Maximum Surface Area

Given a specific volume of a solid with rectangular faces, what is the maximum surface area of the solid?

Start by using cubes that each have a volume equal to one cubic centimeter.

What is the greatest surface area of one cube? Note that there’s only one answer for 1 one-centimeter cube!

Now let’s consider two cubes. When working with more than one cube, at least one face of each cube must meet a face of another cube. What is the maximum surface area of two connected cubes that share a common face? Note there is only one answer again!

Now let’s look at three cubes. You can arrange groups of three (or more) cubes in more than one way. So, what is the maximum surface area with three cubes (remember, at least one face of each cube must meet a face of another cube)? Look for a pattern.

What do you notice? Why do you think this happens? Make a conjecture.