Measuring the Height of a Building Using Shadows

Date: 05/24/2000 at 02:50:10
From: Joan Rapp
Subject: Using shadows to find the height of a building

My daughter, April, has to write a shadow report for math. I can not 
help her with this problem. She needs to know what time of day is best 
to use a shadow to measure the height of a building or object by using 
triangles. Please help me to help her by giving information on shadows 
and the best time and worst time of day to find the size of a 
building. She says it is at sunrise and I say it is at noon. She is 
also confused as to how exactly to use shadows to find the size of a 
building.

Date: 05/24/2000 at 12:33:38
From: Doctor Rick
Subject: Re: Using shadows to find the height of a building

Hi, Joan.

One way to use triangles to find the height of a building makes use of 
similar triangles. Put a vertical post in the ground (or have someone 
hold it vertical), and measure its height and the length of its 
shadow. Measure the length of the building's shadow, before the sun 
has time to move. Now you have two triangles, with three known 
lengths:

       |\
       | \
       |  \
       |   \
     H |    \
       |     \
       |      \
       |       \               |\
       |        \             h| \
     __|_________\_____________|__\______
            L                    l

You know the height h and shadow length l of the post, and the shadow 
length L of the building. The two triangles are similar - why? From 
these facts you can find the height of the building.

Now, let's say it's near noon, so the shadows are very short. Perhaps 
a 30-foot building casts a 10-foot shadow. If your measurement of the 
length of the shadow is off by 1 foot, how far off will your estimate 
of the building's height be?

On the other hand, what if the shadows are very long - that 30-foot 
building casts a shadow 90 feet long. If your measurement of the 
shadow is off by 1 foot, how far off will your estimate of the 
building's height be?

If the shadow is a mile long, when the sun has just risen, you'll have 
other problems. I'll leave the complete answer to you and April.

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