F. Scale Factor and Volume

Volume is the amount of space occupied by an object. Similar to the relationship between the surface area of similar objects and scale factor, the volumes of similar objects can also be compared using scale factors.

When an original object is enlarged, the surface area of the resulting object is the product of the original object's surface area and the square of the scale factor.

Length, width, and height comprise the three dimensions of a 3-D object. What do you suppose is the relationship between scale factor and the volume of similar objects?

Example 1

The base of a mini trophy is enlarged to four times its original size. Determine the surface area and volume of the enlargement.

The scale factor is given as k = 4.

Determine the surface area of the original trophy base.
l = 12 cm, w = 10 cm, h = 3 cm

Determine the volume of the enlargement.

The original trophy base measurements are l = 12 cm, w = 10 cm, h = 3 cm.

Determine the dimensions of the enlargement using the scale factor.

Enlargement length (l2)

Enlargement width (w2)

Enlargement height (h2)

The surface area of the enlarged trophy base is 5 952 cm2 and the volume of the enlarged trophy base is 23 040 cm3.