United States

Department of The Sine Method: An Alternative

Agriculture
Height Measurement Technique
Forest Service

Don C. Bragg, Lee E. Frelich, Robert T. Leverett,

Will Blozan, and Dale J. Luthringer
Southern
Research Station
Research Note
SRS-22 December 2011

Abstract and others 2002). For instance, the modeling of seed

dispersal depends on the trajectory of falling seeds and
Height is one of the most important dimensions of trees, but few observers the distance traveled, which are partially a function of tree
are fully aware of the consequences of the misapplication of conventional height and therefore an accurate measurement of the starting
height measurement techniques. A new approach, the sine method, can
improve height measurement by being less sensitive to the requirements
height of seeds in the canopy is critical (Dovciak and
of conventional techniques (similar triangles and the tangent method). others 2005, Williams and others 2006). More recently, the
We studied the sine method through a couple of comparisons. First, we study of why trees grow as tall as they do has emerged as
demonstrated the validity of the sine method under idealized conditions a research topic (e.g., Domec and others 2008, Nabeshima
by comparing tangent and sine measurements on a stationary object of
a known height. Then, we compared heights collected via climbing and
and Hiura 2008), as has metabolic scaling across size classes
lowering a tape from the highest point of a number of forest-grown trees (Russo and others 2007).
with heights measured with the sine method. The sine method offers a
viable, cost effective alternative to traditional measurement approaches, Popular interest in “big tree” lists (e.g., American Forests
especially for large or leaning trees, and for trees with broadly spreading
crowns.
2010, Forestry Tasmania 2009) has merged with aspects
of science and conservation. Researchers occasionally use
champion trees to define the upper limits of species height
Keywords: Height measurement, hypsometers, similar triangles, sine
method, tangent method, trigonometry. (e.g., Botkin 1992, Bragg 2008a, Parresol 1995, Shifley
and Brand 1984), and some agencies use exceptionally
tall trees to help establish reserves (e.g., Forestry
Tasmania 2009). Height can play a key role in developing
Introduction accurate relationships between tree bole diameter and
aboveground volume and productivity estimates (Newton
Total tree height is an important measure of numerous and Amponsah 2007, Repola 2008). Certain allometric
forest conditions. Height is an indicator of the status of the relationships are embedded within growth functions of
tree within the population and is helpful in predicting stand many forest simulators—the gap models, as an example,
development and successional patterns (Tester and others use an increment function based in part on tree height (e.g.,
1997, Purves and others 2008). The vertical structuring of Botkin and others 1972, Moore 1989, Shugart 1984). Many
adjacent trees largely determines the outcome of gap closure large-scale biomass, carbon storage, and timber volume
and the ability of understory species to reach the canopy estimates are generated from diameter- and height-based
(Webster and Lorimer 2005). Other functional aspects of equations applied to forest inventory data (e.g., Botkin and
forest ecology (e.g., water use by trees, light extinction others 1993, Somogyi and others 2007). Remote sensing
through the canopy, wildlife habitat quality, and seed techniques to derive forest biomass and carbon sequestration
dispersal) depend partly on tree height (Boelman and others require accurate ground-based tree height measurements
2007, Dovciak and others 2005, Ford and Vose 2007, Parker for calibration and verification (Asner and others 2010,
Boudreau and others 2008, Clark and others 2004, Lefsky
and others 2002, Sexton and others 2009, Wang and Qi
Don C. Bragg, Research Forester, USDA Forest Service, Southern 2008).
Research Station, Monticello, AR; Lee E. Frelich, Research Associate,
University of Minnesota, Department of Forest Resources, St. Paul, MN;
Robert T. Leverett, Executive Director, Eastern Native Tree Society, All of these examples suggest the need for reliable
Florence, MA; Will Blozan, Arborist, Appalachian Arborists, Inc., Black quantification of tree size. Such details are not without
Mountain, NC; Dale J. Luthringer, Naturalist, Cook Forest State Park, consequence—for example, the inaccurate measurement
Cooksburg, PA.
of tree diameter has been previously noted as a cause of
overestimates of carbon storage in tropical rainforests (Clark
2002), and differences in height have potentially significant
impacts on individual tree volume computations (Westfall
2008). Nevertheless, height measurement techniques
are often applied without consideration for their basic
assumptions. This paper introduces a new, less sensitive
methodology capable of improving the accuracy and
reliability of height measurement while using conventional
technology.

Mathematics behind Tree Height

Determination

In addition to direct measurement with height poles or tape

drops, there have been two primary methods for ground-
based estimation of tree height. One approach uses the
geometric principle of similar triangles (fig. 1A). This
method, once incorporated in a number of hypsometers, is
very sensitive to tree lean and observer error and has largely
been replaced by other approaches in the forestry profession
(Husch and others 2003). We will not discuss the similar
triangle method (also called the “stick method”) further,
but note that it is still suggested by some tree measuring
organizations as a means to measure champion tree heights
(e.g., American Forests 2010).
Figure 1—Earlier technology for measuring tree heights focused on using
The second method applies trigonometry to estimate tree the geometry of similar triangles A) or the trigonometry of right triangles
height (fig. 1B). For a right triangle, the subsequent relation B).
is always true:
highest point of the tree must be known. Second, the tree
(1)
must be truly vertical (the highest point of the crown is
exactly above the endpoint of the baseline for the horizontal
where the length (height) of the vertical leg of this triangle
plane). Third, the baselines (b1 and b2) represent true
(c) can be determined using the angle α and either the length
horizontal distances. If these conditions are met, HTTAN is an
of the horizontal leg (b) or the length of the hypotenuse
accurate and unbiased representation of total tree height.
(d). Most optical and electronic hypsometers adapt this
relationship using the following approach:
Virtually every depiction of height measurement shows an
idealized tree (usually a conifer) with a straight bole and
(2)
the highest growing tip located directly above the center
of the base. In reality, however, species with strong apical
where HTTAN is the total tree height using the tangent
dominance and a pronounced leader can grow with lean (fig.
method, α1 = degrees of the angle between horizontal line
3A) or develop a branching pattern that offsets the highest
from the observer’s eye to top of the tree; α2 = degrees of
top from the centerline of the bole. Many other species,
the angle from horizontal line from the observer’s eye to
especially decurrent hardwoods growing in relatively
base of the tree (α2 is negative when the base is below the
open conditions (such as the 118-cm diameter at breast
eye level of the observer, thus subtracting a negative value
height (d.b.h.) water oak (Quercus nigra L.) in fig. 3B)
produces a positive height component); and b1 = b2 = the
develop broad, spreading crowns. For the tangent method
horizontal distance from eye of the observer to the midpoint
to accurately estimate height in either of these cases, the
of the trunk of the tree (fig. 2A). The tangent method must
observer first must find and compare the highest apparent
satisfy three main assumptions. First, the base and the actual
extension of the crown by observing it from some distance

2
 away, then find the nadir beneath that exact same point
through an often leafy and branch-filled crown, and finally
must determine the distance between the observation point
and the nadir (the baseline length). The tangent method
involves considerable effort when there are multiple
possible high points in a large, spreading crown (fig. 3B) to
evaluate—to ensure total height is found, several different
spots must be tested with the tangent method before the
maximum value can be determined. Furthermore, other
attributes of a tree may influence the accuracy of height
measurement, such as its growth on a steep slope with lean,
particularly when it has an offset growing tip (fig. 4A). Even
straight trees with thick boles can produce offset errors if the
highest tip does not lie directly above the surface of the stem
where the baseline distance is measured, or if the baseline
distance is not adjusted to reflect the actual configuration
(fig. 4B).

Uneven terrain adds another level of complexity to

the tangent method. Prior to self-correcting electronic
hypsometers, the observer would need to manually adjust
baseline length to reflect the true horizontal distance. On
sloping ground, regardless of the viewing angle (figs. 2B
and 2C), the observer must make a trigonometric conversion
between the uncorrected (slope) distance measured with
their tape (bu) and the true horizontal baseline distance.
Using a tape and clinometer, one possible adjustment is:

(3)

where the angle (αu) is to the point on the stem (u) where
the tape is affixed. This correction assumes that the tape is
stretched tight and held parallel to the slope between the
observer and the tree. After this adjustment, the corrected
baseline distance bc = b1 = b2, and the tangent height can be
determined using bc and the angles to the top and bottom of
the tree (α1 and α2). Figures 2B and 2C illustrate the need
to correct the baseline on very steep slopes, although the
effects of slope are noticeable on even gentle grades and
should be addressed.

For decades, the standard instruments for measuring

tree height have been similar triangle-based mechanical
hypsometers or hand-held clinometers and measuring tapes.
These tools are still popular because they are inexpensive,
easy to learn, and quick to apply in the field, but they
Figure 2—Under idealized circumstances A), tangent height and sine tree
are gradually being replaced by newer technologies. All
heights are exactly the same. For a truly vertical tree, this relationship does commercially available electronic (laser- or sound-based)
not change, regardless if the observer is located upslope B) or downslope hypsometers measure at least slope distance. Those with
C) from the base of the tree. The ground distance from the observer to the integrated clinometers can reduce tangent-based height
tree must be corrected for the slope from true horizontal to produce the
actual baseline distance (bc).
estimation errors by automatically correcting baseline length

3
 Figure 3—The primary source of error with the tangent method arises from the failure to correct for the proper baseline length
(i.e., b2 is used rather than b1), as may happen with a leaning tree A). This error also appears with large crowns B), especially
when rounded or flat-topped, as few observers actually adjust their baseline length when using the tangent method to the point
directly below the crown high point. In this example, each of the arrows indicate potential high spots of the crown that would
need to have their baseline distances calculated after translating these through the thick crown to their respective nadirs. With the
sine method and a laser hypsometer with a continuous scanning mode, it was possible to check all of these points in seconds from
the same observation spot to find the true height of this water oak (dashed circle).

4
Figure 4—Horizontal offset errors arise when the tangent method is not corrected for differences between the appropriate baseline
lengths. If the same baseline length is used (b1 = b2) on a leaning tree without correction A), the observer will generate horizontal
offset errors on both the upper and lower triangles. Horizontal offset can also occur when the highest point on the crown is dis-
placed away from the bole B), again requiring two different baseline distances placed at the appropriate spacing from the observer
in order to calculate the correct height.

5
for vertical trees on uneven ground by converting slope sine methods, respectively, within the given measurement
distance into horizontal distance (figs. 2B and 2C). error of the device (± 0.05 m) and observer error (e.g.,
inconsistent measurement locations). Both techniques
Prior to the advent of reasonably priced electronic performed well, even though the ground surrounding the
rangefinders, it was not practical to measure the slope tower was not entirely level, as the laser was stationed at
distance (d1) from the observer’s eye to the top of the tree elevations from 0.5 to 4.5 m above the base of the clock
(measuring slope distance to the base of the stem (d2) tower. High precision and accuracy of the measurements of
directly with a tape is straightforward). Once this technology this tower were expected, though, as it is located in a part
became available, a new approach to triangulation became of campus with few obstructions and no extraneous factors
feasible: (e.g., wind, lighting, intervening vegetation) affected the
view.
(4)
To provide further evidence of the accuracy of the sine
where HTSIN is the height calculated using the sine method. method, we also compared directly measured heights of
When measuring the height of a perfectly vertical object, climbed trees with sine-based measurements. Forty-two
HTTAN should equal HTSIN (fig. 2A). The sine method yields large, forest-grown trees were climbed and their height
the vertical distance between parallel horizontal planes, determined by lowering a measuring tape from the highest
one touching the base of the tree and the other touching climbable point of the crown to the ground (usually, a pole
the highest point of the tree (fig. 3A). Because of this of known length was extended to get to the very top of the
trigonometry, the sine method does not require the tree to be tree). The sine method was then used to calculate height of
straight, the high point of the tree to be directly above the the climbed trees using the exact same point in the crown.
trunk, or the ground to be level (Blozan 2006, Bragg 2008b). The measurements for this effort were taken with TruPulse®
The sine method is a realization of the hemispherical 200 or TruPulse® 360 hand-held laser hypsometers with
approach of Grosenbaugh (1980), who recognized the manufacturer-stated distance accuracies of ± 0.3 m (out
potential biases of previous technology that assumed trees to 1,000 m) and electronic clinometer accuracy of ± 0.25
were truly vertical. degrees (Laser Technology, Inc. 2010) or recreation-grade
laser rangefinders (e.g., Nikon ProStaff® 440) that are less
Demonstrating the Sine Method exact (± 1 m distance accuracy) and produce only slope
distances (angle readings must be taken with a separate
We demonstrated and validated the sine method by clinometer).
measuring a fixed object of known height with a stationary
distance-measuring device (sensu Bruce 1975). For this test, Tape drop heights of 42 forest-grown trees (table 2) deviated
21 measurements were taken with an Impulse® 200LR laser from heights measured by the sine method by an average
rangefinder on a clock tower on the University of Arkansas- of 0.01 m (range: -0.91 to 0.61 m, standard deviation =
Monticello (UAM) campus from distances of just under 15 0.31 m), with an average relative error of 0.03 percent
m to over 165 m. The Impulse 200LR was affixed to a stable (range: -1.92 to 1.39 percent, standard deviation = 0.64
tripod and the magnified red-dot scope used to locate, target, percent). Hence, with the accurate laser rangefinders and
and measure the pointed top and level base of the clock electronic clinometers available today, instrument error
tower, with a distance-measuring accuracy of 3 to 5 cm and when measuring total tree heights with the sine method
an inclination accuracy of 0.1 degrees (up to 500 m from can be expected to be consistently less than 1 percent for
the target; Laser Technology, Inc. (2010)). We compared the experienced users.
heights determined by the tangent and sine methods with a
paired t-test.
Comparing the Tangent and Sine Methods
Table 1 provides the data collected when both methods
were used to estimate the height of a 12.52-m-tall clock For trees on level ground with straight boles, evenly
tower on the UAM campus. The mean difference for 21 distributed crowns, and a distinct leader located directly
measurements of this tower using the traditional and new above the base of the stem, there are not likely to be any
methods was a statistically insignificant 0.03 m (12.51 significant differences between either height measurement
m (tangent) versus 12.48 m (sine), paired-t = 0.88, P = technique (fig. 2A). This condition exists in nature,
0.3908), verifying that HTTAN = HTSIN for vertical objects. especially in young conifer stands, but is likely the
Each technique had low variance on this structure, with exception rather than the rule in older stands where most
standard deviations of 0.14 and 0.12 m for the tangent and trees have broad crowns with several leaders nearly equal in

6
height above the base of the tree. Large, flat-crowned trees the scanning mode found on most laser rangefinders to
tend to produce height overestimates (Belyea 1931, Husch search for other points higher than the initial observation.
and others 2003) with the tangent method. Under these Implementing this crown-scanning approach from multiple
circumstances, if the proper top is not identified and the viewing angles, then taking the maximum of this collection
correct baseline distance calculated for the tangent method, of sine heights, is the best means to estimate true total height
total height estimates under these circumstances “…are short of direct measurement. Without locating the nadir
of little value” (Husch and others 2003: 109). Another of numerous high points on the ground through the crown
challenge for the tangent method is the need to distinguish and then measuring the length of a series of corresponding
between the true high point of a crown and subordinate baselines, it is virtually impossible to comparably assess
branches projecting towards or away from the observer. all likely high points in a crown with the tangent method
Similar to the leaning tree in figure 3, without correction of (fig. 3B). The choice of the highest apparent point has been
baseline length any subordinate branches facing the observer instilled on measurers from the beginning with the tangent
will thus appear to be taller than what they actually are, method—with a fixed baseline, increasing the angle of
and branches extending away from the observer may seem inclination is the only way to maximize the total height
shorter. of a given tree. For most observers using this approach,
selecting an apparently lower point in the crown might seem
Even if the point measured is a subordinate branch, the counterintuitive, especially given the uncertainty that this
sine method will only underestimate total height. This bias particular adjustment will yield the highest tip.
can be ameliorated, if not completely eliminated, by using

Table 1—Tangent and sine height estimates for the clock tower on the University of Arkansas-Monticello campus
(from the base of the tower to the point on the top)

Horizontal Tangent height Sine height Laser height

Direction distance estimate estimate above tower base

------------------------------------------------------------- m---------------------------------------------------------------
NW 23.25 12.32 12.30 1.10
NW 50.46 12.55 12.39 1.49
NW 69.19 12.64 12.36 2.18
NW 111.63 12.46 12.58 2.51
NE 18.20 12.28 12.13 1.03
NE 39.64 12.65 12.64 0.96
NE 58.67 12.39 12.46 1.00
NE 85.77 12.45 12.46 0.72
NE 166.10 12.28 12.62 0.48
SW 14.86 12.45 12.45 2.05
SW 21.98 12.44 12.45 2.27
SW 34.16 12.78 12.57 2.83
SW 47.79 12.40 12.62 3.55
SW 62.55 12.70 12.44 4.54
SE 22.15 12.67 12.53 2.23
SE 39.15 12.69 12.53 2.91
SE 59.01 12.45 12.53 3.51
E 24.57 12.51 12.45 1.58
E 44.26 12.59 12.50 1.78
E 64.56 12.45 12.54 1.90
E 91.45 12.51 12.49 1.99

Mean 54.73 12.51 12.48 2.03

Standard deviation 36.32 0.14 0.12 1.03

Note: Direct measurement of the clock tower using a height pole yielded a height of 12.52 m.

7
 Table 2—Comparison of sine height with direct measurements (tape drop) for 42 large trees

Sine Tape drop Relative

Species height height Difference error

-------------------------------------- m -------------------------------------- - percent -

Eastern hemlock 52.64 52.76 -0.12 -0.23
Eastern white pine 52.73 52.73 0.00 0.00
Eastern hemlock 51.94 52.46 -0.52 -0.99
Eastern hemlock 52.30 52.33 -0.03 -0.06
Eastern hemlock 52.49 52.27 0.21 0.41
Eastern hemlock 51.42 51.63 -0.21 -0.41
Eastern hemlock 51.48 51.48 0.00 0.00
Eastern hemlock 50.51 51.05 -0.55 -1.07
Eastern hemlock 51.18 50.99 0.18 0.36
Eastern hemlock 51.02 50.96 0.06 0.12
Loblolly pine 50.72 50.90 -0.18 -0.36
Eastern hemlock 50.90 50.81 0.09 0.18
Eastern hemlock 50.84 50.63 0.21 0.42
Eastern hemlock 50.69 50.63 0.06 0.12
Eastern white pine 49.87 50.29 -0.43 -0.85
Eastern hemlock 50.29 50.20 0.09 0.18
Eastern white pine 50.63 50.17 0.46 0.91
Eastern white pine 49.83 49.80 0.03 0.06
Eastern hemlock 49.44 49.32 0.12 0.25
Eastern white pine 49.38 49.04 0.34 0.68
Eastern hemlock 48.65 49.01 -0.37 -0.75
Eastern white pine 48.46 48.86 -0.40 -0.81
Eastern white pine 49.23 48.83 0.40 0.81
Eastern white pine 49.23 48.65 0.58 1.19
Eastern hemlock 48.74 48.65 0.09 0.19
Eastern hemlock 47.85 48.13 -0.27 -0.57
Eastern hemlock 46.73 47.64 -0.91 -1.92
Eastern white pine 47.43 47.24 0.18 0.39
Eastern hemlock 47.18 47.03 0.15 0.32
Eastern hemlock 46.60 46.70 -0.09 -0.20
Eastern hemlock 46.02 46.18 -0.15 -0.33
Eastern hemlock 45.99 45.75 0.24 0.53
Eastern hemlock 44.41 44.74 -0.34 -0.75
Eastern white pine 44.65 44.68 -0.03 -0.07
Eastern hemlock 44.38 44.32 0.06 0.14
Eastern hemlock 44.50 43.89 0.61 1.39
Eastern hemlock 43.92 43.77 0.15 0.35
Eastern white pine 44.04 43.68 0.37 0.84
Eastern hemlock 43.89 43.68 0.21 0.49
Eastern hemlock 36.67 36.67 0.00 0.00
Eastern hemlock 35.36 35.20 0.15 0.43
Southern red oak 28.96 28.96 0.00 0.00

Mean 47.70 47.68 0.01 0.03

Standard deviation 4.85 4.88 0.31 0.64

8
The sine method can be challenging under conditions when suggested for the tangent method. Rather, to minimize the
the canopy or the understory are dense, as the observer time and expense of collecting the most accurate height
must have an unobstructed view of the high point and data possible, most people, it appears, tend to only move
base of the tree. Most laser hypsometers have a fairly to a point that appears to be perpendicular to the lean, and
narrow beam that allows them to penetrate crowns, or can then use that angle and distance to the stem to estimate
be adjusted with either electronic distance “gates” or via height. Such an ad hoc correction is incapable of measuring
the use of special reflectors to minimize the influence of the three-dimensional complexity of most tree crowns, and
intervening vegetation. It is also possible to hybridize the when combined with the observer’s innate tendency to then
sine and tangent methods to facilitate measurement in dense select the highest apparent point when determining total tree
understories and midcanopies. For instance, on flat to gently height, is likely to overestimate this metric. Also, in a further
sloping ground for relatively straight trees, one can use the attempt to reduce height sampling time, it is our experience
following approximation: that observers often measure several trees from the same
viewpoint, regardless of their lean or crown structure. This
(5) tendency has been seen from the first distance-independent
hypsometers (e.g., Krauch 1918) and is likely to be even
The top height component (α1 and d1) should still be done more common with horizontal baseline-correcting electronic
using the sine method, as this is where the greatest potential hypsometers, since the user may incorrectly believe the
for error occurs, but the bottom height (from horizontal to device corrects for all sources of measurement error.
the base of the stem) rarely differs between the sine and
tangent methods unless the ground slopes steeply or the tree
Conclusions
has an extreme lean.

The biggest advantage to the sine method is that it The sine method provides direct (not extrapolated)
eliminates the most problematic assumptions of the tangent measurements of observed points on the tree, thereby
method encountered in the field, and therefore substantially generating tree height as the elevation difference between
increases overall measurement accuracy. Violating the two horizontal planes. This geometric translation of a three-
assumptions of the similar triangle or tangent methods can dimensional object thereby eliminates the need to conduct
produce spectacularly large errors in tree height, especially ad hoc adjustments for tree lean, offset crown high points,
when measurements are taken in close proximity to the and ground slope in the field. When properly applied with
stem. There are numerous examples of champion trees that modern laser technology, the sine method should prove no
have been re-measured with the sine method only to find more onerous to measure than current techniques, and it is
that the similar triangle- or tangent-based errors exceed compatible with all accurately measured tree heights from
15 m (Eastern Native Tree Society 2009). For example, past inventories. This technique has been used to correct
a former national champion bitternut hickory (Carya some overestimated tree height values from champion tree
cordiformis (Wangenh.) K. Koch) in western North Carolina data which had been previously cited by silvicultural and
was first reported at 57.9 m with the tangent method and ecological texts as authoritative. Whether or not the sine
later re-measured by the sine method at 37.5 m. Likewise, method will supplant current approaches to measuring tree
a red maple (Acer rubrum L.) from Michigan originally height has yet to be determined—however, its accuracy,
measured at 54.6 m was eventually measured at 36.6 m with reliability, and repeatability suggest that it can be considered
the sine method. In both cases, these trees did not have their a standard for any science-based studies of forest conditions
tops killed or broken between the two measurements, but that include height as a parameter.
rather the observers failed to correctly apply conventional
height measurement techniques.
Acknowledgments
The sine method is sometimes more time-consuming to
apply because it requires that the laser beam directly strikes We thank Cook Forest State Park; the Eastern Native Tree
the object being measured. However, the sine method is a Society; the USDA Forest Service; and the University of
reliable and useful alternative to the tangent method because Minnesota for supporting this effort. We also thank Mike
of the freedom it offers to observe and measure the object Shelton (Southern Research Station, USDA Forest Service,
from any direction and distance without concern for tree retired), Nancy Koerth (Southern Research Station, USDA
lean or crown displacement. We believe few observers spend Forest Service), and Jamie Schuler (University of Arkansas-
the time needed to follow the textbook corrective procedures Monticello) for their reviews of this manuscript.

9
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