Directions: What is the least and greatest amount of surface area possible on a rectangular prism with a volume of 64 cubic inches and whole number side lengths?

### Hint

How do you find the Surface Area of a Rectangular Prism?

### Answer

minimum: 96 square inches (4 x 4 x 4)

maximum: 258 square inches (1 x 1 x 64)

maximum: 258 square inches (1 x 1 x 64)

Source: Marie Isaac, Katrin Marti, and Ryan Turner

We have been working on volume in my high school equivalency class, but we haven’t discussed surface area yet. I scaffolded this problem by breaking it into two questions:

1) What are the dimensions of three different rectangular prisms that all have a volume of 64 square inches?

2) Which of your shapes from question #1 would take the most material to make?

It turned out that the students who showed up had enough thinking with just figuring out number 1. For one student who had missed a lot of class, we talked about the formula v=lwh and rewrote it as 64=_ x _ x _. Part 2 was a bit over their heads, so we worked it as a class by finding the area of the sides of each shape and adding them together. If class had been bigger today or if a different group had shown up, I think some of them could have figured it out on their own. I might try this again another day with a different. The students who were here today could still get more out of the problem.

* different volume

I think you mean 96 square inches and 258 square inches? Surface area is measured as a 2-D measurement. Volume is 3 dimensional and you had stated that it must be 64 cubic inches so a 1x1x64 would have a volume of 64 cubes but its surface area could be a variety of options of which 258 would be the greatest.

Thank you for bringing this to our attention, Heidi. The units have been corrected.

I did a similar activity with my students in 7th grade, with a couple modifications to help those on the lower end. I did with 60 cubic units and gave the student groups some hands on “help” – 60 unifix cubes so they could build their different prisms and count the “outside squares” for surface area if the formula was “too much”. NOTE: Remind them to have volume of 60 units need to use ALL the cubes..

I’d also highly recommend assigning roles in the group to minimize arguments on who does what (all want to build). The hands on nature of the unifix cubes representing volume does help the students differentiate surface area from volume formulas.