ENTS,

A recognized method for measuring tree height is the method of similar triangles. Basically, this method works because corresponding sides of similar triangles are in direct proportion to one another. The method works as follows. The measurer holds up a measuring stick to mask a tree so that a line from the measurer's eye touches the top end of the measuring stick and the high point of the crown, i.e. the three points are in alignment. A second line from the measurer's eye touches the bottom of the measuring stick and the base of the tree. The second three points are in alignment. The measuring stick is held vertically.

If:
   h = length of measuring stick,
   d = distance from eye to bottom of stick,
   H = height of tree
   D = distance from eye to top of tree,

then h:d = H:D stated as a ratio and proportion.

In the algebraic form this relationship can be stated as:

h/d=H/D

and solving for H gives us H = D(h/d).

The method of similar triangles requires that h and H be parallel to one another, but what if they aren't. If a line from the top of the tree to the base is not vertical, then the method of similar triangles will fall prey to the infamous horizontal offset problem that plagues the tangent method. Problem#3 in the attachment addresses this situation. In the problem, the lengths of the small triangle and  the distance from the eye to the base of the tree are known. We assume an angle d between a vertical line through the trees base and t he line from the base to the top of the tree. We then derive the equations needed to:

1. Calculate the height of the tree  by the measurer's method,

2. The true tree height of the tree,

3. The error.

Of course, the measuer isn't going to know the value of the angle d. So, this exercise is more to show what kind of errors are made with the method of similar triangles when there is a horizontal offset and to investigate the relationships between the variables. We will derive the formulas necessary to calculate errors and the values of particular variables that lead to specified errors.

As a final note, the process may seem unnecessaarily involved. It is what it is. Problems with tree measuring often result from measurers ignoring variations in tree shape, assumptions implicit in a particular technique, and the mathematical processes needed to solve problems. To put it simply, there ain't no free lunch in this stuff. 

Bob  

Problem#3: A measurer sets out to determine the height of a tree using the method of similar triangles. The measurer establishes a small triangle with known lengths of its sides and creates a second triangle of similar shape. One side of the larger triangle represents the height of the tree. In the diagram below, variables l₁, l₂, and l₃ represent the sides of the small triangle and L₁, L₂, and L₃ represent the sides of the large triangle. The intersection of l₁ and l₂ establish the vertex of the small triangle where the measurer’s eye is positioned. The angle between l₁ and l₂ is designated by the variable a. The measurer establishes the distance L₁ from the point of measurement to the base of the tree. The measurer assumes that the line from the high point of the tree to the base represented by L₂ is vertical and parallel to l₂. The similar triangle method is to be used to determine the value L_2, the assumed height of the tree. However, unknown to the measurer, the high point of the tree is actually offset toward the measurer. A line from the base to the high point is represented by the variable L₄. The vertical line L₂ through the base forms an angle of d degrees with L₄. The following diagram shows the variables to be used in this problem.

            Determine the height of the tree as calculated by the measurer, the actual height, and the error.

Solution: In the diagram above, a, b, c, d, and e represent angles. The remaining variables represent distances. The measurer supplies l₁, l₂, l₃, and L₁.The angle d is assumed. The objective is to show the process needed to determine H and L₂-H.

            We begin by determining the value of the angles a, b, and c using the law of cosines. The law allows us to compute the length of a side, if we known the lengths of the adjacent sides and their included angle. For example, to compute the length of l₂, we state the law of cosines as follows:   

We can algebraically solve this equation for the angle a. The result is as follows: 

 By a similar process, we get angle b as:

To compute c, we utilize the fact that the sum of the angles of a plane triangle equals 180 degrees. Therefore,

 Similarly, we recognize that the angle c also applies to the large triangle as shown in the diagram. To compute e in the diagram, we use the sum of the angles formula again to get

Here, we remember that the value of d is assumed. Applying the ratio and proportion of similar triangles, we compute L₂ as follows.

 We note that L₂ is what the measurer determines to be the height of the tree, because the measurer assumes the l₂ and L₂ are parallel, but from the diagram, we can see that it is actually the line H that is parallel to l₂. We proceed by first determining the value of L₄ using the law of sines, which states that the ratio of the sine of an angle to the length of the opposite side is equal to the ratio of the sine of either remaining angle to its respective opposite side. Relating side L₄ and c to side L₂ and angle e, we can write the equation

Rearranging, we get

Finally, appealing to the small right triangle that includes the side H and L₄, we can write

 Determining the magnitude of the measurer’s error, we calculate

Comments: The above process can be reduced to a single equation that involves the original known quantities. Making substitutions from above, we get

Rearrannging,

 and

There is no particular advantage to boiling the process down to a single equation for H except to reduce the number of steps required.

Sample Problem: Assume the following values:

l₁ = 2.2

l₂ = 2.0

l₃= 2.4

L₁=100

d = 11

Using the above process, compute the height L₄ obtained by the measurer, the actual height H, and the error.