# How Tall Is Our School’s Library?

A few days ago, my Honors Precalculus students finished up their study of trigonometry. In this activity, they used what they learned to measure the height of the library on our campus. I’ve done this activity before with my Geometry students and a flagpole (see this post). The girls needed to measure the angle of elevation from the student observer to the library roof, and the horizontal distance from the front of the library to the student. Then, using the tangent function, they can calculate the vertical height of the library:

Here’s a shot of the sextant we made and the geometry involved in calculating the angle of elevation:

It was beautiful day – sunny and in the mid-60s – so nobody complained about going outside! I divided the class into three groups, and each group took their measurements from a different spot along the front of the building.

Measuring the horizontal distance

Measuring the angle of elevation

All three teams came up with fairly consistent results: between 27.5 feet and 29 feet!

I’ve been teaching math long enough to know that a year from now these girls won’t remember very many of my wonderful lectures, but hopefully they won’t forget the day they used trigonometry to figure out how tall their library is.

# Using Trigonometry to Measure a Flagpole

Today in geometry, we used trigonometry to estimate the height of a flagpole. First, I made a homemade sextant, using a protractor, some thread, and a piece of paper that I rolled around a pencil and taped to the protractor:

I weighted the thread with four large paper clips. You can see from the picture that the line of sight makes a 50 degree angle with the vertical thread. To estimate the flagpole height, we needed the angle of elevation, which is measured from the horizontal. So, some geometry needs to be applied to figure out the angle of elevation:

Angle of elevation is 40 degrees

Now that we knew what to do, we went outside and measured 10 meters from the base of the flagpole. Then, one of my students used the sextant to sight the top of the pole, while another student read off the angle indicated by the thread:

Sighting the top of the flagpole

The angle was 51 degrees, which meant the angle of elevation must be 90-51 = 39.

So here’s the picture the students drew on the Smartboard:

Figure illustrating the problem

Notice that they realized they had to take into account the height of the person taking the sighting. My other geometry class did the same activity, and they got a result of 9.7 meters. Pretty consistent!