The purpose of this exercise is to use trigonometry to indirectly measure the height of tall objects.
To indirectly measure the height of tall objects (such as trees or buildings) using trigonometry, you need the following:
- Height of observer to eye level
- Distance from object to observer
- Angle of elevation as measured by a clinometer
A clinometer is a tool that is used to measure the angle of elevation, or angle from the ground, in a right triangle. A clinometer can be constructed using a plastic protractor (available at stores that sell school supplies), or you can construct a homemade protractor using a protractor photocopy (provided below) and a piece of cardboard. If you choose to construct a homemade protractor, follow these steps:
- Cut out the protractor photocopy below.
- Glue it to a piece of cardboard or poster board.
- Cut out the cardboard around the protractor so it has the shape of the protractor.
- With a pen or pencil, poke a hole through the center of the protractor.
Once you have your protractor (either store bought or homemade), follow these steps to create your clinometer:
- Cut a piece of string or strong cotton that is approximately 8 inches long.
- Push the string through the hole in your protractor and tie a large knot on the other side so it won’t pull through. Note: If you purchased a plastic protractor that does not have a center hole, you may tape the string instead.
- Tie or tape a weight – such as a metal nut, paper clips or pennies – to the other end of the string
- Tape a drinking straw to the straight edge of your protractor.
Your clinometer should look like this:
To use the clinometer:
- Find a tall object (tree, building, flag pole etc.) where there is plenty of space to move away from the object that you are measuring. Your object should be tall enough that its height cannot be easily measured with a ladder. You will need a measuring tape long enough to measure the distance between your feet and the base of the object.
- Hold the straw to your eye and aim the straw so that you can see the very top of the object you are measuring (in the picture below it is a wall); the string line will naturally fall straight down (this is called plumb). Below is a diagram of how this will work.
- Read the angle being recorded on the clinometer by holding down the string where it touches the protractor and record this value. In the diagram below, this is marked angle B. Note that angle B and angle A are congruent since the string line is parallel to the object (angles A and B are corresponding angles).
- Measure the distance between where you are standing and the base of the object (preferably in feet) and record this value.
- Measure the distance from your eyes to the ground and record this value.
- Construct a right triangle that can be used to find the height of the object.
- For demonstration purposes, in the diagram below, the right triangle is labeled as triangle XYZ, and the recorded measurements needed to complete the project are labeled as angle A, line segment ZY, and line segment YG.
- Calculate the vertical distance of the height of your object to the height of your eye (in the diagram above, this is labeled line segment XY) by using one of the trigonometric ratios we have studied (sine, cosine, or tangent) and record this value.
- Find the height of your object by adding the distance recorded above to the height from your eye to the bottom of the ground (in the diagram above, this would be the sum of the lengths of line segment XY and line segment YG).
Report your clinometer findings:
To complete the project, you must turn in a photo of your clinometer and one page of work that contains the following:
- On the top half of your paper, create a detailed, labeled diagram of the situation. Your right triangle does not have to be drawn precisely to scale but it should be representative of the situation. A ruler should be used so that the sides of the triangle and other line segments drawn are straight. All known lengths and angles must be labeled. Unknown lengths (those you had to find) must be marked with a variable. You must also include yourself and the tall object (identify it) in your diagram. Neatness will be a factor in determining your grade.
- On the bottom half of your paper, show all work (formulas, equations, and computations) used to find the height of your object. Show all work in a neat and organized fashion.
