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.
Jason Marshall, PhD
Listen
How to Use Trigonometry to Measure the Height of a Tree
My 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!.
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?
How to Calculate the Height of a Tree
Armed with this equation, all I had to do to calculate the height of the tree was:
- Come up with an estimate of the angle between the ground and the top of the tree from the point at which I was standing.
- Walk off (or use a tape measure to find) the distance from myself to the tree.
So that’s exactly what I did. I started by asking a friend to estimate the angle between the ground and my arm as I pointed to the top of the tree. Once I had this angle estimated in degrees, I simply converted it into radians, used my calculator to find the tangent of the result, and then multiplied this by the distance from myself to the tree.
The answer I got certainly wasn’t exact (since my estimate of the angle and distance to the tree were nowhere near exact), but it was good enough to tell me that my house wasn’t going to be crushed in the next wind storm!
Super Bowl Scoring Puzzle
Before finishing up, did you get a chance to think about the Super Bowl scoring puzzle that I posed last week? In case you missed it, the question was: What scores are possible in a football game? Or, put another way, are there any impossible scores?
As I mentioned in the episode on Super Bowl Fun Facts, to answer this, you need to know the different numbers of points that can be awarded. Namely:
2 points for the relatively rare safety- 3 points for a field goal
- 6 points for a touchdown
- 7 points for a touchdown and a successful extra-point kick
- 8 points for a touchdown and a successful two-point conversion
2 points for the relatively rare safetySo, what do you think? Given all of these possibilities, are there any point totals that a team simply cannot finish the game with?
Well, right off the bat, it’s clear that a team can’t finish with a single point—since there are no plays that are awarded 1 point. So we can immediately see that final scores like 1 to 0, 1 to 1, or 1 to anything else for that matter are all impossible. But are there any other impossible scores?
One way to go about answering this question is to simply start working your way up the point total chart. For example, we know that 2 and 3 point totals are possible, but how about 4? Indeed, that can be done by scoring two 2-point safeties. How about 5 points? One field goal and one safety will do the trick. And on and on you can go combining various numbers of safeties, field goals, and touchdowns to add up to any point total.
But instead of doing all of that work figuring out how to combine different numbers of plays to make up a point total, we can also simply note that since a safety is worth 2 points, any even score is possible. And since a field goal is worth 1 more point than a safety, any odd score is possible too since we can always remove one safety in exchange for a field goal.
Which means that the only impossible final point total in football is 1!
Wrap Up
Okay, that’s all the math we have time for today.
Please be sure to check out my book The Math Dude’s Quick and Dirty Guide to Algebra. And remember to become a fan of the Math Dude on Facebook where you’ll find lots of great math posted throughout the week. If you’re on Twitter, please follow me there, too.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!
Palm tree image courtesy of Shutterstock.
About the Author
Jason Marshall is the author of The Math Dude’s Quick and Dirty Guide to Algebra. He provides clear explanations of math terms and principles, and his simple tricks for solving basic algebra problems will have even the most math-phobic person looking forward to working out whatever math problem comes their way.