Generally when I go for runs my mind is clear as my body can only focus on my working muscles. This gave an ever present complication to the fact that I wanted to come up with an idea. However oddly enough, this time around a mantra of "perspective, perspective" was ingrained into my head.
As I looked at a small tree in the distance it all clicked. If I was aware of how far I was from the tree, I could calculate the height of the tree. The method? Utilizing similar triangles.
Similar triangles are a fundamental trigonometric idea that is taught in high schools. It is the concept that if you have two triangles both with the same internal angles, you could write a ratio for Triangle A in terms of a ratio for Triangle B.
Part 1: Learning a Little Math Before
Using words to describe mathematical concepts is difficult so I drew some helpful diagrams to describe what similar triangles are. But first you must understand what cosθ and sinθ are. cosθ and sinθ are trigonometric functions that relate the angle of a triangle to the length of the adjacent side and hypotenuse or the length of the opposite side and hypotenuse, respectively. That is a little heavy to digest if you have no math backing so here is a picture:
Using these two pictures you can see how cosθ and sinθ are defined. cosθ equals the length of the opposite side to the angle, divided by the length of the hypotenuse of the triangle. sinθ equals the length of the adjacent side to the angle, divided by the length of the hypotenuse.
Interestingly enough this is all the info you need to derive what similar triangles are. Starting with a simple triangle like the one in the last picture you can draw a smaller triangle inside of that one. Thus creating two triangles with the same angles on the inside of each but with different side lengths.
In both (1) and (2) we found a separate definition for what cosθ equals. Because cosθ always equals cosθ, we can write that (1) = (2) which is simply A/C = a/c.
This method is not a good way to calculate heights or distances accuracy but it a method for sure. I hope you enjoyed my interesting perspective into the world of measurements. I find it incredibly funny that I thought all this up randomly on a run.
Remember "perspective, perspective," that's all it is. And to end this blog on an upbeat note life is also all about perspectives. No matter the situation you're dealing with it all comes down to how you handle it, what perspective you have. You got this.
Cheers.
Using these two pictures you can see how cosθ and sinθ are defined. cosθ equals the length of the opposite side to the angle, divided by the length of the hypotenuse of the triangle. sinθ equals the length of the adjacent side to the angle, divided by the length of the hypotenuse.
Interestingly enough this is all the info you need to derive what similar triangles are. Starting with a simple triangle like the one in the last picture you can draw a smaller triangle inside of that one. Thus creating two triangles with the same angles on the inside of each but with different side lengths.
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| If you notice I labeled the larger triangle "1" and the smaller triangle "2." I also labeled the sides of each triangles. Triangle "1" has uppercase letters and triangle "2" has lowercase letters. |
*Remember the goal is to relate the sides of triangle 1 to the sides of triangle 2*
Now for the most important step, writing our equations. For triangle 1 you can write the basic cosθ equations, but you may notice by looking at the diagram the adjacent side is defined as "A" and the hypotenuse is defined as "C". Therefore instead of writing cosθ = adj/hyp you may write cosθ = A/C, which is what I did in the diagram below in number 1.
(For triangle 2 you can write the basic cosθ equations, but you may notice by looking at the diagram the adjacent side is defined as "a" and the hypotenuse is defined as "c". Therefore instead of writing cosθ = adj/hyp you may write cosθ = a/c, which is what I did in the diagram below in number 2)
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| The 3 dots signifies the word "therefore," it is used commonly in mathematics. |
In both (1) and (2) we found a separate definition for what cosθ equals. Because cosθ always equals cosθ, we can write that (1) = (2) which is simply A/C = a/c.
These similar triangle ratios can also be written. I won't derive them today but if you know some trigonometry I think you could do it your self! Give it a try!
Part 2: My Actual Idea
The fact I showed in Part 1 is the back bone to my idea on how to measure the height of a tree. Here is the picture explaining what it is:
If you notice this picture looks awfully similar to the previous triangle diagram with some slight changes on what the letters represent. "B" is the height of the tree, "b" is the height of my thumb, "A" is the distance from the tree and "a" is the length my arm.
To set this up properly what you need to do is extend your arm all the way out and stick your thumb up. (This next sentence is a bit funky but bear with me) Now from your perspective walk forwards or backwards in an attempt to make your thumb the same height as the tree. It is like the silly game of "squishing people" from afar using your index finger and thumb, it's all about preservative. Same idea for this. This is incredibly important as doing this insures the angle between your eye and your thumb is the same angle between your eye and the tree.
Fantastic we are almost done! Measure the height of your thumb, measure the length of your arm and walk a known distance away from the tree. You can find out this distance by stepping heel to toe a counted number of times away from the tree and then look up your shoe size (in a unit of length) and multiply the number of steps by the length of your foot.
With all this data known you can now successfully (not the most accurately) calculate the height of the tree. Using a derived similar triangles equation which is A/B = a/b you can substitute in the known values.
I.E.
and rearrange for the variable "A," which is the height of the tree,
It is finally at this stage that you can calculate the height of a tree, just make sure all the measurements are in the same units and substitute in your data! As well, you are not just limited to trees or finding out specifically heights of things. Hypothetically if you knew the height of a building and you used this same method of lining up your thumb you could re-arrange this equation to calculate how far away you are from the building!
Part 3: The Conclusion
I hope this made sense to you, if it didn't and you want to learn I would suggest finding someone that understands similar triangle because that is all you need to know!This method is not a good way to calculate heights or distances accuracy but it a method for sure. I hope you enjoyed my interesting perspective into the world of measurements. I find it incredibly funny that I thought all this up randomly on a run.
Remember "perspective, perspective," that's all it is. And to end this blog on an upbeat note life is also all about perspectives. No matter the situation you're dealing with it all comes down to how you handle it, what perspective you have. You got this.
Cheers.
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