## Learning Math: Measurement

# Indirect Measurement and Trigonometry

## Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.

### In This Session

**Part A:** Indirect Measurement With a Transit

**Part B:** Measuring Heights of Tall Objects

**Part C:** Steepness and Trigonometry

**Homework**

How do people determine lengths when they can’t use standard measuring tools, such as a tape measure? How do they find the distance across a river if, for example, they don’t have a boat? Measurements of very large objects or of long distances are often made indirectly, using similar triangles and proportions. Such indirect methods link measurement with geometry and number.

For information on required and/or optional materials for this session, see **Note 1**.

### Learning Objectives

In this session, you will do the following:

- Explore a number of methods for indirect measuring, such as using similar triangles, shadows, and transits
- Learn about right-triangle trigonometry
- Learn about the relationships between steepness, angle of elevation, and height-to-distance ratio (tangent ratio)
- Use trigonometry ratios to solve problems involving right triangles

### Key Terms

**Previously Introduced:**

**Ratio: **A ratio is a comparison between two quantities. A measurement is a type of ratio — it is a comparison with a unit. When we state that a**n object is eight inches long, we mean in comparison to the unit of one inch.**

**Scale Factor: **A scale factor is a constant used to enlarge or reduce a figure. For example, if the sides of a triangle are enlarged to twice the length of the original triangle, we say the scale factor is 2.

**New in This Session:**

**Proportion: **A proportion is an equation that states that two ratios are equal, for example 2:1 = 6:3 (or, 2/1 = 6/3).

**Similar Triangles: **Similar triangles are triangles that have the same shape but may be different sizes. In similar triangles, the corresponding angles are congruent, and the corresponding sides are in proportion.

**Tangent: **The tangent of an acute angle in a right triangle (∠α) is the length of the side opposite to ∠α divided by the length of the side adjacent to ∠α. We often abbreviate this as tan.

∠α = (opposite)/(adjacent).

### Notes

**Note 1**

**Materials Needed:**

- Drinking straw*
- Metric ruler*
- Protractor*
- Pushpin*
- Tape
- Ruler
- Tape measure
- Scientific calculator or trigonometric function table

Trundle wheel (a plastic wheel, usually graduated in 5 cm intervals, designed to measure lengths by counting the number of clicks, each of which equals 1 m. It can be purchased from:

ETA/Cuisenaire

500 Greenview Court

Vernon Hills, IL 60061

Phone: 800-445-5985/800-816-5050 (Customer service)

Fax: 800-875-9643/847-816-5066

http://www.etacuisenaire.com

* You’ll need one of these for each homemade transit you make in Part A