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?

To address the reservation of Julie Wright regarding those students who does not have a prerequisite knowledge that the cube is optimal, maybe it would be better to start off with an analog problem at a lower dimension: Given a specific area of a rectangle, what whole number dimensions of the rectangle gives the least perimeter? The student can do a mathematical investigation and explore the various possibilities similar to the one presented by Julie above. Thus, the natural next level question would be to answer the question you presented. In this way, the students also get to have a conceptual understanding about the geometric structures between polygons (2D) and solids (3D). This helps also in honing the students’ mathematical and inductive reasoning (making conjectures about geometric properties in 3D based on specific observations made in 2D).

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?

i dont get it makes no since