In the previous example, the length, width, and height of the original trophy base were each multiplied by the scale factor of 4 to determine the corresponding dimensions of the enlarged trophy base.

And, the volumes of the two similar objects can be compared as follows:

As can be seen in the volume comparison, the volume of the enlarged trophy base was 4 × 4 × 4 = 43 times larger than the volume of the original trophy base. What is the relationship between the volumes of the objects and the scale factor?

When the three dimensions (length, width, and height) of an object are each increased or decreased by the same scale factor, k, the volume of the object is increased or decreased by k3.

Let's verify this relationship using the same trophy base scenario from Example 1.

Example 2

The base of a mini trophy is enlarged to four times its original size. Determine the volume of the enlargement using the volume of the original trophy base and the scale factor. Compare your findings with the volume determined in the previous example.

The scale factor is given as k = 4.

Determine the surface area of the original base.

Determine the volume of the enlargement.

Compare with the volume determined in previous example.

Using both methods, the volume of the enlarged trophy base was found to be 23 040 cm3.

When only the dimensions of an original object and the scale factor are known, and the volume of the resulting object is desired, determining the volume of the resulting object using the method from Example 2 is more efficient (the individual dimensions of the resulting object are not required). Both methods are acceptable and as shown, give the same final result.