Easy Way to Determine the Height of a Tree

I was talking with Fred recently when I mentioned a very tall, very dead tree in the forested common area near my house. I see this tree every time I walk out our front door and often wonder if it’s tall enough to hit my house should it fall. It’s so tall that I really can’t estimate it’s total height. Fred reminded me of some simple geometry and told me an easy way to find the height of the tree.

Similar Triangles

The idea is that if you have two similar triangles, the corresponding sides are proportional and you can deduce unknown lengths (such as the height of a tree). For a more visual explanation, check out this Math Reference site.

• Given: ΔABC ~ ΔDEF
• Then: AB/BC = DE/FE, AB/AC = DE/DF, and so on

Setup

So how do we create two similar triangles? Follow these steps:

• Grab a shallow bowl or pan. Fill it with water.
• Set your pan on the ground somewhere moderately level.
• Step back from the pan and arrange yourself up so that it’s a straight line from you to the pan, to the tree.
• Move around until you find the very top of the tree reflected in the water.
• After finding the tree top, measure these lengths:
  • Your height, from the ground to eye level
  • Distance from you to the pan
  • Distance from the pan to the tree

This is a pretty sweet mock up, eh?

These two triangles have the same shape but not necessarily the same size. That means that corresponding angles are equal (i.e.  angle c = angle f). It also means that the sides are proportional (i.e. side AB is proportional to side ED). We’ll use side AC and FD to determine the ratio and then find out the height of our tree.

The Math

Here are my numbers.

• Side AB = 70 inches = 5.83 feet
• Side AC = 87 inches = 7.25 feet
• Side FD ≈ 30 yards ≈ 120 feet

So let’s setup our equation. Side AB / side AC = side DE / side DF. Substituting real numbers gives us: 5.83 / 7.25 = side ED / 120. Using some simple math reveals that side ED = 96.5 feet. According to my numbers, the tree is about 100 feet tall and should be just shy of damaging my house. Hurray!

Tips

Here are a couple things to keep in mind.

1. I used a pan with black interior. It made finding the reflected tree top easier.
2. Don’t use a mirror or you’ll have to spend time making it perfectly level. Water, on the other hand, will level itself.
3. Watch your units. Don’t mix inches and feet. Pick one and convert everything.
4. Setup next to your house to avoid having to measure another distance (from the tree to your house).
5. This method assumes you are on relatively flat ground. It doesn’t account for hills or valleys.

What do you think? Ever think high school math would have easy application like this?
Image courtesy of guy schmidt

by: Ethan | June 8, 2009 | filed in: General
Related Tags: height, Tree

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