Create a prism with each dimension measuring 5 units or less and record the surface area and volume. Now, change the height of the prism so it is twice as high. Record the surface area and volume for the new prism. How does the surface area and volume of the second prism relate to the first prism?
Now, double the length of the prism. Record the surface area and volume for the new prism. How does the surface area and volume of this prism relate to the previous prism?
Now, double the width of the prism. Record the surface area and volume for the new prism. How does the surface area and volume of this prism relate to the previous prism?
Start over and create a 2 \times 2 \times 4 prism. Record the surface area and volume. Choose one of the dimensions and multiply it by a number of your choice to create a new prism. How does the surface area and volume change?
What is the relationship between the scale factor used to change one of the dimensions and the surface area and volume?

Adjusting one dimension of a solid by a scale factor of d will affect both the surface area and the volume of the solid. This means that:

the volume will scale by a factor of d
the surface area will change, but not by a scale by a factor of d

A 4x3x2 rectangular prism and 3 rectangular prisms resulting from doubling each dimension. The 3 new rectangular prisms have the same volume at 48 units but different surface areas. See your teacher for more info.

The rectangular prisms in the image show the affect of multiplying each dimension by a scale of 2. Regardless of which dimension changed, the volume was increased by a scale of 2. The surface area for each prism increased, but the amount that the surface area increase varied based on the dimension that was scaled by 2.

A 4x3x2 rectangular prism with surface area of 52 square units. An 8x6x4 rectangular prism is shown after scaling all the dimensions of the first prism by 2, giving a surface area of 208 square units

\\\\\,\\\\\\\,If we had scaled all 3 dimensions by the same factor d then the surface area would have scaled by d^{2} but the same is not true when only scaling one dimension.

For example, scaling all of the dimensions of the original prism by 2 gives a new surface area of 52 \cdot 2^2=52 \cdot 4= 208 \text{ units}^2.

Examples

Example 1

A rectangular prism had a volume of 96 cubic centimeters. The height of this prism was changed from 9 centimeters to 3 centimeters to create a new rectangular prism. The other dimensions stayed the same. What is the volume of the new prism?

Worked Solution

Create a strategy

First find the scale factor by dividing the new height by the original height.

Since only one dimension was scaled, the volume will change by the same factor.

Apply the idea

First we will find the scale factor: \text{Scale factor} = \dfrac{\text{New height}}{\text{Original height}}=\dfrac{3 \text{cm}}{9 \text{cm}}=\dfrac13

Next, we will multiply the volume by the scale factor.

96 \text { cm}^3 \cdot \dfrac{1}{3} = 32 \text { cm}^3

The volume of the new rectangular prism will be 32 \text { cm} ^3

Example 2

A gift box with a length of 10 \text{ cm}, a width of 20 \text{ cm}, and a height of 5 \text{ cm} needs to be wrapped in paper. If the height of the box is increased by a factor of 2, how much additional wrapping paper is needed?

Worked Solution

Create a strategy

First, we will need to calculate the surface area of both the original box and the box after the height is increased. We can use the formula SA=2lw+2lh+2wh to calculate these. Then, we will find the difference in the areas to determine how much more paper will be needed.

Apply the idea

Find the surface area of the original box:

\displaystyle \text{Surface Area}	\displaystyle =	\displaystyle 2 \left(10\cdot 20\right)+2 \left(10\cdot 5 \right)+2 \left(20\cdot 5\right)	Substitute the given values
	\displaystyle =	\displaystyle 2\left(200\right)+2 \left(50\right)+2\left(100\right)	Evaluate the multiplication in parentheses
	\displaystyle =	\displaystyle 400+100+200	Evaluate the multiplication
	\displaystyle =	\displaystyle 700	Evaluate the addition

The surface area of the original box is 700 \text{ cm}^2.

Next, find the surface area of the new box. The height was increased by a factor of 2, so we will substitute 10 for the height instead of 5.

\displaystyle \text{Surface Area}	\displaystyle =	\displaystyle 2\left(10\cdot 20\right)+2\left(10\cdot 10\right)+2\left(20\cdot10\right)	Substitute the given values
	\displaystyle =	\displaystyle 2\left(200\right)+2\left(100\right)+2\left(200\right)	Evaluate the multiplication in parentheses
	\displaystyle =	\displaystyle 400+200+400	Evaluate the multiplication
	\displaystyle =	\displaystyle 1000	Evaluate the addition

The surface area of the new box is 1000 \text{ cm}^2.

Last, we need to find the difference between the two surface areas to determine how much more paper will be needed for the new box.

1000-700=300

The new box will require 300 \text{ cm}^2 more paper than the original box.

Idea summary

Adjusting one dimension of a rectangular prism by a scale factor of d will affect both the surface area and the volume of the solid. This means that:

the volume will scale by a factor of d
the surface area will change, but not by a scale by a factor of d

Outcomes

7.MG.1

The student will investigate and determine the volume formulas for right cylinders and the surface area formulas for rectangular prisms and right cylinders and apply the formulas in context.

7.MG.1d

Describe how the volume of a rectangular prism is affected when one measured attribute is multiplied by a scale factor of 1/4,1/3,1/2, 2, 3, or 4, including those in contextual situations.

7.MG.1e

Describe how the surface area of a rectangular prism is affected when one measured attribute is multiplied by a factor of 1/2 or 2, including those in contextual situations.

7.06 Effect of scale factors on volume and surface area

Ideas

Effect of scale factors on volume and surface area

Exploration