1.2.6: Units in Scale Drawings

Lesson

Let's use different scales to describe the same drawing.

Exercise \(\PageIndex{1}\): Centimeters in a Mile

There are 2.54 cm in an inch, 12 inches in a foot, and 5,280 feet in a mile. Which expression gives the number of centimeters in a mile? Explain your reasoning.

1. \(\frac{2.54}{12\cdot 5,280}\)
2. \(5,280\cdot 12\cdot (2.54)\)
3. \(\frac{1}{5,280\cdot 12\cdot (2.54)}\)
4. \(5,280+12+2.54\)
5. \(\frac{5,280\cdot 12}{2.54}\)

Exercise \(\PageIndex{2}\): Card Sort: Scales

Your teacher will give you some cards with a scale on each card.

1. Sort the cards into sets of equivalent scales. Be prepared to explain how you know that the scales in each set are equivalent. Each set should have at least two cards.
2. Trade places with another group and check each other’s work. If you disagree about how the scales should be sorted, work to reach an agreement.
   Pause here so your teacher can review your work.
3. Next, record one of the sets with three equivalent scales and explain why they are equivalent.

Exercise \(\PageIndex{3}\): The World's Largest Flag

As of 2016, Tunisia holds the world record for the largest version of a national flag. It was almost as long as four soccer fields. The flag has a circle in the center, a crescent moon inside the circle, and a star inside the crescent moon.

1. Complete the table. Explain or show your reasoning.

   Table \(\PageIndex{1}\)

   	flag length	flag height	height of crescent moon
   actual	\(396\) m		\(99\) m
   at \(1\) to \(2,000\) scale		\(13.2\) cm

2. Complete each scale with the value that makes it equivalent to the scale of 1 to 2,000. Explain or show your reasoning.
   1. 1 cm to ____________ cm
   2. 1 cm to ____________ m
   3. 1 cm to ____________ km
   4. 2 m to _____________ m
   5. 5 cm to ___________ m
   6. ____________ cm to 1,000 m
   7. ____________ mm to 20 m
3. 1. What is the area of the large flag?
   2. What is the area of the smaller flag?
   3. The area of the large flag is how many times the area of the smaller flag?

Exercise \(\PageIndex{4}\): Pondering Pools

Your teacher will give you a floor plan of a recreation center.

1. What is the scale of the floor plan if the actual side length of the square pool is 15 m? Express your answer both as a scale with units and without units.
2. Find the actual area of the large rectangular pool. Show your reasoning.
3. The kidney-shaped pool has an area of 3.2 cm² on the drawing. What is its actual area? Explain or show your reasoning.

Are you ready for more?

1. Square A is a scaled copy of Square B with scale factor 2. If the area of Square A is 10 units², what is the area of Square B?
2. Cube A is a scaled copy of Cube B with scale factor 2. If the volume of Cube A is 10 units³, what is the volume of Cube B?
3. The four-dimensional Hypercube A is a scaled copy of Hypercube B with scale factor 2. If the “volume” of Hypercube A is 10 units⁴, what do you think the “volume” of Hypercube B is?

Summary

Sometimes scales come with units, and sometimes they don’t. For example, a map of Nebraska may have a scale of 1 mm to 1 km. This means that each millimeter of distance on the map represents 1 kilometer of distance in Nebraska. Notice that there are 1,000 millimeters in 1 meter and 1,000 meters in 1 kilometer. This means there are \(1,000\cdot 1,000\) or 1,000,000 millimeters in 1 kilometer. So, the same scale without units is 1 to 1,000,000, which means that each unit of distance on the map represents 1,000,000 units of distance in Nebraska. This is true for any choice of unit to express the scale of this map.

Sometimes when a scale comes with units, it is useful to rewrite it without units. For example, let's say we have a different map of Rhode Island, and we want to use the two maps to compare the size of Nebraska and Rhode Island. It is important to know if the maps are at the same scale. The scale of the map of Rhode Island is 1 inch to 10 miles. There are 5,280 feet in 1 mile, and 12 inches in 1 foot, so there are 63,360 inches in 1 mile (because \(5,280\cdot 12=63,360\)). Therefore, there are 633,600 inches in 10 miles. The scale of the map of Rhode Island without units is 1 to 633,600. The two maps are not at the same scale, so we should not use these maps to compare the size of Nebraska to the size of Rhode Island.

Here is some information about equal lengths that you may find useful.

Customary Units

1 foot (ft) = 12 inches (in)
1 yard (yd) = 36 inches
1 yard = 3 feet
1 mile = 5,280 feet

Metric Units

1 meter (m) = 1,000 millimeters (mm)
1 meter = 100 centimeters
1 kilometer (km) = 1,000 meters

Equal Lengths in Different Systems

1 inch = 2.54 centimeters
1 foot \(\approx\) 0.30 meter
1 mile \(\approx\) 1.61 kilometers

1 centimeter \(\approx\) 0.39 inch
1 meter \(\approx\) 39.37 inches
1 kilometer \(\approx\) 0.62 mile

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 \(\PageIndex{1}\)

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 \(\PageIndex{2}\)

Practice

Exercise \(\PageIndex{5}\)

The Empire State Building in New York City is about 1,450 feet high (including the antenna at the top) and 400 feet wide. Andre wants to make a scale drawing of the front view of the Empire State Building on an \(8\frac{1}{2}\)-inch-by-\(11\)-inch piece of paper. Select a scale that you think is the most appropriate for the scale drawing. Explain your reasoning.

1. 1 inch to 1 foot
2. 1 inch to 100 feet
3. 1 inch to 1 mile
4. 1 centimeter to 1 meter
5. 1 centimeter to 50 meters
6. 1 centimeter to 1 kilometer

Exercise \(\PageIndex{6}\)

Elena finds that the area of a house on a scale drawing is 25 square inches. The actual area of the house is 2,025 square feet. What is the scale of the drawing?

Exercise \(\PageIndex{7}\)

Which of these scales are equivalent to 3 cm to 4 km? Select all that apply. Recall that 1 inch is 2.54 centimeters.

1. 0.75 cm to 1 km
2. 1 cm to 12 km
3. 6 mm to 2 km
4. 0.3 mm to 40 m
5. 1 inch to 7.62 km

Exercise \(\PageIndex{8}\)

These two triangles are scaled copies of one another. The area of the smaller triangle is 9 square units. What is the area of the larger triangle? Explain or show how you know.

Figure \(\PageIndex{3}\)

Exercise \(\PageIndex{9}\)

Water costs $1.25 per bottle. At this rate, what is the cost of:

1. 10 bottles?
2. 20 bottles?
3. 50 bottles?

Exercise \(\PageIndex{10}\)

The first row of the table shows the amount of dish detergent and water needed to make a soap solution.

1. Complete the table for 2, 3, and 4 batches.
2. How much water and detergent is needed for 8 batches? Explain your reasoning.

Table \(\PageIndex{2}\)

number of batches	cups of water	cups of detergent
1	6	1
2
3
4