Lesson 7

Discover

There are 27 individual cubes in the original 3 × 3 Rubik’s Cube. These “cubelets” or “cubies” are similar to the Rubik’s Cube itself. The scale factor between the length of a side of a cubelet and the length of a side on the Rubik’s Cube is 3.

The 4 × 4 Rubik’s Revenge is made up of 56 small cubes. The scale factor between the length of a side of a cubelet and the length of a side on the 4 × 4 cube is 4. The 5 × 5 Rubik’s Professor’s Cube contains 98 cubelets. If the smaller cubelets and the larger cubes are similar, what do you think the scale factor is between the dimensions of the cubelets and the dimensions of the 5 × 5 cube? That’s right—it’s 5!

As you know, each cube is made up of similar, smaller cubes. What do you think the relationship is between the scale factor and the surface areas of these two similar objects? What about the relationship between the scale factor and the volumes of the original cube and the cubelets? If there is a relationship, do you think it applies to other 3-D objects such as rectangular prisms, right cylinders, or cones?

Math Lab: Scale Factors and Areas and Volumes of 3-D Objects