7.5 Converting Measurements Using Scale Factor

Learning Objectives

You will be able to recognize the ratio of corresponding side lengths of similar figures as the scale factor.
You will be able to use proportions to find unknown measures of corresponding parts of similar figures.
You will be able to use diagrams of similar figures to find unknown lengths without directly measuring.
You will solve real-world problems involving indirect measurement.

Introduction

“Look at this!” Josh exclaimed to his friend Carlo while they were in the computer lab. Carlo rolled his chair over to Josh’s computer to see what he was looking at.

“What?”

“This is a picture of Everest from space. The space shuttle Endeavor took it on October 10, 1994. It was a clear day too, so it should be pretty accurate. They wanted to measure the area of the mountain,” Josh said smiling.

“You and Mount Everest, but it is pretty cool,” Carlo said looking at the picture.

“I’m going to make a drawing of this,” Josh said taking out a piece of paper to make some notes.

In looking at the website, Josh discovered that the space shuttle Endeavor figured out that the length of Mount Everest from space is 43 miles and the width of Everest from space is 24 miles long. Josh wrote down the measurement $\frac{1}{4}^{\prime\prime} = 1 \ mile$ .

If Josh uses this measurement, what will the dimensions of his drawing be? Will it fit on $11^{\prime\prime} \times 14^{\prime\prime}$ paper? What is the actual area of Everest according to the space shuttle?

Guided Learning

Solve Real-World Problems Involving Indirect Measurement

You may be surprised how often we use similar figures that are related by a scale factor. Maps, architectural blueprints, and diagrams are just some examples. In most of these cases, the scale factor is given so that we know how to enlarge the items in the drawing to their real sizes. Take a look at the floor plan below. It shows where the furniture is located in a living room.

The size of everything in the drawing has been made smaller from a real size by the scale factor. What is the scale factor for the floor plan?

It tells us that one inch in the drawing is equal to two feet in actual size.Therefore, if we know the size in inches of any object in the floor plan, we can find its actual size in feet. Let’s give it a try.

How many feet long is the sofa?

Let’s find the sofa on the floor plan. Then we can use a ruler to find its length in inches. How many inches long is the drawing of the sofa? The sofa in the floor plan is 2 inches long. Imagine this is like knowing the length of one side in a similar figure. Now we need to use the scale factor as we would to find the length of the corresponding side in a similar figure (in this case the “corresponding side” is the actual sofa). We simply multiply the length we know by the scale factor:

$\text{sofa drawing} \times \text{scale factor} &= \text{actual sofa size}\\2 \ inches \times 2 &= 4 \ feet$

The sofa is four feet long.

How long is the fireplace?

Use a ruler to measure the fireplace in the drawing. It is 2.5 inches long. We multiply this by the scale factor to find the length in feet.

$\text{fireplace drawing} \times \text{scale factor} &= \text{actual fireplace length}\\2.5 \ inches \times 2 &= 5 \ feet$

The real length of the fireplace is 5 feet.

We can also reverse the process to take an actual size and reduce it.

Example A

Chris is making a drawing of his school and the grounds around it. The basketball court is 75 feet long and 40 feet wide. If Chris uses a scale factor in which 1 inch equals 10 feet, what should the dimensions of the basketball court be in his drawing?

First of all, what do we need to find?

We need to know the dimensions (length and width) that the small version of the basketball court should be.

What information have we been given?

We know the actual size of the basketball court, and we know the scale factor Chris is using for his drawing.We can set up an equation to find the drawn dimensions. We’ll have to find the length first and then the width.

$\text{drawing length} \times \text{scale factor} &= \text{actual basketball court length}\\\text{drawing length} \times 10 &= 75 \ feet\\ \text{drawing length} &= 75 \div 10\\\text{drawing length} &= 7.5 \ inches$

The length of the basketball court in Chris’s drawing should be 7.5 inches.

Now let’s use the same process to find the width Chris should draw.

$\text{drawing width} \times \text{scale factor} &= \text{actual basketball court width}\\\text{drawing width} \times 10 &= 40 \ feet\\ \text{drawing width} &= 40 \div 10\\\text{drawing width} &= 4 \ inches$

Great! Now we know that Chris should represent the basketball court as a 4 by 7.5 inch rectangle on his drawing.

Using a Map's Scale

A map is another type of scale drawing of a region. Maps can be very detailed or very simple, showing only points of interest and distances. You can read a map just like any other scale drawing—by using the scale.

On the map below, the straight-line distance between San Francisco and San Diego is 3 inches. What is the actual straight-line distance between San Francisco and San Diego?

Set up a proportion. Write the scale as a ratio.

$\frac{0.5 \ inch}{75 \ miles}$

Now write the second ratio, making sure it follows the form of the first ratio, inches over miles.

$\frac{0.5 \ inch}{75 \ miles} = \frac{3 \ inches}{x \ miles}$

Now cross-multiply and solve for .

$(0.5)x &= 3(75)\\0.5x &= 225\\x &= 450$

The straight-line distance between San Francisco and San Diego is 450 miles.

Note: The straight-line distance is also known as “as the crow flies.” If you were actually traveling from San Francisco to San Diego, it would be farther than 450 miles, since you would need to drive on highways that are not a straight line.

Now let's go back to the dilemma from the beginning of this lesson.

First, let’s figure out the scale dimensions of the drawing. We will need to use the scale to calculate a length and a width.

The scale is $\frac{1}{4}^{\prime\prime} = 1 \ mile$ .

Next, let’s write a proportion for length. We know that the actual length = 43 miles.

$\frac{\frac{1}{4}^{\prime\prime}}{1 \ mile} &= \frac{x}{43 \ miles}\\x &= 10.75^{\prime\prime}$

Now we can use a proportion for width. The actual width is 24 miles.

$\frac{\frac{1}{4}^{\prime\prime}}{1 \ mile} &= \frac{x}{24 \ miles}\\x &= 6 \ inches$

The dimensions of Josh’s drawing will be $10.75^{\prime\prime} \times 6^{\prime\prime}$

This drawing will fit on an $11 \times 14^{\prime\prime}$ piece of paper.

Finally, we can figure out the area of Everest according to the picture.

$A &= lw\\A &= (43 \ miles)(24 \ miles)\\A &= 1032 \ square \ miles$

Review

When converting measurements, make sure you cancel out units that are diagonal from each other in the ratios that you are multiplying to determine the unit your solution has.

Indirect Measurement

Indirect measurement is a technique that uses proportions to find a measurement when

direct measurement is not possible.

Proportion

A proportion is two ratios that are equal to each other.

Scale Factor

The scale factor is the ratio that determines the proportional relationship between the

sides of similar figures.

Similar Figures

Similar figures are shapes that exist in proportion to each other. They have congruent angles, but their sides are different lengths.