# 4.10: Understand Three-Dimensional Scale Models and Designs

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

**Practice**Similar Solids

Have you ever thought about different types of maps? Take a look at this dilemma.

Josh spent most of his Saturday morning at the library looking at different books on Mount Everest. After completing his drawings of the Mountain, and figuring out the scale for his model, Josh wanted to look at some maps that other people had created of the mountain.

He began looking in books, but most of the maps weren’t drawn in very high detail. Finally, after a lot of searching, he began using the computer.

Right away, Josh discovered this map on a website.

http://upload.wikimedia.org/wikipedia/commons/6/66/MountEverestRelief.png

“What did you find?” his sister Karen asked. She had also been at the library writing a book report.

“I found this map. It is called a relief map,” Josh said.

“What a “relief” that you found it!” Karen joked.

“Not really. It is called a relief map because of what is on it. Look,” Josh began to explain all about the map.

**Have you ever seen a relief map? Before Josh explains about relief maps, use this Concept to learn all about them. When finished, you will be able to explain the difference between a relief map and a two-dimensional map.**

### Guidance

There are many different types of maps and models. In other words, real-world distances or objects can be represented in many different ways. If we worked on interpreting two-dimensional maps, scale drawings and floor plans, we would use a scale to interpret measurements the actual dimensions and the scale dimensions. These are two – dimensional representations, this means that the things being represented could be easily shown in a flat plane.

**What happens is something can’t be shown in a two – dimensional way?**

When this happens, we have to use a ** three – dimensional** method of display. Whereas a

**image takes into account length and width,**

*two – dimensional***a three – dimensional figure contains the length, width and height or depth.**

When we wanted to represent a two – dimensional space, like a map, we used a scale drawing or a scale map. When we want to represent a three-dimensional space, we use a ** scale model** to represent the space.

**A scale model is a model used to represent a three – dimensional space.**

Yes. You can find the actual dimensions of the space in the same way as in a scale drawing.

**First, let’s think about how we can find the actual dimensions.**

In order to find the actual dimensions from a scale model, you can set up and solve a proportion. The scale given in the model is the first ratio. The unknown length and the scale length is the second ratio. We compare the scale in the first ratio and we compare the two lengths in the second ratio.

**Brianna is making a scale model of the White House using the scale 1 cm = 0.5 m. If the height of Brianna’s model is 42 cm, what is the height of the actual White House?**

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{42 \ cm}{x \ m}\end{align*}

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

\begin{align*}(1)x &= 42(0.5)\\ x &= 21\end{align*}

**The actual height of the White House is 21 meters.**

**What about maps?**

You will notice that if you look at a two – dimensional map, that while it is excellent for measuring distances, it isn’t as helpful when measuring mountains or other features.

When we want to show a map in a three-dimensional way, we use a *topographic map.*

A *topographic map***is a type of map that shows not only the distances on the ground, but also the relief features of the area, such as mountains.** The map uses *contour lines***to show the elevation of the area**. Each contour line is a line of equal ** elevation** or height. They show the general shape of the

**or land. When contour lines are spread farther apart, the elevation is not as steep. Where contour lines are bunched close together, the elevation is steeper.**

*terrain*
**Topographic maps may also use colors to represent different features.** Blue represents water, green represents vegetation, and brown lines represent topographic contours.

You can interpret these maps by using the scale. There will be a scale to show what each distance on the map represents, just like other maps. There will also be a scale to tell you what each contour line represents.

**This map shows a mountain from a national park in California. What is the height of the mountain?**

**First, look at the map scale. It states that the contour interval is 40 feet. That means that each contour line represents 40 feet of elevation.**

**Count the number of contour lines that make up the mountain. There are 10 contour lines.**

**Write a proportion to find the height of the mountain.**

\begin{align*}\frac{1 \ line}{40 \ feet}=\frac{10 \ lines}{x \ feet}\end{align*}

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

\begin{align*}(1)x &= 10(40)\\ x &= 400\end{align*}

**The mountain is 400 feet high.**

**Use the contour lines above to answer each question.**

#### Example A

What height would be represented by 5 contour lines?

**Solution: 200 feet**

#### Example B

What height would be represented by 8 contour lines?

**Solution: 320 feet**

#### Example C

What height would be represented by 3 contour lines?

**Solution: 120 feet**

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

**A two – dimensional map is created on a flat surface and only shows the dimensions length and width. There aren’t any other three-dimensional features included on the map. When Josh drew the area of the Everest from space in an earlier lesson, he drew it in a two – dimensional way. He only showed the length, width and area in the scale drawing.**

**A relief map uses a scale just like any other map, but other features are included on the map. A relief map uses different colors and textures to show the contour of the terrain. It also includes bodies of water and other landmarks. In addition, the map shows contour lines which measure the elevation of a natural land mass.**

### Vocabulary

- Two – Dimensional
- A figure drawn in two dimensions is only drawn using length and width.

- Three – Dimensional
- A figure drawn using length, width and height or depth.

- Scale Model
- a model that represents a three – dimensional space.

- Topographic Map
- a map that shows distances on the ground, but also relief features of the map such as mountains.

- Contour Lines
- lines on a map to show elevation. Each contour line represents the same measure of elevation.

- Elevation
- the measure of height

- Terrain
- the land

### Guided Practice

Here is one for you to try on your own.

Mike is building a scale model of an airplane using the scale \begin{align*}\frac{1}{4} \ inch = 1 \ foot\end{align*}. If the actual length of the airplane is 150 feet, what will the length of the scale model be?

**Solution**

**First, set up a proportion.**

The scale for the model is \begin{align*}\frac{1}{4} \ inch = 1 \ foot\end{align*}. So set up a ratio using these values: \begin{align*}\frac{0.25 \ inch}{1 \ foot}\end{align*}.

**Now write the second ratio.**

You know the actual length is 150 feet. 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{0.25 \ inch}{1 \ foot}= \frac{x \ inches}{150 \ feet}\end{align*}

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

\begin{align*}(1)x &= 150(0.25)\\ x &= 37.5\end{align*}

**The length of the scale model is 37.5 inches.**

### Video Review

### Practice

Directions: Solve each problem.

Kevin built a scale model of a pool. He used the scale \begin{align*}\frac{1}{2}^{\prime\prime} = 5 \ ft\end{align*}. Use this information to answer the following questions.

- The width of the pool on the scale model measures 1.5 inches. What is the actual width of the pool?
- The length of the pool on the scale model measures 2.5 inches. What is the actual length of the pool?
- The depth of the pool on the scale model measures 0.5 inches. What is the actual volume of the pool.

This is a map of a national park. Use this information to answer the following questions.

- The map distance of the length of the distance across the lake is 1.5 cm. What is the actual distance across the lake?
- Explain why the contour lines on the map are closer together at some points and farther apart at other points.

Directions: Answer each of the following questions as true or false.

- A topographic map would include lakes and rivers.
- A two – dimensional map could also be a topographic map.
- Three – dimensions means including length, width and height.
- Depending on what you are measuring, height might be replaced by depth.
- There is a proportional relationship between length and the area of a figure.
- A two – dimensional map also includes contour lines.
- Contour lines can be different sizes if there is a different elevation involved.
- Elevation also means height.
- A topographic map can be built in three dimensions.
- Two – dimensional maps and three – dimensional maps both include the same information.

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Contour Lines

Contour line are drawn on a map to show elevation. Each contour line represents the same measure of elevation.Elevation

Elevation is the measure of height.Scale Model

A scale model is a model that represents a three-dimensional space.Terrain

The terrain on a map is the land.Three – Dimensional

A figure drawn in three dimensions is drawn using length, width and height or depth.Topographic Map

A topographic map is a map that not only shows distances on the ground, but also features of the area such as mountains.Two – Dimensional

A figure drawn in two dimensions is only drawn using length and width.### Image Attributions

Here you'll understand three-dimensional scale models and designs.