1. Angle of Elevation
By Karen Borst

2. Goal
Develop an understanding of the trigonometric ratios and their real life applications SOH CAH TOA
sin = 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
cos= 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
tan= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

3. Objectives
Given materials, students will successfully build a clinometer which is:
An object used to measure angles to find the height of an object
Using the clinometer, students will find the height of objects that are unmeasurable using trig ratios.
Using the clinometer, students will understand the application of trig ratios.
Tangent used in
these cases

4. Grade Level
Generally this would be a middle school or high school lesson.
Building and reading the clinometer: elementary level
Using tangent to find the height: middle/high school students

5. Standards High School Geometry
Similarity, Right Triangles, & Trigonometry
Define trigonometric ratios and solve problems involving right triangles
6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

6. Previous Class Introduce right triangle trig
SOH CAH TOA
Compare to Pythagorean Theorem
Can only be used to find sides
Determine how to solve for a side or an angle

7. Example
To measure the height of an inaccessible TV tower, a surveyor paces out a base line of 200 meters and measures the angle of elevation to the top of the tower to be 60°. How high is the tower?
tan= 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
tan(60)= 𝑥 200 x
200 tan 60 ≈346 𝑚𝑒𝑡𝑒𝑟𝑠
60°
200 meters

8. Building my Clinometer
Protractor
Ruler
String
Scissors
Tape
Button
Cardboard
Straw

10. protractor
Finished Product
straw
cardboard
String with button

11. How do you use it? Need two people
Measure the distance from the base of the object to where you are standing.
Look at the object you are measuring
Have partner look at what degree value is shown on clinometer
Use trig ratios to determine the height of the object
DON’T FORGET! Add the height from the ground to your eye level since that is a part of the height of the object.

12. What I did using a Clinometer
Emergency Blue Light pole
15, 25, and 35 paces away
1 pace ≈ 10.6 inches
159, 265, and 371 inches away
Found the degree values to be
18°, 10°, and 8° respectively
The height from the
ground to my eye = 62 inches

13. Too small – must add my height Eye to ground since my measurement
Using tangent, I estimated the heights to be ≈51.7, 51.5, 𝑎𝑛𝑑 52.1 𝑖𝑛𝑐ℎ𝑒𝑠 ℎ𝑖𝑔ℎ .
Too small – must add my height
Eye to ground since my measurement
was from my eyes
62 inches
Real heights for the emergency blue light pole
113.7, 113.5, and inches high
Actual Height = 114 inches

14. Percent Error
113.5− ∗100=.43859%

15. Example 2 – No Parking Sign
Paces
Conversion to Inches
Degrees
Height
Final Height (+62) 10 18
≈34.538
≈96.538 20
≈37.487
≈99.486 30 5
≈33.625
≈95.625
Actual Height = 99 inches
Approximate Average Percent Error: 2.12%

16. Example 3 – Lamp Post
~ Chose something I couldn’t measure using a ruler being that I knew my method was generally accurate
Inches Conversion
Degree Value
Height
Final Height (+62)
120 44
≈115.88
≈177.88
240 28
≈127.61
≈189.61
360 21
≈138.19
≈200.19
Average Approximate Height: inches

17. Example 4 - Stairs

18. For Students
Have students stand at different intervals from the base (10, 20, 30 feet away)
Record the degree values by looking through the straw on the clinometer.
Find the height for each trial.
Average those values to determine the approximate height of the object.

19. Possible Errors Taping the string to the cardboard
Error in measurements
Shaking hands
Not look at same area
Under/Over estimating degree values

20. Challenge Problem
If you know the height of the object and the angle of elevation, how would you be able to determine how far away from the object you are?
If you know the height of the object and how far away from it you are, how would you determine the angle of elevation?

21. What they will learn next…
After angle of elevation, students will look at the angle of depression and determine how they would be able to the find the height of objects smaller than them and how this process would be different than using the clinometer.

22. Thanks to my helpers!

23. Reference