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How Tall Is That Tree?

© Copyright 2000, Jim Loy

How can you measure the height of a tree. We could cut the tree down, and then measure it. But we want to avoid such drastic "measures." The most obvious method is to climb the tree, carrying one end of a tape measure, measuring the tree directly, still too drastic for my tastes. The second most obvious way may be to use trigonometry (see below for some definitions). Trig is mainly involved in the measurement of right triangles, which are triangles with one right angle. In the diagram at the left, the height of the tree is 50 m. times the tangent of 17 degrees. We can use trig tables (or a scientific calculator) to find the tangent of 17 degrees, which is 0.30573. That makes the height 15.2865 m. We probably want to make that 15 m. or (15.3 m.), because we probably haven't made our measurements very accurately.

Trigonometry is based upon similar triangles. We could easily draw a scale model of the above situation (in fact, that is what my diagram is) on paper, making a right triangle with one angle of 17 degrees. Then we can use proportions. Let's say that the base of the triangle in my drawing is 8 cm., and the height of my model tree is 2.5 cm. Again, I measure the base of the actual right triangle to be 50 m. Then the tree is 15.6 m (or 16 m).

The Boy Scout manual showed this method. I take a stick, and hold it up so that it looks like it is the same height as the tree. I may have to adjust my distance from the tree, or shorten the stick somehow (like holding my thumb at the place that makes the rest of the stick look the same height as the tree). Then I rotate the stick 90 degrees, and see that it hits the ground at the red flower (or some other landmark, such as a friend) in the picture. Then I use a tape measure to measure the distance from the flower to the tree.

Trigonometry:

Trig deals with certain proportions in right triangles. In particular, we define these three functions (sine, cosine, and tangent):

sinA=a/c (opposite/hypotenuse)
cosA=b/c (adjacent/hypotenuse)
tanA=a/b (opposite/adjacent)

A popular mnemonic is to remember Indian Chief Soh-cah-toa. All right triangles with the same angle A are similar to each other. That means that their sides are proportional. So the above three functions have the same value, no matter what size triangle we have. If the angle is different, then (usually) the three functions have different values. And, depending on the angle, we can get the values of these functions from trig tables (or from a scientific calculator).

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