1.2.4: Changing Scales in Scale Drawings

Lesson

Let's explore different scale drawings of the same actual thing.

Exercise \(\PageIndex{1}\): Appropriate Measurements

1. If a student uses a ruler like this to measure the length of their foot, which choices would be appropriate measurements? Select all that apply. Be prepared to explain your reasoning.

Figure \(\PageIndex{1}\)

1. \(9\frac{1}{4}\) inches
2. \(9\frac{5}{64}\) inches
3. \(23.47659\) centimeters
4. \(23.5\) centimeters
5. \(23.48\) centimeters

2. Here is a scale drawing of an average seventh-grade student's foot next to a scale drawing of a foot belonging to the person with the largest feet in the world. Estimate the length of the larger foot.

Figure \(\PageIndex{2}\)

Exercise \(\PageIndex{2}\): Same Plot, Different Drawings

Here is a map showing a plot of land in the shape of a right triangle.

Figure \(\PageIndex{3}\): By American Fact Finder. Public Domain. American Fact Finder. Source.

1. Your teacher will assign you a scale to use. On centimeter graph paper, make a scale drawing of the plot of land. Make sure to write your scale on your drawing.
2. What is the area of the triangle you drew? Explain or show your reasoning.
3. How many square meters are represented by 1 square centimeter in your drawing?
4. After everyone in your group is finished, order the scale drawings from largest to smallest. What do you notice about the scales when your drawings are placed in this order?

Are you ready for more?

Noah and Elena each make a scale drawing of the same triangular plot of land, using the following scales. Make a prediction about the size of each drawing. How would they compare to the scale drawings made by your group?

1. Noah uses the scale 1 cm to 200 m.
2. Elena uses the scale 2 cm to 25 m.

Exercise \(\PageIndex{3}\): A New Drawing of the Playground

Here is a scale drawing of a playground.

Figure \(\PageIndex{4}\)

The scale is 1 centimeter to 30 meters.

1. Make another scale drawing of the same playground at a scale of 1 centimeter to 20 meters.
2. How do the two scale drawings compare?

Summary

Sometimes we have a scale drawing of something, and we want to create another scale drawing of it that uses a different scale. We can use the original scale drawing to find the size of the actual object. Then we can use the size of the actual object to figure out the size of our new scale drawing.

For example, here is a scale drawing of a park where the scale is 1 cm to 90 m.

Figure \(\PageIndex{5}\)

The rectangle is 10 cm by 4 cm, so the actual dimensions of the park are 900 m by 360 m, because \(10\cdot 90=900\) and \(4\cdot 90=360\).

Suppose we want to make another scale drawing of the park where the scale is 1 cm to 30 meters. This new scale drawing should be 30 cm by 12 cm, because \(900\div 30=30\) and \(360\div 30=12\).

Another way to find this answer is to think about how the two different scales are related to each other. In the first scale drawing, 1 cm represented 90 m. In the new drawing, we would need 3 cm to represent 90 m. That means each length in the new scale drawing should be 3 times as long as it was in the original drawing. The new scale drawing should be 30 cm by 12 cm, because \(3\cdot 10=30\) and \(3\cdot 4=12\).

Since the length and width are 3 times as long, the area of the new scale drawing will be 9 times as large as the area of the original scale drawing, because \(3^{2}=9\).

Glossary Entries

Definition: Scale

A scale tells how the measurements in a scale drawing represent the actual measurements of the object.

For example, the scale on this floor plan tells us that 1 inch on the drawing represents 8 feet in the actual room. This means that 2 inches would represent 16 feet, and \(\frac{1}{2}\) inch would represent 4 feet.

Figure \(\PageIndex{6}\)

Definition: Scale Drawing

A scale drawing represents an actual place or object. All the measurements in the drawing correspond to the measurements of the actual object by the same scale.

Figure \(\PageIndex{7}\)

Practice

Exercise \(\PageIndex{4}\)

Here is a scale drawing of a swimming pool where 1 cm represents 1 m.

Figure \(\PageIndex{8}\)

1. How long and how wide is the actual swimming pool?
2. Will a scale drawing where 1 cm represents 2 m be larger or smaller than this drawing?
3. Make a scale drawing of the swimming pool where 1 cm represents 2 m.

Exercise \(\PageIndex{5}\)

A map of a park has a scale of 1 inch to 1,000 feet. Another map of the same park has a scale of 1 inch to 500 feet. Which map is larger? Explain or show your reasoning.

Exercise \(\PageIndex{6}\)

On a map with a scale of 1 inch to 12 feet, the area of a restaurant is 60 in². Han says that the actual area of the restaurant is 720 ft². Do you agree or disagree? Explain your reasoning.

Exercise \(\PageIndex{7}\)

If Quadrilateral Q is a scaled copy of Quadrilateral P created with a scale factor of 3, what is the perimeter of Q?

Figure \(\PageIndex{9}\): A trapezoid labeled P. The parallel sides form the top and bottom of the shape. The length of the top side is labeled 7 units and the length of the bottom side is labeled 25 units. The two legs that join the two parallel sides together are each labeled 15 units.

(From Unit 1.1.3)

Exercise \(\PageIndex{8}\)

Triangle \(DEF\) is a scaled copy of triangle \(ABC\). For each of the following parts of triangle \(ABC\), identify the corresponding part of triangle \(DEF\).

• angle \(ABC\)
• angle \(BCA\)
• segment \(AC\)
• segment \(BA\)

Figure \(\PageIndex{10}\): Two triangles labeled ABC and DEF. Triangle ABC has horizontal side AB with point B to the right of point A and point C is directly above side AB. Triangle DEF has horizontal side DE with point E to the right of point D and point F is directly above side DE.

(From Unit 1.1.2)