Useful Formula List   Robert Leverett
  Apr 19, 2007 11:10 PDT 


      As a consequence of the mathematics that I've been wading through
as of late, it occurred to me that a useful byproduct of the effort
might be a listing of our most frequrntly used formulas and measurement
processes, a top ten list or something along those lines. Well, as a
first crack, the simple H = Lsin(A) leads the pack, where A is the angle
from the eye to the target and L is the straightline distance from eye
to target. Of course, we use this formula twice to compute the above and
below eye level components of a tree's height.

     In terms of frequency of usage, D = Lcos(A) might be second, where
D is the straightline, eye level distance to the point vertically below
(or above) the target point. This frequently represents the distance to
the trunk of a tree at eye level.

     I have to throw in the tangent formula out of recognize for its
value in places where it fits: H = Dtan(A), where H and D are defined
as above.

     A 4th formula of great use to me is A^2 = B^2 + C^2 - 2*B*C*cos(a)
or the law of cosines. It allows us to compute the straightline distance
between two points in space using a triangle connecting the two points
and our eye. B and C are the distances to the two points from our eye, a
is the angle between B and C, and A is the connecting distance between
the two points.

     A 5th formula stated somewhat generally is any algebraic derivation
of a/b = A/B. This formula is expresses the proportionality of sides for
similar triangles.

     Number 6 relates circumference (C) to the diameter (D) of a circle:
C = PI*D.

     Number 7 is the formula for the cross-sectional area of a circle: A
= PI*R^2.

     Number 8, the elliptical equivalent is A = PI*a*b, where a and b
are the semi-major axes of the ellipse. Going from two to three
dimensions, the most useful volume formula for us is the one for the
frustum of a cone.

     So, #9 is: V = H/3*(A1 + A2 + SQRT(A1*A2) where H is the height of
the frustum, and A1 and A2 are the cross-sectional areas of the top and
bottom of the frustum.

     Finally, I would include as #10,the Pathagorean relation for the
length of the hypotenuse of a right triangle in terms of the lengths of
the other two sides. If C is the hypotenuse and A and B are the other
two sides, then we have C^2 = A^2 + B^2.

     So, let's now summarize these 10 formulas, noting the definitions
of the individual variables for each formula as defined above.

   1. H = Lsin(A)

   2. D = Lcos(A)

   3. H = Dtan(A)

   4. A^2 = B^2 + C^2 - 2*B*C*cos(a)   
      or A = SQRT[A^2 + + C^2 - 2*B*C*cos(a)]

   5. a/b = A/B

   6. C = PI*D

   7. A = PI*R^2

   8. A = PI*a*b

   9. V = H/3*(A1 + A2 + SQRT(A1*A2)

10. C^2 = A^2 + B^2

      I am taking onto myself the project of keeping a running list of
useful formulas to ENTS work for periodic dissemination through our
e-mail list and on the website. I'll work out a better format for future
submissions, one that is consistent and always defines the variables
being used in the formulas.


Robert T. Leverett
Cofounder, Eastern Native Tree Society
RE: Useful Formula List   Paul Jost
  Apr 20, 2007 18:33 PDT 


4 and 10 are really the same equation with 10 being a special case of 4 with
a term eliminated at 90 degrees.... The Pythagorean theorem is really just
a special case of the law of cosines applied to a right triangle...

RE: Useful Formula List    Robert Leverett
   Apr 23, 2007 06:50 PDT 


   Yes, I understand. For others on the list who may not see this,

    C^2 = A^2 + B^2 - 2*B*C*cos(c) Law of Cosines

    C^2 = A^2 + B^2                 Pathagorean relation
                                      for right triangles

    If c = 90 degrees, then cos(c) = 0

and through substitution of the latter into

    C^2 = A^2 + B^2 - 2*B*C*cos(c)

we get, C^2 = A^2 + B^2 - 2*B*C*cos(90)

    This then simplifies to:

    C^2 = A^2 + B^2 - 2*B*C*0

    and finally to

    C^2 = A^2 + B^2

    Paul, like almost all electrical engineers is extremely strong in
mathematics. He was the first of us to make use of the law of cosines to
check on tree height calculations. Thanks, Paul. I have a feeling that
if I make a mistake in my derivations, you'll be among the first to catch

10 going to 11 and beyond   Robert Leverett
  Apr 23, 2007 10:13 PDT 


Exploring the features of
trees in the field mathematically and discovering dimensional attributes
that may be obvious only in hindsight is where we are headed. The
challenge is to bring our whole group along. The ENTS motto should be:
Leave no Ent Measurer Behind".

      However, the discovery mission just articulated does not mean that
we will need to launch a "new formula of the week" program. Our progress
needs to be methodical and practical and that was the purpose of the big
ten list. What should be #11? The formula that I propose is the one to
measure tree height using the tangent function and two vantage points as
exhibiting significant potential, but not yet proven practically. The
formula referred to is:

      H = [Dtan(A)tan(B)]/[tan(A)-tan(B)].

      This formula assumes that we have lined up the point being
measured in the crown with two vantage points on the ground, i.e. the
three points fall within the same vertical plane. We take the horizontal
distance at eye level between the two vantage points as D and we measure
the angles of the crown point and eye level from the two vantage points
as A and B and then apply the above formula. Jess Riddle believes that D
must be at least 20 feet to get sufficient differentiation for angle
that are otherwise close together in value. I would bet that he is
right. So, I will add the above formula to the previous list of 10, duly
noting Paul's point that 10 is really a variation of 4. I've employed
the computer symbols for math operations to make the formulas a little


1. H = L*sin(A)

2. D = L*cos(A)

3. H = D*tan(A)

4. A^2 = B^2 + C^2 - 2*B*C*cos(a)
or A = SQRT[A^2 + + C^2 - 2*B*C*cos(a)]

5. a/b = A/B

6. C = PI*D

7. A = PI*R^2

8. A = PI*a*b

9. V = H/3*(A1 + A2 + SQRT(A1*A2)

10. C^2 = A^2 + B^2

11.   H = [D*tan(A)*tan(B)]/[tan(A)-tan(B)].

   Unless someone else sees it differently, the two variations of #11
for handling situations where the baseline is not level, but slanted,
will not be added to the list until #11 proves its value. City parks
with big spreading trees that can be seen from a distance on grassy
fields should provide a satisfactory testing ground for formula #11. If
they don't we are positioned to lay down and take a nap.


Robert T. Leverett
Cofounder, Eastern Native Tree Society
12-14 for your viewing pleasure   Robert Leverett
  Apr 25, 2007 06:56 PDT 


    The parade of formulas continues. Since we, in ENTS, use the sine
top-sine bottom formula primarily to measure tree height, a fairly
simple formula that reflects the sensitivity of changes in height to
changes in hypotenuse distance, angle, or both would be extremely
useful. One can attack the problem directly and compute the change in
tree height based on a direct application of the sine formula to the new
values and subtract the sine application to the old values to get the
change or differential. The following development shows that result.

         L = original measured hypotenuse distance to target
         j = a small change in distance
         A = original measured angle to target
         k = a small change in angle
         H1 = height based on L and A
         H2 = height based on L + j and A + k

         H1 = L*sin(A)
         H2 = (L + j) * sin(A + k)
         h   = H2 - H1

      Or stated as a single formula:
         h = (L + j) * sin(A + k) - L*sin(A)

    This formulaic statement of the height difference is fairly obvious.
However, another way of approaching the problem is through total
differentials, a concept in calculus that computes the change in a
dependent variable associated with small changes in one or more
independent variables:

        dH = sin(A)*dL + L*cos(A)*dA,


       dH = change in H associated with,

           dL, the change in L, and

           dA, the change in A.

    From the meaning of H2-H1, i.e. the full change in height, we
defined h. We acknowledge dH to be an approximation of h. But the
difference between h and dH is miniscule for relatively small values of
dL and dA. So, the differential dH can be used to estimate the error in
H from being a little off in distance or angle or both. Please keep in
mind that in the dH formula, angles are measured in radians - an angular
measure that represents the ratio of the length of an arc of a circle to
the radius of that circle. There are 2 * PI radians in a circle.
Therefore, one radian = 360/(2*PI) = 57.29578 degrees. To convert
degrees to radians, use the simple formula:

               R = A * (PI/180)

       where A = the angle expressed in degrees
                PI = 3.1415926
                R = the angle expressed in radians.

    I especially like the differential formula dH because it highlights
the quantities dL and dA and their respective roles. You can easily
trace the separate effects of the angle and hypotenuse changes for
different values of L and A.

    In terms of usefulness, or at least potential usefulness, I would
add the above formulas to our growing list, which presently stands at
11. To summarize the new formulas:

    12.       R    = A *(PI/180)

    13.       A    = R*(180/PI)

    14.      dH = sin(A)*dL + L*cos(A)*dA

    I note that the derivation of dH is a straightforward application of
the definition of a total derivative applied to the formula: H =
L*sin(A). I would include the simple derivation here, but notation of
partial derivatives or differentials is a problem in Topicaís minimal

    I will pass with this explanation of what the recent deluge of
formulas means. The intensity of the mathematics isnít a new devotion
for me. Mathematical modeling, practiced at one level or another, goes
on all the time in my noodle. It was my bread and butter while still in
the U.S.A.F and at the Pentagon. With no false modesty, my penchant for
math and math modeling is where the ENTS approach to sine-based
mathematics originated. But until recently, Iíve hesitated to put much
formula derivation into these e-mails with the exception of sine-based
measuring along with a scattering of other stuff. Iíve been afraid of
turning off the Ents who are not into the mathematics of tree measuring.
However, I realize that those of you who arenít oriented in that
direction will just skip the math intensive e-mails. However, to give
initial warning, I will supply e-mail titles that communicate their
contents as that of tree measuring.

    So here is a recapitulation of the full list of hot-to-trot

   1. H = L*sin(A)

   2. D = L*cos(A)

   3. H = D*tan(A)

   4. A^2 = B^2 + C^2 - 2*B*C*cos(a)   
       or A = SQRT[A^2 + + C^2 - 2*B*C*cos(a)]

   5. a/b = A/B

   6. C = PI*D

   7. A = PI*R^2

   8. A = PI*a*b

   9. V = H/3*[A1 + A2 + SQRT(A1*A2)]

10. C^2 = A^2 + B^2

11. H = [D*tan(A)*tan(B)]/[tan(A)-tan(B)].

12. R    = A *(PI/180)

13. A    = R*(180/PI)
14. dH = sin(A)*dL + L*cos(A)*dA

    My current guess is that the full list of formulas most useful to
our tree measuring will grow to around 25. Algebraically derived
relationships for computational convenience will double or triple this
number. The key to the utility of the list will be a clear statement of
what each formula does and where it is most useful for our purposes. As
a final comment, there are statistical formulas with value to us such as
arithmetic and perhaps geometric means, standard deviation, standard
error of the estimate, median, etc. They will be individually discussed
in the e-mails at the appropriate times and included in total in the
book on dendromorphometry.


Robert T. Leverett
Cofounder, Eastern Native Tree Society
Back to Beth
  Apr 25, 2007 13:30 PDT 

     One of our ENTS objectives is to create a cookbook recipe of formulas for tree measuring with very explicit instructions on how to best apply each formula in the field. Ed Frank is particularly good at writing crystal clear recipes for applying our methods. Hopefully, Ed will be heavily involved in that phase.

     In terms of what gets posted, some of the e-mails will be the conceptual stuff and some of it more practical. Please don't hesitate to ask questions on how to apply the formulas in real situations. There is no bad question.


-------------- Original message --------------
From: beth koebel 


I do read your math heavy emails, although I have to admit most of it
goes right over my head. I will someday have to sit down and fiddle with
them to try to understand what you talking about. It took me no time at
all to understand formula #1 after I looked in my copy of "Pocket Ref"
(my "little black book"). I guess I'm more of "here's the formula and
how you use it" then I get it type person.

#15 with reference to #16   Robert Leverett
  May 02, 2007 10:25 PDT 


     Formula #15 for our list of high use (or potentially high use)
formulas is the differential for height using the tangent method. For
investigating the magnitude of height errors, the tangent differential
formula has the same utility as the differential for height formula
based on the sine method. The height differential formula for tangent

dH = [D/(cos(A))^2] *dA + tan(A)*dD,    

where D = baseline distance, A = angle, and dA and dD are the
differentials corresponding to measurement errors in angle and distance.

     Recalling the corresponding differential for sine is:

dH = L*cos(A)*dA   + sin(A)*dL,

where L = hypotenuse distance, A = angle, dA and dL are the

     How do errors for the two height measuring methods compare using
the differentials to calculate error? The examples below show dHt, the
height differential for the tangent process, and dHs for the sine
process. Distances of 100 and 150 feet and angles of 30 and 60 degrees
are used in these examples.

     D   100.0    150.0

     L    115.5    300.0

     A     30.0      60.0

     dA     1.0       1.0

     dL     1.0       1.0

     dHt    2.95    12.20

     dHs    2.25      3.48

      At relatively low angles and short baselines, the differentials
for an error of 1 degree and a distance of 1 foot are almost equal for
the sine and tangent methods. But the differentials diverge as the angle
gets steeper and/or the distance gets longer. At 40 degrees and a
224-foot baseline, an angle and distance needed to produce the height of
a Boogerman Pine type tree, the errors are 4.55 for sine and 7.50 feet
for tangent. At very steep angles the error becomes even more pronounced
for the tangent method. For example, suppose the Boogerman Pine were
arrow straight with a top visible from 50 feet out. If the top is
directly over the base and our eye is level with teh base of the tree,
the top would be at 75 degrees. If this seems unrealistic, just imagine
that broken branches provide us with a window of visibility to the top.
Under this configuration, a 1-degree angle error and no distance error
would produce a height error of 13.03 feet for the tangent method. A
1-foot distance error without angle error would add another 3.73 feet.
Thus, the maximum error for the combination of the 1-foot distance error
and 1-degree angle error is 16.76 feet - nothing to blow off. By
contrast, the corresponding maximum height error for the sine method
under the same configuration is a very modest 1.84 feet. This should be
enough to catch the attention of even the most lethargic-thinking tree

     Turning our attention from combined errors to just distance, the
maximum error that can ever be made in the height dimension for a
hypotenuse error of 1 foot IS 1 foot and that error is made shooting
straight up. So the height error for a 1-foot error in hypotenuse varies
from 0 to 1 foot as the angle goes from 0 to 90 degrees. There is no
corresponding situation for the tangent method. The tangent value is
infinite at 90 degrees. However, assuming that 75 degrees is the highest
angle for which the tangent method can reasonably be applied to a tree
like the Boogerman Pine, then the height error for a 1-foot baseline
error is 3.73 feet as shown previously. The corresponding error for the
sine method at 75 degrees is 0.97 feet. The difference in errors for the
two methods based on distance error alone is 2.76 feet. Obviously angle
error can be the big problem with the tangent method.

      A more realistic scenario for a Boogerman-sized tree is an angle
of 40 degrees Ė assuming the Boogermanís height to be around 188 feet
tall by now. With both a 1-degree angle error and a 1-foot distance
error, the sine method results in a height error of 4.55 feet and 7.50
feet for the tangent method. The difference of 2.95 feet suggests that
around 3 feet is the largest likely difference in the height error made
attributable to the method of measurement, i.e. sine versus tangent when
the angle error is 1 degree and the distance error is 1 foot where the
angles arenít too steep. Note that the distance error is made in the
baseline for the tangent method and in the hypotenuse for the sine

     So now we have 15 formulas. I should add Lee Frelichís crown volume
formula as #16. Weíll probably identify 3 or 4 more before weíre done to
come out with about 20 highly useful ENTS formulas that deal directly
with tree dimensions. Adding formulas for Rucker indexing, champion tree
programs, etc. should put us at around 25. That sounds like a lot, but
it really isnít.



Robert T. Leverett
Cofounder, Eastern Native Tree Society