Similar Triangles   Robert Leverett
  Feb 07, 2007 08:10 PST 


     The geometrical concept of similar triangles provides a powerful
tool for tree measurers. However, as with the percent slope method, it
is often misapplied to tree height. But, used correctly, the method
gives us one more tool to work with. Let's look at the concept of
similar triangles.
     Similar triangles are triangles that have the same overall shape.
One is just a blown-up or reduced version of another. Because similar
triangles have the same shape, corresponding angles of similar triangles
are equal and corresponding sides are proportional in length. It is the
condition of proportionality of side length that allows us to compute
tree height. But how does the process work?
     Assume we have two triangles that are similar. Triangle ABC is the
larger and abc the smaller where the lengths of the sides are designated
by A, B, and C for the larger and a, b, and c for the smaller.   The
condition of proportionality leads to the following relationships:
     A/B = a/b, A/C = a/c, B/C = b/c.

     These are all algebraic expressions and can be manipulated
algebraically to get equivalent forms such as B/A = b/a, B/b = A/a, and
b/B = a/A, etc. Now suppose that we form a big triangle ABC such that A
is the tree's height and B is the baseline from the measurer to the
trunk. Now, if we can form a smaller similar triangle abc where we can
measure sides a, b, and c, we can measure the baseline to the tree B and
can compute the tree's height by using the A/B = a/b proportionality
relationship. Algebraically rearranging, we get A = B(a/b). This last
formula is what often accompanies diagrams showing how to measure tree
height using similar triangles. If side A is vertical and the two
triangles are truly similar, then the process works. However, what
happens when side A is not vertical (the line from the crown-point to
the base is not vertical)? Then the process does not work and that will
be the case when the crown-point is not directly over the base of the
tree. Sound familiar?     

     Similar triangles can also be used very productively for
determining crown-point offset. However, that determination requires a
mult-step process that will be explained with diagrams at the April ENTS
event at Cook Forest SP. In fact, all the material in these e-mails will
be brought together in what I hope will be our first crack in producing
an ENTS Guide to Dendromorphometry. BTW, Dendromorphometry was a term
created by Gary Beluzo, partly in jest and partly in seriousness to
describe the energy with we Ents expend pursuing our passion. Gary, is
of course, a first class dendromorphometrist. Gary once made up
certificates for John Knuerr and myself. However, the idea of
formalizing Dendromorphometry as an official ENTS discipline has merit.
ENTS could issue certificates to bonafide dendromorphometrists - both in
fun and in all seriousness.


Robert T. Leverett
Cofounder, Eastern Native Tree Society