# 4.4: Scale in Two Dimensions

**At Grade**Created by: CK-12

## Introduction

*Drawing Everest*

“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 \begin{align*}\frac{1}{4}^{\prime\prime} = 1 \ mile\end{align*}.

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

**This lesson is all about scale drawings. Use what you learn to help you to answer these questions.**

**If you want to see the pictures Josh and Carlo saw go to**

www.solarviews.com/cap/earth/everest2.htm

*What You Will Learn*

In this lesson, you will learn how to complete the following skills.

- Read and interpret scale drawings and floor plans.
- Read and interpret maps for distance and area.
- Compare scale relationships of distance and area.
- Solve real-world problems involving two-dimensional scale drawings and maps.

*Teaching Time*

I. **Read and Interpret Scale Drawings and Floor Plans**

In the last lesson, we looked at scale factors and how to use that ratio to find scale and actual dimensions. In this lesson, we’ll look at how you can set up a proportion using scale factors from scale drawings and maps.

**A** *scale drawing***or floor plan is a representation of an actual object or space drawn in two-dimensions.** For a floor plan, you can imagine that you are directly above the building looking down. The lines represent the walls of the building, and the space in between the lines represents the floor.

**In order to find the actual dimensions from a floor plan, you can set up and solve a proportion. The scale given in the drawing is the first ratio. The unknown length and the scale length is the second ratio.**

Example

*This floor plan shows several classrooms at Craig’s school. The length of Classroom 2 in the floor plan is 2 inches. What is the actual length, in feet, of Classroom 2?*

**Set up a proportion.** The scale in the drawing says that \begin{align*}\frac{1}{2} \ inch = 3 \ feet\end{align*}. So set up a ratio using these values: \begin{align*}\frac{0.5 \ inch}{3 \ feet}\end{align*}.

**Now write the second ratio.** You know the scale length is 2 inches. The unknown length is \begin{align*}x\end{align*}. Make sure that the second ratio follows the form of the first ratio: inches over feet.

\begin{align*}\frac{0.5 \ inch}{3 \ feet} = \frac{2 \ inches}{x \ feet}\end{align*}

Now cross-multiply to solve for \begin{align*}x\end{align*}.

\begin{align*}(0.5)x &= 2(3)\\ 0.5x &= 6\\ x &= 12\end{align*}

**The actual length of the classroom is 12 feet.**

Example

*This scale drawing shows the fountain in front of a hotel. The diameter of the fountain in the scale drawing is 4 centimeters. What is the actual diameter of the fountain?*

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

\begin{align*}\frac{1 \ cm}{0.5 \ m}\end{align*}

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

\begin{align*}\frac{1 \ cm}{0.5 \ m} = \frac{4 \ cm}{x \ m}\end{align*}

Now cross-multiply and solve for \begin{align*}x\end{align*}.

\begin{align*}(1)x &= 4(0.5)\\ x &= 2\end{align*}

**The actual diameter of the fountain is 2 meters.**

II. **Read and Interpret Maps for Distance and Area**

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.

Example

*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.**

\begin{align*}\frac{0.5 \ inch}{75 \ miles}\end{align*}

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

\begin{align*}\frac{0.5 \ inch}{75 \ miles} = \frac{3 \ inches}{x \ miles}\end{align*}

Now cross-multiply and solve for \begin{align*}x\end{align*}.

\begin{align*}(0.5)x &= 3(75)\\ 0.5x &= 225\\ x &= 450\end{align*}

**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.*

**We can also use a scale to find the area of a space or region. First, we need to figure out the length and width then we can complete any necessary calculations.**

Example

*This floor plan shows Bob’s apartment. The scale dimensions of the living room are 3 inches by 2.5 inches. What is the actual area of his living room?*

**First use a proportion to find the actual length of the living room.** Write the scale as the first ratio, and the scale and unknown actual length of the living room as the second ratio.

\begin{align*}\frac{0.25 \ inch}{1 \ foot} &= \frac{3 \ inches}{x \ feet}\\ (0.25)x &= 3(1)\\ 0.25x &= 3\\ x &= 12\end{align*}

**Next find the actual width of the living room.**

\begin{align*}\frac{0.25 \ inch}{1 \ foot} &= \frac{2.5 \ inches}{x \ feet}\\ (0.25)x &= 2.5(1)\\ 0.25x &= 2.5\\ x &= 10\end{align*}

**Now that you know the actual length and width of the room, you can find the area.**

\begin{align*}A &= lw\\ A &= (12 \ feet)(10 \ feet)\\ A &= 120 \ feet^2\end{align*}

**The area of the living room is 120 square feet.**

*Note: Remember just like with any other word problem, you have to figure out what you are solving for and what information has already been given.*

III. **Compare Scale Relationship of Distance and Area**

Sometimes, we will have two different distances or areas that we are working to compare. When this happens, we can use proportions to compare the differences and similarities. Take a look at this example.

Example

*Marta has a square with a side length of 4 inches. She has a similar square with dimensions that are twice the first square. How does the area of the larger square compare to the area of the smaller square?*

**First, find the dimensions of the larger square. The problem states that the dimensions are twice the first square. We can use this information to figure out the scale factor, and this means they are scaled up by a factor of 2. The side length of the larger square is \begin{align*}4 \ inches \times 2 = 8 \ inches\end{align*}.**

**Next find the area of both squares and compare.**

Area of smaller square:

\begin{align*}A &= lw\\ A &= (4 \ inches)(4 \ inches)\\ A &= 16 \ inches^2\end{align*}

Area of larger square:

\begin{align*}A &= lw\\ A &= (8 \ inches)(8 \ inches)\\ A &= 64 \ inches^2\end{align*}

**Now compare the two areas. You want to know how the area of the larger square compares to the area of the smaller square. Write a ratio comparing the two areas.**

\begin{align*}\frac{64 \ inches^2}{16 \ inches^2} = 4\end{align*}

**The area of the larger square is 4 times larger than the area of the smaller square.**

In the previous example, the scale factor that changed the smaller square to the larger square was 2, but the area was 4 times as large. This leads to a rule when comparing areas of similar figures.

*Write this rule down in your notebook.*

Let’s look at another example.

Example

*Sandy has a garden with a length of 6 feet and a width of 10 feet. He wants to scale up his garden by a factor of 3. How will the area of his new garden compare to the area of the original garden?*

**First, find the dimensions of the new garden.**

\begin{align*}6 \times 3 &= 18\\ 10 \times 3 &= 30\end{align*}

The dimensions of the new garden will be 18 feet by 30 feet.

**Now find the area of both gardens and compare.**

Area of original garden:

\begin{align*}A &= lw\\ A &= (6 \ feet)(10 \ feet)\\ A &= 60 \ feet^2\end{align*}

Area of new garden:

\begin{align*}A &= lw\\ A &= (18 \ feet)(30 \ feet)\\ A &= 540 \ feet^2\end{align*}

**Next compare the two areas.**

\begin{align*}\frac{540 \ feet^2}{60 \ feet^2} = 9\end{align*}

**The area of the new garden will be 9 times greater than the area of the original garden. You can check that your answer is correct using the rule: the ratio of areas of similar figures is the square of the scale factor.**

**The scale factor is 3, and \begin{align*}3^2 = 9\end{align*}.**

IV. **Solve Real – World Problems Involving Two – Dimensional Scale Drawings and Maps**

We have been working with real-world examples now let’s look at another one.

Example

*The local community is planning to put a playground in the space provided. This is the drawing that they have created. You can see that the width of the playground is 7 inches wide in the drawing. If this is the case, what is the actual width of the playground?*

**To work on this problem, first, let’s write a ratio to show the scale.**

\begin{align*}\frac{1^{\prime\prime}}{20 \ ft}\end{align*}

**Next, we can write a proportion showing the scale to the inches on the drawing.**

\begin{align*}\frac{1^{\prime\prime}}{20 \ ft} &= \frac{7^{\prime\prime}}{x}\\ x &= 140 \ feet\end{align*}

**The width of the playground is 140 feet.**

**Notice that whether you are working with floor plans, scale drawings or maps, you are still working to create a proportion and solve for the missing measurements.**

## Real-Life Example Completed

*Drawing Everest*

**Here is the original problem once again. Reread it and then answer the three questions at the end of the problem. There are three parts to your answer.**

“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 \begin{align*}\frac{1}{4}^{\prime\prime} = 1 \ mile\end{align*}.

*Remember, there are three parts to your answer.*

**Solution to Real – Life Example**

**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 \begin{align*}\frac{1}{4}^{\prime\prime} = 1 \ mile\end{align*}.**

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

\begin{align*}\frac{\frac{1}{4}^{\prime\prime}}{1 \ mile} &= \frac{x}{43 \ miles}\\ x &= 10.75^{\prime\prime}\end{align*}

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

\begin{align*}\frac{\frac{1}{4}^{\prime\prime}}{1 \ mile} &= \frac{x}{24 \ miles}\\ x &= 6 \ inches\end{align*}

**The dimensions of Josh’s drawing will be \begin{align*}10.75^{\prime\prime} \times 6^{\prime\prime}\end{align*}**

**This drawing will fit on an \begin{align*}11 \times 14^{\prime\prime}\end{align*} piece of paper.**

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

\begin{align*}A &= lw\\ A &= (43 \ miles)(24 \ miles)\\ A &= 1032 \ square \ miles\end{align*}

## Vocabulary

Here are the vocabulary words that are found in this lesson.

- Ratio
- a way of comparing two quantities using a colon, fraction form or by using the word “to.”

- Proportion
- showing two equal ratios. Two equal ratios form a proportion.

- Similar Figures
- figures that are the same shape but different sizes.

- Scale Drawing
- a drawing that is done with a scale so that specific small units of measure represent larger units of measure.

## Time to Practice

Directions: Using the scale \begin{align*}1^{\prime\prime} = 5.5 \ miles\end{align*}, figure out the number of inches needed to map each number of miles. Use proportions to figure out your answers.

- 16.5 miles
- 11 miles
- 27.5 miles
- 8.25 miles
- 33 miles
- 60.5 miles
- 13.75 miles

Directions: Using the scale \begin{align*}\frac{1}{2}^{\prime\prime} = 100 \ miles\end{align*}, figure out the number of actual miles represented by each scale measurement.

- \begin{align*}1^{\prime\prime}\end{align*}
- \begin{align*}2^{\prime\prime}\end{align*}
- \begin{align*}3^{\prime\prime}\end{align*}
- \begin{align*}\frac{1}{4}^{\prime\prime}\end{align*}
- \begin{align*}\frac{3}{4}^{\prime\prime}\end{align*}
- \begin{align*}1 \frac{1}{2}^{\prime\prime}\end{align*}
- \begin{align*}2 \frac{1}{2}^{\prime\prime}\end{align*}
- \begin{align*}5 \frac{1}{2}^{\prime\prime}\end{align*}
- \begin{align*}7^{\prime\prime}\end{align*}

Directions: This floor plan shows Bonnie’s house. Use it to answer the following questions.

- The width of the bedroom on the floor plan measures 1.5 inches. What is the actual width of the bedroom?
- The length of the kitchen on the floor plan measures 3 inches. What is the actual length of the kitchen?
- The study measures 2 inches by 1.5 inches on the floor plan. What is the actual area of the study?

Directions: In Gary’s town, 2 cm = 1 mile, use this information to answer these questions.

- The map distance from the bank to the post office is 4 centimeters. What is the actual distance from the bank to the post office?
- The map distance from the high school to the library is 5.5 centimeters. What is the actual distance from the high school to the library?
- The park on the map measures 2 centimeters on all sides. What is the actual area of the park?

Directions: Answer each question.

- The scale of a map of the United States is 2 cm = 500 km. The distance from New York City to Los Angeles is about 4,000 km. How far apart should the two cities be on the map?
- A square measures 10 inches on each side. A similar square has dimensions that are 1.5 times as large. How does the area of the larger square compare to the area of the smaller square?
- A classroom measures 21 feet by 15 feet. A second classroom has a similar size but the dimensions are \begin{align*}\frac{1}{3}\end{align*} as long. How does the area of the second classroom compare to the area of the first classroom?
- Yuri’s backyard has an area of 1,000 square feet. The dimensions of Kyle’s backyard are all \begin{align*}\frac{1}{5}\end{align*} the size of Yuri’s. What is the area of Kyle’s backyard?

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