HOW DO YOU ESTIMATE THE HEIGHT OF A TREE?

Trees are tall. Humans, relatively speaking, are not and we're not the most agile climbers in the animal kingdom, either. But we are crafty, and we can out-think even the most wizened redwood. 

If you have ever tried to guess the height of a tree, you’ll know that it is difficult. Well, there are many simple methods to estimate the height of a tree, which doesn’t require any guesswork or tree climbing. Or cutting the tree down.

How do we measure things that are too big for our measuring tools?  Get outdoors and harness the power of ratios and geometry to estimate the height of trees and other tall objects. But how do you do so when all you've got is a pencil, chalk, a mirror, or a smartphone? Let’s find out how math can help!

5 EASY METHODS TO FIND THE HEIGHT OF A TREE

The Stick Method

This old but simple method only works on level ground.  It just requires a stick and a distance measuring tape.  The stick must be the same length as your arm or grasped at a point where the length of the stick above your hand equals that of your arm.  The stick is held pointing straight up, at 90 degrees to your outstretched, straight arm.  

 

Carefully walk backwards until the top of the tree lines up with the top of your stick.  Mark where your feet are.   The distance between your feet and the tree is roughly equivalent to the height of the tree.  You might find it interesting to compare your results using this simple method with the standard methods described below.

 

The trick uses the geometry of triangles to make this estimate. In geometry, similar triangles are triangles that have the same shape, but one is a larger or smaller version of the other. If triangles are similar, their corresponding angles are equal, and the lengths of the corresponding sides will be proportional too. An isosceles right triangle is one that has a 90-degree angle and two sides of equal length. Here, you used similar isosceles triangles to help you find and measure the distance from the tree that was equal to the tree’s height.

The Shadow Method

Find a tree or tall pole whose shadow is on a flat surface that you can see and reach (not down a hill, or onto a house or busy street). Mark the top of the tree’s shadow on the ground with chalk.

 

Measure the distance from the base of the tree to the top of the tree’s shadow. Use a measuring tape, if you have one long enough. Or, walk that distance. Multiply the number of steps you took by the average stride distance you calculated. This number is the height of the tree’s shadow.

 

Now stand up tall so you can see your own shadow on a flat surface. Have another person put a small chalk mark at the top of your shadow and one at the back of your heels. Measure that distance; that number is the height of your shadow. Have the other person measure how tall you are when standing up straight. That number is your height.

 

Now for the math trick: the ratio of the tree’s height to its shadow is going to be the same as the ratio of your height to your shadow. In other words:

 

Tree’s height      =      Your height  x  Tree’s shadow

                                                  Your shadow

Using A Smartphone

Stand away from the tree so that you can see its top.  The method works best if your angle of elevation is about 45 degrees.  In other words that your distance from the tree is equivalent roughly to the height of the tree.

 

If you are working with someone else they can help you measure the distance from where you’re standing to the tree.  If you’re on your own, drop something where you’re standing to mark the position so that you can return once you’ve paced or measured the distance to the tree.

 

Standing at your spot, open an angle measuring app on your smartphone. Bring the smartphone to your eye and sight along its edge, as if you’re looking down a gunsight, aiming at the very top of the tree.  You will need to hold the phone so that your fingers are not in the way.

 

Calculate tree height. Open the calculator on your smartphone.  You will need to access the scientific calculator. It doesn’t matter whether you use metres or feet as long as you use consistently the same units throughout 

With a little more calculations we have the height of the tree by putting the values in the equation:

 

Tree’s Height = (Tan ∠ to tree top x distance to the tree) + (Tan ∠ to tree base x distance to the tree)

Comparing With A Measurable Object

Comparing with a measurable object nearby the tree can also be a great way of finding the height, eg. a pole or a house of which you know or can measure the height and by looking from a distance how many times that objects fit the tree. 

 

You could also do this on a photograph that was preferably taken from a distance as large as possible (to have the smallest perspective distortion of the tree possible) with the largest zoom factor your camera has to have the least lens distortion possible.

Pole Method

Before starting the method let’s assume one person will be the called the “pole bearer” and the other will be called the “seer”. The pole bearer will hold a pole.

To begin the activity, your team will measure a distance of 20 meters from the base of the tree and mark that spot on the ground; the seer will then stand on that spot. The bearer will then make a mark on the pole which corresponds to the eye level of the seer. The pole bearer will then measure the two lengths on the pole: the length corresponding to the eye-level.

Now the pole bearer will walk slowly away from the seer and toward the tree. Here is the part where the seer comes into play. It is the seer’s job to align the top of the tree with the top of the pole. As the pole bearer moves away from the seer, the top of the pole will appear to move downward in relation to the top of the tree the seer will need to tell the pole bearer to stop at the exact place where the top of the pole and the top of the tree are in alignment. The pole bearer will mark that spot on the ground.

Now the seer will measure the distance on the ground from the original spot (20 meters from the base of the tree) to the spot which the pole bearer has marked. These are all of the measurements that are needed to find the height of the tree. Do the calculations. Make a diagram showing the tree, the pole in its final position and the seer. Put the corresponding measurements you found on your diagram. Using similar triangles and proportionality, you can easily find the height of the tree.

In Conclusion

Estimation of the height can be both easy and fun if done through the right tricks and methods. Measurements become more reliable the greater the distance you are from the tree for the stick and shadow method (the distance you are away from the tree must be greater than the total tree height).  In dense forests, it can be challenging to get a clear view of the treetop.  The slope of the ground can also make measurement difficult.  Trees that are leaning significantly should be measured with the lean to the right or left, not with the lean toward or away from you.  

 

 In challenging forest situations try to make more than one attempt to measure height.  If possible try and remeasure from a different viewpoint, and always double-check your measurements.