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
5-minute read
Episode #185

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:

  1. 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.
  2. 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

So, 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, PhD

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.