**Problem #12:** A tree growing near the bottom of a
ravine is to be measured for its height. The measurer has a clinometer and
tape, but no laser. From the side of the ravine, the measurer looks down toward
the base, which cannot be reached, and upward toward the crown. The measurer is
able to line up the high point of the crown with the trunk from a spot on the
side of the ravine having uniform, linear slope. How might the measurer
determine the height of the tree, using an exterior baseline and angles to the
top of the tree from the ends of the baseline and the angle to the bottom from
the lower end of the baseline.

**Solution:** Diagrams and accompanying equations will illustrate
the solution. Angles a_{1}, a_{2}, a_{3}, and a_{7 } as
defined above are measured by the measurer - also distance d. Angles a_{4},
a_{5}, and a_{6} are computed using the rule that the angles of
a plane triangle add up to 180 degrees. The length of side L of the triangle
from the measurer’s eye to the crown-point, then down to eye level, and back to
the eye is computed using the law of sines that says that the ratios of a sides
of the triangle to the sines of the respective angles are all equal. Therefore,
if A, B, and C are the sides of a triangle and the opposite angles are
designated by a, b, and c, then the law of sines is illustrated below. The law
of sines illustration is followed by the tree diagram.

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**Tree Diagram**

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In the above diagram, H = full tree height, h_{1}
= height above eye level and h_{2}=height below eye level.

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**Derivation of equations and calculation of tree height**

In the above diagram, let L be the distance from the eye to the high point of the crown. Then by the law of sines

In the above diagram, let D be the level distance from the eye to a point directly beneath the highpoint of the crown.

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By algebraic substitution, we can derive a formula for H that involves only the known quantities angles a1, a2, a3, and a7 and distance d.

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**Summary comments:** If a_{2}=0, the baseline is
level, so the formula for H serves two measurement scenarios, i.e. where P_{1}
is above P_{2} and where they are on the same level. An Excel workbook
accompanies this problem that automates the solution.

measure/problems/Problem_12.xls