Measuring Immeasurable Objects with Trigonometry
Introduction:
Original measurements of the earth required people to measure objects that were very difficult to
measure. For example, one method required a person to know the height of a nearby mountain in
order to calculate the radius of the earth. Let’s take a look at one way to measure that without
using computers, satellites, GPS, or maps.
Activity:
1. Attach a long string to the top of the wall in your classroom.
2. Pick two points on the floor that extend out in a straight line from the attachment point. Make
sure one of the points is somewhat close to the wall and the other is significantly farther.
3. Using the string and a protractor, measure the angles θ and ϕ and distance d as depicted
below (we won’t measure m because when measuring a mountain, this is a very difficult
measurement to make).
4. By creating a right triangle with the floor and the wall from each of the two points you
picked, you can set up these two equations:
𝑡𝑎𝑛𝜃 =
ℎ
𝑑+𝑚
𝑡𝑎𝑛𝜑 =
ℎ
𝑚
Using the numbers you have for θ, ϕ and d, solve these two equations for h. Compare your
answer with the actual height of the wall.
Extension Activity:
Try to measure the height of a larger object like your school or a tree.