In earlier activities, we saw that if we dilate a two-dimensional shape, the area of the dilated shape is the area of the original shape multiplied by the square of the scale factor. What happens when we dilate three-dimensional solids?

Here is a rectangular prism with side lengths 3, 4, and 5 units. When we dilate the prism using a scale factor of 3, the lengths become 9, 12, and 15 units.

Because these are three-dimensional shapes, we can look at both volume and surface area. The volume of the original prism is 60 cubic units because . The volume of the dilated prism is 1,620 cubic units because . The volume became 27 times larger! Why? Because the side lengths tripled, when we calculated the volume we were really finding . The volume was multiplied by the cube of the scale factor, or by .

Now let’s look at surface area. How will the surface area change when the prism is dilated?

In the original prism, 2 faces have area 12 square units, 2 have area 20 square units, and 2 have area 15 square units for a total surface area of 94 square units. The corresponding faces of the dilated prism have areas 108, 180, and 135 square units. The surface area of the new prism totals 846 square units, 9 times that of the original. Just like the area of two-dimensional shapes, the surface area of the prism changed by the square of the scale factor.

In general, when we dilate any three-dimensional solid by a scale factor of , the surface area is multiplied by , and the volume is multiplied by .