a theoretically good problem and highlight tweaks to improve it. Could you please submit these new problems for the site?

what is the greatest volume you can make with a rectangular prism that has a surface area of 20 square units ??

What y’all are very smart…While i am just looking for answers so I can read the Twilight series…

Finished? What is the least amount of surface area for a rectangular prism with a volume of 729 cm², how can you do this quickly?

Finished again? Can you generalise? What is the least amount of surface area for a rectangular prism with a volume of 𝑥?