Indirect Measurement: Definition & Examples

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  • 0:00 Indirect Measurement
  • 0:50 Pythagorean Theorem
  • 1:55 Proportions and…
  • 3:16 Thales Shadow Theorem
  • 4:10 Lesson Summary
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Lesson Transcript
Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

Learn what indirect measurement is and see what is involved when we use this measuring tool. Become comfortable applying indirect measurement through explanation and examples.

Indirect Measurement

Assume we wanted to build a bridge across a small river where the land on one side of the river is higher than the land on the other. We know the river is 30 ft. wide where we want to build the bridge, and we know that the higher side of the river is 5 ft. from the water. How can we figure out how long our bridge needs to be?

This is a situation where we can't measure this distance using a ruler or some other measurement device. In other words, we can't measure it directly. When situations like this arise, we use what is called indirect measurement. Indirect measurement is an approach to measuring things using alternative measurements and properties to find our desired measurement.

Indirect measurement normally involves properties pertaining to the Pythagorean theorem, proportions, similar triangles or polygons, and others. Let's review these topics and look at some examples where we need to measure something indirectly using these tools.

Pythagorean Theorem

The Pythagorean theorem states that for any right triangle with side lengths a, b, and c, where c is the hypotenuse or longest side, it is always true that a^2 + b^2 = c^2. This theorem comes in extremely handy when it comes to indirect measurement. For instance, consider our initial example of building a bridge across a small river. Let's draw a picture of the scenario.

We see that the bridge is the hypotenuse of a right triangle, and we know the distance across the river and the distance between the water and the land on the higher side of the river. We can use the Pythagorean theorem to find out how long our bridge should be. We plug our side lengths for a and b into a^2 + b^2 = c^2 to get:

52 + 302 = c^2

Simplifying the left-hand side of this equation gives:

925 = c^2

We then take the square root of both sides to solve for c to get:

c = ?925 = 30.414

Thus, it looks like we want our bridge to be about 30.414 ft. long.

Proportions and Similar Triangles

In mathematics, proportions are two ratios that are equal. Proportions are written as a / b = c / d. When two quantities are proportional, they satisfy this equation. For example, 1/2 = 4/8. Reducing this we get 4/8 = 1/2. Therefore, 1/2 is proportional to 4/8.

Now that we have reviewed what proportions are, we can review what similar triangles are. Similar triangles show up frequently in indirect measurement. Two triangles are said to be similar if their corresponding angles have equal measure and their sides are proportional. They have the same shape, but may be different sizes.

We see that similar triangles and proportions go hand in hand. These are both frequently used in indirect measurement.

Let's look at an example of this. Suppose you are at the beach and you want to know how deep the water is 70 meters from shore. You know that the water gets deeper and deeper at a constant rate, and when you are 10 meters from shore, the water is 2 meters deep.

Beach Example
indirect measurement beach example

Notice that the two triangles shown are similar. Therefore, we can represent our depth at 70 meters from shore as x and set up a proportion of the side lengths of our triangles. We have:

x / 2 = 70 / 10

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