05, 2007 07:58 PST
A few months ago, several of us decided
to present ENTS measuring
methods to our new members in something approximating a cookbook
format. Our challenge is to present each ENTS measurement method
by step process that can be applied by someone who shudders at
thought of formulas and algebraic processes. I've seen that Ed
particularly good at developing measurement recipes. His
are extremely clear, so I will defer to him for the last word on
aprocedure. Sometimes, I function best to get the ball rolling,
that spirit, this e-mail deals with the concept of the slope of
and how slope, as a concept, relates to the trigonometric
function. Readers of the ENTS e-mails devoted to measuring will
frequently hear us speak of slope and tangent basically as
First let's define the slope
of a straight line. Slope is a
measure, expressed as a percent, of the degree of inclination of
straight line from the horizontal. Let's consider an example.
you concurrently move forward (horizontally) 100 feet as you are
upward (vertically ) 80 feet. The line connecting your starting
ending points obviously slopes upward. How could we best measure
degree of slope? In mathematical terms, the horizontal distance
feet is called the "run" and the vertical distance of
80 feet is called
the "rise". In our example, using simple arithmetic,
the rise distance
represents 80% of the run distance. This is in fact the slope of
line. So the rule is to get the slope of the line, the rise
simply divided by the run distance and the result multiplied by
the above example, this is 80/100 = 0.80 and 0.80 x 100 = 80%.
implication is that for every foot you go out, you go up 0.8
Grades on highways are usually expressed in slope percentages.
encounter a sign at the start of a steep hill going downward
grade 10% for the next 1/4 mile. This means that for the
distance of a quarter of a mile, you will be descending 10
for every 100 horizontal feet you travel. In these cases, the
an average. Most people understand this, but the highway example
good way to explain the concept of slope for those who donít
application to trees. However, as we often discuss, the slope
can be applied to compute the height of a tree, albeit in a
obscure method that is often applied inaccurately.
Letís now go to tangent. The
definition of the tangent of an angle
of a right triangle is the length of the opposite side (height)
by the length of the adjacent side (base). But, isnít this
over run? Of course, the third or long side of the triangle is
hypotenuse. In the above example the opposite side is 80 feet
adjacent side is 100 feet. The angle formed by the hypotenuse
base sides creates the height leg, i.e. it is the angle included
base and hypotenuse. The tangent of this included angle is
80/100 or 0.80. So, the slope percent of the hypotenuse line is
tangent of the angle times 100, i.e. the tangent expressed as a
Slope expressed as a decimal IS the tangent. Tangent turned into
percent is slope. Slope = tangent x 100. That is the connection.
In the above example, the
hypotenuse distance (direct line from
eye to crown-point) can be calculated by the Pythagorean
SQRT(80^2 + 100^2) = 128.1. If we walked up a straight road that
80 feet in elevation for a horizontal distance of 100 feet, we
cover a distance of 128.1 feet and the slope of the road would
In terms of a tree, if we walked forward toward the trunk from
away, staying level for the 100 feet, we would wind up directly
the crown-point, which would then be 80 feet above our head. We
also bump into the trunk were we not watching.
To apply the method of using
percent slope to calculate tree
height, we need a clinometer that has a percent slope scale.
clinometer at a point such as the top of a tree and reading the
scale gives us the tangent x 100. It also makes the statement
height of the point we are aiming at is such and such a percent
horizontal distance of the point. In the case of a tree, the way
measurers apply the technique, the trunk and crown-point must be
vertical alignment. Otherwise, the percent slope is taken
baseline that is too long or too short, generating and over or
Many folks who measure trees
still do not understand this last
point. Others may understand it and try to compensate, but when
to locating where the crown-point of a tree is relative to the
using Kentucky windage isnít something a novice can be
expected to do
reliably. Using a laser rangefinder and clinometer combination
sine-based mathematics, eliminates the doubt as to where in
crown-point actually is. Nonetheless, the tangent-based method
measuring presents us with one of our tools.
In using clinometer-based measurements,
clinometers with a degree
and slope scale are the most useful for our purposes. As Will
others have pointed out, clinometers can be bought with other
scales, but they are problematic to correctly measuring tree
most important scale on a clinometer to have is degrees.
Robert T. Leverett
Cofounder, Eastern Native Tree Society
to Ed on Slope and Tangent
05, 2007 11:17 PST
Yes, good point: arctan(% slope/100)= degrees. Then sin(degrees)
distance = height. The distance is, of course, the hypotenuse
from eye to crown-point, as opposed to baseline distance.
So, I think we are saying that if someone has
a laser rangefinder, a
clinometer with a percent slope scale (but not necessarily
a scientific calculator with an arctan function, they are set.
don't have to buy a new clinometer just to get a degrees scale.
use the percent scale to derive degrees via the relationship:
= arctan(% slope/100)
That is a good point to stress. However, I'm
unsure if there are
clinometers out there that have percent scales that compute
percentages based on something other than slope. For example, do
them treat the percentage as the percentage of some fixed
as 66 feet? It seems to me that an old Haga Altimeter I once
did just that. If so, that kind of percent scale wouldn't work.
someone knows. Will Fell? Don Bertolette? Don Bragg?
In the slope versus tangent narrative, I don't
mean to sound as if
I'm shifting gears. I'm just unsure if everyone is truly clear
definitions that we use.
Back to Ed on Slope and Tangent
05, 2007 12:36 PST
As I think about it, the scale used for a
66-foot baseline yielded
full height in feet instead of a percent of baseline. The instrument
set up to allow its user to read height directly from one scale,
as the baseline was exactly 66 feet. I now recall that the
on the instrument was meant to be used against a baseline of any
So in that case, %/100 would be the tangent as we have been
This subject merits further research. If percent scales are
yielding the tangent of the angle x 100, then it is true that we
always derive the angle using arctan. I'll look at my Forestry
Catalog of their writeups on Clinometers.
Edward Frank wrote:
Is there some simple way to convert one with a 66 foot