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.
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:
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
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
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 measurerís 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 L2,
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
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
Determining the magnitude of the
measurerís 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
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
l1 = 2.2
l2 = 2.0
d = 11
Using the above process, compute the height L4
obtained by the measurer, the actual height H, and the error.