Measuring by Shadows

Date: 05/22/2001 at 07:55:02
From: Andreza Rio Branco
Subject: Measuring by shadow

How can I measure a tree using its and my shadows?

Date: 05/23/2001 at 08:21:25
From: Doctor Rick
Subject: Re: Measuring by shadow

Hi, Andreza, thanks for writing to Ask Dr. Math!

The secret is in the sun's rays: they fall on both you and the tree 
from the same direction. We say the rays are parallel. Also, the tree 
stands straight and so do you, so you and the tree are parallel. 
Finally, if the ground is flat, the tree's shadow and your shadow are 
parallel.

We can draw the tree, its shadow, you, and your shadow as triangles. 
The top of the tree is joined to the shadow of the top of the tree by 
a line that points back up to the sun. The top of your head and the 
shadow of the top of your head are joined by another line pointing 
back up to the sun.

     TREE
       |\
       | \
       |  \
       |   \
     ? |    \
       |     \
       |      \               YOU
       |       \               |\
       |        \            5'| \
     __|_________\_____________|__\______
           50'                  2.5'

When each side of one triangle is parallel to a side of another 
triangle, the triangles are SIMILAR. When we use this word in everyday 
life, we just mean "they are sort of the same, but not quite." When we 
use it in math, it has a special meaning: "their shapes are exactly 
the same, though their sizes don't have to be the same."

The "tree triangle" and the "you triangle" are similar.

The sides of two similar triangles have equal proportions. If the 
horizontal side of one triangle is twice as long as the horizontal 
side of the other triangle, then the vertical side of the first 
triangle is twice as long as the vertical side of the other, and the 
diagonal sides follow the same pattern.

These equal proportions let you figure out one length if you know 
three others. In the picture above, I marked "your height" as 5 feet, 
and your shadow as 2 1/2 feet. The tree's shadow is 50 feet. How long 
is the tree's shadow compared to your shadow?

      50   100
     --- = --- = 20
     2.5    5

The ratio is 20 to 1; that is, the tree's shadow is 20 times as long 
as your shadow. Since the triangles are similar, the ratio of the 
tree's height to your height is also 20 to 1. If your height is 5 
feet, and the tree is 20 times as tall, then the tree's height is

     20 * 5 feet = 100 feet

(The "*" is the multiplication sign.) This is how you can use shadow 
lengths and your own height to measure the height of a tree!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/