Using trigonometry to measure a really, really tall building

Trigonometry is often a part of mathematics that students struggle with because they find it too abstract to understand the practical applications of this field. My 10th grade mathematics class was recently introduced to right-angled trigonometry, and as we talked about its various applications including bearings, 3-D shapes, and angles of elevation and depression, I decided to take them outside of the classroom to apply these skills.

The task that was given to them was to create a “How-To” ebook on finding the height of a really tall building without physically measuring it. The students were told that they had one week before they would be taken to a very tall building in the downtown Boston area and asked to measure its height. I assigned my students into groups of three to four and gave each group a budget of $15 to buy the materials required for this activity.

The groups came up with various ways of calculating the height of the building. One group initially planned on taking a picture of a student pointing a ruler at the top of the building and then later use that picture to calculate the height. Another group went with the more traditional approach of creating a clinometer to measure the building’s angle of elevation from a certain distance, and then using that angle to measure its height.

Before taking the students to the building, I decided to use a class period for them to test out their methods by measuring the height of the classroom. The students had purchased some of the necessary materials already and were able to do a mock trial of the task. This trial was extremely beneficial for the groups, and after the activity the groups found ways to tweak their plans in order to make the measurements more accurate.

Photo by Alejandro R.

On the day of the activity, I took the students on a short walk to the Federal Reserve Bank of Boston. When choosing a building to measure, I looked for an especially tall building with a safe and large area surrounding it from which the students could reach a good distance away and measure the angle of elevation. What I did not consider was the fact that the Federal Reserve Bank’s security guards would find people measuring distances around the bank as suspicious activity! Luckily for us, the guard that came out to inquire about what we were doing was okay with the idea of students performing a math activity based on the bank.

It was very interesting to see the students carry out the activity. One group started by measuring a perpendicular distance of 100 feet away from the bank while the other group walked on a perpendicular route far away in order to determine a distance that would give them approximately a 60 degree angle of elevation. This allowed the groups to share their 100-foot measuring tape that they had been able to purchase by combining their individual budgets together. The groups did very well at measuring the angle of elevation to the building from various distances and submitting the averages of their calculations to me.

Photo by Alejandro R.

I promised a prize to the groups that were able to get within 5% of the actual height of the building. One group measured the building to be 181 meters (593.4 feet) tall while the other group submitted a measurement of 184 meters (603.7 feet) tall. The actual height of the building (according to a Google search) is 187 meters (614 feet), which means that the group that calculated the height closest to the actual height of the building had a percentage error of only 1.6%!

Overall, the students really enjoyed the activity and came away with a better understanding of the applications of trigonometry. The resulting ebooks will reflect this enthusiasm and hopefully spread the excitement for mathematics to any and all learners interested!