We are occasionally faced with having to make measurements of things that would be difficult to measure directly: the height of a tall tree, the width of a wide river, the height of the moon’s craters, even the distance between two cities separated by mountainous terrain. In such circumstances, measurements can be made indirectly, using proportions and similar triangles. Such indirect methods link measurement with geometry and numbers. In this lesson, we will examine some of the methods for making indirect measurements.
A map is a two-dimensional, geometrically accurate representation of a section of the Earth’s surface. Maps are used to show, pictorially, how various geographical features are arranged in a particular area. The scale of the map describes the relationship between distances on a map and the corresponding distances on the earth's surface. These measurements are expressed as a fraction or a ratio.
In the last lesson, you learned the different ways to write a ratio: using the fraction bar, using a colon, and in words. Outside of mathematics books, ratios are often written as two numbers separated by a colon (:). Here is a table that compares ratios written in two different ways.
Is Read As
one to twenty
two to three
one to one-thousand
If a map had a scale of 1:1000 (“one to one-thousand”), one unit of measurement on the map (1 inch or 1 centimeter, for example) would represent 1000 of the same units on the ground. A 1:1 (one to one) map would be a map as large as the area it shows!
Example : Anne is visiting a friend in London and is using the map above to navigate from Fleet Street to Borough Road. She is using a 1:100,000 scale map, where 1 cm on the map represents 1 km in real life. Using a ruler, she measures the distance on the map as 8.8 cm. How far is the real distance from the start of her journey to the end?
The scale is the ratio of distance on the map to the corresponding distance in real life and can be written as a proportion.
dist.on mapreal dist.=1100,000
By substituting known values, the proportion becomes:
8.8 cmreal dist.(x)880000 cmx=1100,000=x=8800 mCross multiply.100 cm=1 m.1000 m=1 km.
The distance from Fleet Street to Borough Road is 8800 m or 8.8 km.
We could, in this case, use our intuition: the 1 cm=1 km scale indicates that we could simply use our reading in centimeters to give us our reading in km. Not all maps have a scale this simple. In general, you will need to refer to the map scale to convert between measurements on the map and distances in real life!
Example 1: Oscar is trying to make a scale drawing of the Titanic, which he knows was 883 feet long. He would like his drawing to be 1:500 scale. How long, in inches, must his sheet of paper be?
Solution: We can reason that since the scale is 1:500 that the paper must be 883500=1.766 feet long. Converting to inches gives the length at 12(1.766) in=21.192 in.
The paper should be at least 22 inches long.
Not everything has a scale. Architecture such as the St. Louis Arch, St. Basil’s Cathedral, or the Eiffel Tower does not have a scale written on the side. It may be necessary to measure such buildings. To do so requires knowledge of similar figures and a method called indirect measurement.
Similar figures are often used to make indirect measurements. Two shapes are said to be similar if they are the same shape and “in proportion.” The ratio of every measurable length in one figure to the corresponding length in the other is the same. Similar triangles are often used in indirect measurement.
Anatole is visiting Paris, and wants to know the height of the Eiffel Tower. Since he's unable to speak French, he decides to measure it in three steps.
- He measures out a point 500 meters from the base of the tower, and places a small mirror flat on the ground.
- He stands behind the mirror in such a spot that standing upright he sees the top of the tower reflected in the mirror.
- He measures both the distance from the spot where he stands to the mirror (2.75 meters) and the height of his eyes from the ground (1.8 meters).
Explain how Anatole is able to determine the height of the Eiffel Tower from these numbers and determine what that height is.
First, we will draw and label a scale diagram of the situation.
The Law of Reflection states, “The angle at which the light reflects off the mirror is the same as the angle at which it hits the mirror.” Using this principle and the figure above, you can conclude that these triangles are similar with proportional sides.
This means that the ratio of the long leg in the large triangle to the length of the long leg in the small triangle is the same ratio as the length of the short leg in the large triangle to the length of the short leg in the small triangle.
500 m2.75 m1.8⋅5002.75327.3=x1.8 m=x1.8⋅1.8≈x
The Eiffel Tower, according to this calculation, is approximately 327.3 meters high.
Example 2: Bernard is looking at a lighthouse and wondering how high it is. He notices that it casts a long shadow, which he measures at 200 meters long. At the same time, he measures his own shadow at 3.1 meters long. Bernard is 1.9 meters tall. How tall is the lighthouse?
Solution: Begin by drawing a scale diagram.
You can see there are two right triangles. The angle at which the sun causes the shadow from the lighthouse to fall is the same angle at which Bernard’s shadow falls. We have two similar triangles, so we can again say that the ratio of the corresponding sides is the same.
200 m3.1 m1.9⋅200 m3.1 m122.6=x1.9 m=x1.9 m⋅1.9≈x
The lighthouse is approximately 122.6 meters tall.
Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Basic Algebra: Scale and Indirect Measurement (10:44)
- Define similar figures.
- What is true about similar figures?
- What is the process of indirect measurement? When is indirect measurement particularly useful?
- State the Law of Reflection. How does this law relate to similar figures?
- A map has a 1 inch : 20 mile scale. If two cities are 1,214 miles apart, how far apart would they be on this map?
- What would a scale of 1 mile : 1 mile mean on a map? What problems would the cartographer encounter?
- A woman 66 inches tall stands beside a tree. The length of her shadow is 34 inches and the length of the tree’s shadow is 98 inches. To the nearest foot, how tall is the tree?
- Use the scale diagram above to determine:
- The length of the helicopter (cabin to tail)
- The height of the helicopter (floor to rotors)
- The length of one main rotor
- The width of the cabin
- the diameter of the rear rotor system
- On a sunny morning, the shadow of the Empire State Building is 600 feet long. At the same time, the shadow of a yardstick (3 feet long) is 1 foot, 514 inches. How high is the Empire State building?
- Omar and Fredrickson are 12.4 inches apart on a map with a scale of 1.2 inches : 15 miles. How far apart are their two cities in miles?
- A man 6 feet tall is standing next to a dog. The man’s shadow is 9 feet and the dog’s shadow is 6 feet. How tall is the dog?
- A model house is 12 inches wide. It was built with a ratio of 3 inches : 4 meters. How wide is the actual house?
- Using the diagram below and assuming the two triangles are similar, find DE, the length of DE¯¯¯¯¯.
- A 42.9-foot flagpole casts a 253.1-foot shadow. What is the length of the shadow of a woman 5 feet 5 inches standing next to the flagpole?
- Solve for n: −(7−7a)+4a=−23+3a.
- How is evaluating different from solving? Provide an example to help you illustrate your explanation.
- Simplify 243−−−√.
- Simplify: 2(8g+2)−1+4g−(2−5g).
- Draw a graph that is not a function. Explain why your picture is not a function.
- Jose has 23 the amount of money that Chloe has. Chloe has four dollars less than Huey. Huey has $26. How much does Jose have?