Tree Height Measuring Example

Quote of the Day

Your grades, whatever is your GPA, rapidly becomes irrelevant in your life. I cannot begin to impress upon you how irrelevant it becomes. Because in life, they aren’t going to ask you your GPA ....If a GPA means anything, it’s what you were in that moment — and it so does not define you for the rest of your life.

— Neil deGrasse Tyson, astrophysicist, during a commencement address at the University of Massachusetts Amherst. I often cite quotes like this to young people who have a grade problem. A low GPA may limit some options early on, but it should not be viewed as an insurmountable obstacle.


Introduction

Figure 1: Silver Maple, a Common Tree in My Area.

Figure 1: Silver Maple, a Common Tree in My Area (source).

I have been testing a number of Android applications that are intended to measure the size of objects knowing their range or vice versa. One application that I have found particularly useful is called Baumhöhenmesser – Tree Height Meter (my translation) – which is an application written by a German developer. I have found this application particularly useful, and I thought I would review its operation here. It is part of a suite of Android applications intended for forestry management. This app makes excellent use of the Android's ability to measure angles.

My plan in this post is to present one of the formulas used in the application and illustrate this formula's use with an example. While the app is focused on measuring the height of trees, I have been using this application to measure the height of many objects. I do occasionally need to determine the height of a tree – usually just before I cut it down. I want to know the tree's height so that I can clear a spot on the ground of appropriate size.

Background

There are two formulas used in this application. My focus here will be on the formula used to measure the height of a tree using a fixed height reference that is placed against the tree and three angle measurements. Equation 1 shows the formula used in this measurement scenario.

Eq. 1 \displaystyle h=L\cdot \frac{{\tan \left( {{{\alpha }_{1}}} \right)-\tan \left( {{{\alpha }_{3}}} \right)}}{{\tan \left( {{{\alpha }_{2}}} \right)-\tan \left( {{{\alpha }_{3}}} \right)}}

where

  • h is the tree height
  • α1 angle from horizontal to the top of the height reference (counterclockwise positive)
  • α2 angle from horizontal to the top of the height reference (counterclockwise positive)
  • α3 angle from horizontal to the bottom of the tree (counterclockwise positive)
  • L is length of the height reference.

I show how to derive this formula in Figure 2.

Example

Figure 2 shows a simple example of how Equation 1 can be used to determine the height of a tree. In Figure 2, I define a range term R, but it will be used only as a temporary variable. The tree height is computed in terms of the reference length L.

Figure 2: Tree Height Measurement Example.

Figure 2: Tree Height Measurement Example.

Conclusion

Just a quick example illustrating how a useful height-measurement application works.

 
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3 Responses to Tree Height Measuring Example

  1. Ronan Mandra says:

    I assume R, in the diagram, is the horizontal distance to the tree.

    Nice article

     
  2. Pingback: Samsung S5 Field of View | Math Encounters Blog

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