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How to Use Trigonometry to Measure the Height of a Tree

Have you ever wondered if sines, cosines, and tangents are actually useful in the real world? If so, wonder no more! Today we're going to learn how the wonderful tools of trigonometry can be used to estimate the height of a tree.

By
Jason Marshall, PhD,
February 8, 2014
Episode #185

Page 1 of 2

Palm Tree on Tropical BeachMy beloved hometown of Los Angeles is known for its warm, sunny, and pretty much perfect weather nearly year-round (depending on your taste, of course). While I won't argue with the accuracy of this picture, there is another—much less pleasant—side to the weather in the particular part of L.A. in which I live: gnarly wind.

No, I'm not talking Winnie the Pooh style "blustery days," I'm talking about days of wind with gusts of 70 MPH and beyond. My point in teling you this isn't to complain (because I admit I don't have much to complain about), it's to give you a bit of background to help you understand why I'm so interested in estimating the heights of trees.

You see, my neighbor's yard features another one of those L.A. icons—a mildly menacing tall and gangly palm tree. Understandably, at some point I began to wonder exactly how tall it is? And, in particular, is it tall enough to leave a large dent in my roof if those winds should ever prove too mighty for its shallow roots.

So, how did I figure out the height of the tree…without having to resort to climbing it? Keep on reading to find out!

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Real World Triangles

There are, of course, many ways that I could have measured the height of my nemesis palm tree. The most direct—but also most difficult, dangerous, and dumb—method would have been to climb the tree and stretch a giant tape measure down its trunk. While that sounds like a hoot, I thought that using math while safely planted on the ground was a much better option.

So I dug into my bag of math tricks and pulled out the tangent function from the wonderful world of trigonometry. Why tangent and not sine or cosine? Well, let's think about the right triangle made by drawing an imaginary line between myself, the base of the palm tree, and its tippy-top. One side of this triangle is the trunk of the tree itself, the other side is the line drawn from me to the base of the tree, and the hypotenuse of the triangle is formed by the line from me up to the top of the tree.

Real World Trigonometry

As we learned when talking about sine, cosine, and tangent, the tangent of an angle in a right triangle is the ratio of the length of the side of the triangle "opposite" the angle to the length of the side "adjacent" to it.

If you think about it, you'll see that the side "opposite" the angle formed between the ground and the line running from me to the top of the tree is the height of the palm tree. And the length of the side "adjacent" this angle is simply the distance from me to the base of the tree. Which means that:

tan( angle ) = height / distance

If we turn this equation around, we can solve for the height of the tree in terms of the tangent of the angle and the distance to the tree:

height = tan( angle ) x distance

Bingo! This equation was my key to finding the height of the tree.

Umm…how exactly?

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