# Trig-o-nom-e-tree!

## Using Clinometers and Trig Ratios to Measure Unknown Heights

K20 Center, Matthew McDonald, Jennifer Kornell | Published: June 2nd, 2022 by K20 Center

• Grade Level 9th, 10th, 11th, 12th
• Subject Mathematics
• Course Geometry, Precalculus
• Time Frame 55-70 minutes
• Duration 2 class periods

### Summary

In this lesson, students will learn how to use a clinometer and trigonometric ratios to solve problems involving right triangles.

### Essential Question(s)

How can you determine the height of an object without actually measuring it?

### Snapshot

Engage

Students brainstorm ideas for how to determine the height of a tree without measuring it.

Explore

Students build a clinometer and discuss the mathematics involved with its use.

Explain

Students learn about trigonometric ratios and connect them to prior knowledge of special right triangles.

Extend

Students apply their knowledge of trigonometric ratios and clinometers to determine the heights of some objects around the school.

Evaluate

Students reflect on what they learned using the What? So What? Now What? strategy. As an additional option, they create a presentation that explains the process they used to determine the heights of objects around the school.

### Materials

• Lesson Slides (attached)

• Record Sheet (attached; one per student)

• Right Triangle Problems (attached; one per student)

• Clinometer Cutout (attached; one per student to be cut out on cardstock)

• Straws

• Tape

• String

• Washers or paper clips

• Tape measures

• Calculators

### Engage

5 Minute(s)

Go to slide 5.

Assign students to work in pairs.

Using the Think-Pair-Share strategy, give students thirty seconds to think about the following question: How can you determine the height of a tree or a building without actually measuring it? Ask students to discuss their thinking with their partner. Have groups share out by writing their answers on the board. Discuss the merits and shortcomings of each solution.

### Explore

15 Minute(s)

Go to slide 6.

Have students work in pairs to build a clinometer.

Before class, print copies of the Clinometer Cutout on card stock. Pass out a copy to each student and have them cut out the protractor that's printed on the page. Next, students attach a straw along the base of the protractor with tape and cut it to fit the length of the base. Finally, students place a small hole in the center of the base and secure a string in a knot through the hole and tie a washer to the other end of the string to keep the string taut.

Discuss with students how the clinometer can be used to determine the heights of objects.

### Explain

15 Minute(s)

Go to slide 7. Now is the time to present trig ratios and connect them to using the clinometer. Draw a right triangle on the board and label one angle THETA. Ask students to do the same on their papers. Ask students to recall SOHCAHTOA and, still working in pairs, define what it means. Call on three pairs to define SOH, CAH, and TOA. Have students write down the sine, cosine, and tangent ratios on their paper and label the sides of their triangle appropriately. You might want to model diagram this on the board if students struggle. Students might also discuss special right triangles and angle of depression or elevation from geometry.

Go to slide 8. Give students two special right triangles 45-45-90 and 30-60-90. Connect this information back to the clinometer.

Here are some questions for students to consider:

• Where do you see right triangle connections in your clinometer?

• How does the clinometer help with measuring trig ratios?

• How can you use trig ratios to measure heights of objects?

### Extend

15 Minute(s)

Go to slide 9.

Students should now extend their basic understanding of clinometers and trig ratios to further explore how the two connect to determine heights of objects.

Option 1: Exploring heights around the school

Students continue to work in pairs to determine the heights of different objects. Pass out the attached Record Sheet for students to use to record their measurements. Have students perform these steps to get their measurements:

1. Find the angle of elevation from the student to the top of an object and the distance from the object to the student as they are standing looking through the clinometer at the object.

2. Record the height of the student using the clinometer from ground to eyes (the height of the clinometer). For the first two clinometer angles, students walk toward the object or away from the object to the location where the clinometer registers 45 degrees for the first problem and 30 degrees for the second problem. They repeat this process for three more objects at any clinometer angle and record their measurements.

3. Based on their knowledge of special right triangles, students should be able to find the height of the objects for the first two angles. Ask them to try to devise a way to find the other three objects' heights.

Option 2: Worksheet

Pass out the attached Right Triangle Problems worksheet. The worksheet covers real-world scenarios where students can explore angles of elevation and depression.

### Evaluate

5 Minute(s)

Go to slide 10.

Use the What? So What? Now What? instructional strategy to evaluate student understanding.

• What? What did you do in this lesson?

• So what? What mathematics did you use and why do you think it was important?

• Now what? What will you take away from this lesson?