Session 5, Part B: Measuring Heights of Tall Objects (35 minutes)
Measuring With Shadows
What other methods exist for measuring indirectly? One such method is to use shadows. For tall objects that are difficult to measure directly, such as skyscrapers and giant redwood trees, shadows can be very useful, as they lie on the ground and are fairly easy to measure. The "shadow" method also relies on similar triangles.
Go outside on a sunny day with a friend, and find a tall lamppost that is casting a shadow. (If you can't find a lamppost, use another tall object.) You need to be able to measure the length of the shadow, so be wary of shadows that fall onto roadways. A large parking lot or field is ideal. Note 4
a.
Measure the length of the shadow of the lamppost. Make sure that the lamppost is on flat, level ground, since the lamppost and the shadow should be perpendicular to each other. Record the measurement.
b.
Now measure the shadow your friend casts on level ground.
c.
Make a sketch of the lamppost and shadow and label what you know. Also draw a right triangle that shows your friend and his or her shadow as the legs of another triangle. Again, label what you know. Your sketch should look something like this:
Problem B2
Why are the two triangles (lamppost/shadow and friend/shadow) similar? Think about the angles in each of the triangles. How were those angles formed?
Do the sunbeams form an angle? Also, remember that if two of the angles in both triangles are congruent, the triangles must be similar. Close Tip
Problem B3
If you know that the triangles are similar, how can you find the height of the lamppost?
Problem B4
Determine the height of the lamppost. Discuss the different proportions that you might use to calculate its height. Note 5
Why is the height of the lamppost a derived measure (i.e., measured indirectly)?
b.
Let's assume that your measurements were accurate to the nearest 0.25 m. Use proportions to calculate an upper and lower limit on the height of the lamppost.
c.
What do you think is the best value for the height of the lamppost?
Problem B6
Imagine that you have a tall tree in your yard that needs to be cut down. You want to make sure that the tree won't hit your house when it falls. How might you approximate the height of the tree?
Video Segment In this segment, Katy and Lombi use the method described in this section to find out the height of a tree in the schoolyard. They set up a similar triangle by measuring the length of the shadow of a meterstick.
What are some of the advantages and disadvantages of using a shadow to measure lengths?
If you are using a VCR, you can find this segment on the session video approximately 8 minutes and 28 seconds after the Annenberg Media logo.
