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Maximizing Surface Area

Directions: The following prism is made up of 27 identical cubes. What is the greatest possible surface area the prism can have after removing 1 or more cubes from the outside?



How many cubes (and which ones) would you remove? How could you record the information to help you see a pattern?



Removing the center cube in each face of the prism results in a surface area of 78 square units.

Source: Brian Lack

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  1. That’s a cool problem, but I don’t understand the title. Isn’t the point NOT to keep the surface area constant? Puzzled…

  2. A colleague has found an additional solution with a surface area of 78 square units:
    Remove four cubes on each face – the center cube on the top and bottom row; the two cubes on the ends of the middle row.
    This additional solution prompted conversations about the comparison of surface area to volume, as well as real-world instances where maximizing (or minimizing) the ratio of surface area to volume is important.

  3. Could a student think that the phrase “removing cubes from the outside” means removing a layer therefore the student would remove 9 cubes.

  4. Can you remove the X on all sides and still get 78? So the middle and all corners are gone. I believe it is the same solution and could set up a greatest number of blocks you can remove and still have the largest surface area possible.

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