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The Height of a Distant Tree

Date: 07/08/98 at 22:26:23
From: Kostia
Subject: How can I calculate the real size of a tree?

Let's say a tree stands 1 mile from me, and I see it as 5cm tall. If 
you need an angle, the angle is 10 degrees. How tall is the tree?

Date: 07/22/98 at 10:50:11
From: Doctor Rick
Subject: Re: How can I calculate the real size of a tree?

Hi, Kostia - 

Let's see if we can figure out the height of that tree without climbing 
it or even walking that mile.

There are two ways to do this sort of thing. If I use the angle you 
gave me (and it's good that you figured out that I would need more 
information), I need to use a trigonometric function, the tangent 
function on a calculator. If you don't know what a trigonometric 
function is, don't worry, it's pretty easy. "Trig functions" are 
ratios of the sides of a right triangle. When you look at the top and 
bottom of the tree, you are one corner of an invisible triangle:

                                                      - ^
                                                -      /|\
                                          -           / | \
                                    -                /  |  \
                              -                     /___|___\
                       -                               |||
                 -                                     |||
          -                                            |||

The tree goes straight up, so the angle of the triangle marked "Bottom" 
is a right angle. The angle you measure is the angle between the line 
from You to Top and the line from You to Bottom. And the tangent of 
that angle is the ratio of the height (Bottom to Top) to the distance 
(You to Bottom).

Since you know the angle, you can calculate the tangent on a scientific 
calculator, or you could do it the old-fashioned way and look it up in 
a book. (People spent their whole lives figuring out those numbers 
before calculators were invented.) The tangent of 10 degrees is 0.1763. 

Now we have :

    height        height
   --------  =  ---------  = 0.1763
   distance     5280 feet

So height = 0.1783 * 5280 feet = 931 feet. That's a pretty tall tree.

There is another number you could have told me instead of the angle. 
If you told me the other number, I would not need a scientific 
calculator. Also, the other number is easier to measure than an angle.

I think that what you meant when you said the tree looked 5 cm tall, 
is that you held up a ruler and lined up the markings with the tree, 
and it measured 5 cm. But how far away did you hold the ruler? I would 
hold my arm out all the way, and that would make it about 27 inches 
from my eye. You aren't grown up yet, so let's just pretend it's 25 
inches, or 61 cm.

Why do I want to know this? The secret is that you made another 
triangle the same shape as the one we saw before, but a lot smaller. 
We can compute the tangent for ourselves by measuring this triangle. 
This triangle has the ruler instead of the tree, and your arm instead 
of a mile of ground. We compute the tangent by dividing the ruler 
measurement by the length of your outstretched arm. This should give 
the same result as dividing the height of the tree by the distance to 
the tree, because the two triangles are exactly the same shape (we say 
they are "similar"):

    height      ruler measurement
   --------  =  -----------------
   distance        arm's length

    height        5 cm
   --------  =  -------  =  0.082
   5280 feet     61 cm

This time, we find that the height = 0.082 * 5280 feet, or 433 feet. 
It's not the same as the height we found the other way. Either the 
angle wasn't right, or I guessed wrong about how far away you held the 
ruler. Anyway, that's still a pretty tall tree.

By the way, those people who spent their whole lives computing trig 
tables didn't do it by drawing triangles and measuring them. There are 
ways to calculate tangents, but we won't go into that today.

Try measuring some tall things this way. See if you can measure 
something whose height you know, so you can tell if you were right. 
Write and let us know how it goes.

- Doctor Rick, The Math Forum
Check out our web site!   
Associated Topics:
High School Trigonometry

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