How to use trigonometry to measure the height of a tree

How to Use Trigonometry to Measure the Height of a TreeYou only need three pieces of equipment to properly measure a tree: a measuring tape, a calculator (with cosine and tangent functions), and an inclinometer to measure angles.  If purchasing an inclinometer (Abney level, clinometer, etc.) is beyond your budget, or you can’t borrow one, there are mobile phone apps that allow you to use a smartphone as an inclinometer. Here are two possible options: Smart Measure and iHandy Carpenter.  See Gabriel Hemery’s helpful instructions on using iHandy Carpenter to measure tree height.

Today Foresters use a Hypsometer – an all in one tool that measures distance, angles and even calculates tree height for you.  This equipment hastens the measurement process, but it is not necessary.

For fun!

How to Use Trigonometry to Measure the Height of a TreeThe stick method- how they built the pyramids!
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.

How to Use Trigonometry to Measure the Height of a Tree

The math used in height calculations

Working on level ground
Calculating tree height requires the use of basic trigonometry: h = Tan A x d, where h is the tree height, d is the distance from tree, and A is the angle to the top of the tree.  Since your measurements will be made at eye level, you need to know your eye height (height of your eye above the ground). The equation then becomes h = Tan A x d + eye height.

How to Use Trigonometry to Measure the Height of a Tree

Working on moderately sloped terrain
If the only option available to you is to stand either up or down slope of the tree, and the gradient is such that the base of the tree is above or below eye level, additional angles need to be measured.   In addition to tree top angle, you need to measure the angle to the tree base.  These angles are either subtracted or added depending on whether you are above (added) or below (subtracted) the tree.

How to Use Trigonometry to Measure the Height of a Tree

Tree base is obscured or hidden from view
Often obstacles such as shrubs, rocks, or fallen trees can obscure the tree base from view. In this case, you will need to measure the angle to a mark on the trunk that is a known height from the ground.

One method, shown in the illustration below, is to have someone stand at the base of the tree and measure the angle to the top of their head (height x).

If you can’t see them through the tall bushes, try having them hold a flag or bright coloured stick above their head at a known height.

How to Use Trigonometry to Measure the Height of a Tree

Working on steep terrain
On very steep terrain it is almost impossible to accurately determine your horizontal distance from the tree. In situations where the ground is sloped (up or down) more than 6 degrees (10% slope) you will need to measure slope distance. Once you measure slope angle and slope distance, horizontal distance can be calculated.

How to Use Trigonometry to Measure the Height of a Tree

Activity: How High?

For this activity you will need:

  • A piece of cardboard or thin sheet of wood, about the size of a sheet of paper.
  • A piece of string about 30cm or 1 foot long:

How to Use Trigonometry to Measure the Height of a Tree

  • A drawing pin (thumbtack) or a small nail:
  • A small rock (one you can tie the string to):

How to Use Trigonometry to Measure the Height of a Tree

Aim

The aim of this activity is to measure the height of a tall building or tree. I will illustrate this with a tree, but the method is just the same for a building. How to Use Trigonometry to Measure the Height of a Tree

 

Suppose you want to measure the height

Can I just measure it with
a tape measure?
That won't be easy. You may have to climb the tree to do that. Dangerous!

But there is another, easier, way!

Measure a length along the ground and measure the angle θ to the top of the tree:

How to Use Trigonometry to Measure the Height of a Tree

Surveyors and engineers use special equipment to measure these distances and angles accurately. The instrument they use to measure the angle is called a theodolite. In this activity you will make a simple theodolite. You won't get accurate answers, but it will help you to understand how surveyors do their work.

There are three parts to this activity:

  • Making a simple theodolite.
  • Measuring the angle and the distance along the ground.
  • Using the measurements to find the height of the tree.

Making a Simple Theodolite

Step 1: Print out this image of a protractor onto A4 paper. You will need to change the page orientation to landscape.

Step 2: You do not need to cut out the picture of the protractor, but you will need to cut as near as you can to the top edge – just leave a small space, enough for you to attach the drawing pin or nail.

Step3: Glue your picture of the protractor with its straight-edge lying along the top edge of your cardboard or timber:

Step 4: Tie the small rock to one end of the piece of string:

How to Use Trigonometry to Measure the Height of a Tree

Step 5: Attach the other end of the string to the top middle point of the protractor using the drawing pin or nail:

How to Use Trigonometry to Measure the Height of a Tree

When you allow the rock to pull the string downwards, gravity will make it hang vertically. Even if you tilt the piece of cardboard or timber at an angle, the string will still hang vertically:

Can you see how the angle of 30° is related to the angle shown by the string on the protractor?

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Measuring the Angle and the Distance Along the Ground

First of all, find a good place to stand  about 100 feet or 30m away from the tree. Make sure that you are standing on level ground and in a safe place. Mark the place with a stick. Now you are ready to measure.

To measure the angle, you just point your 'theodolite'' towards the top of the tree. Put your eye as close as possible to one edge of the protractor; then, looking along the straight edge, point it directly at the top of the tree:

Hold the string and weight in position before you lower the theodolite. Then read the angle from the theodolite as the angle the vertical weight on the string  makes with the protractor. Adjust your answer to find the angle of elevation, θ, from your eye to the top of the tree.

To find the distance, d, use your tape measure to measure from the stick to the base of the tree, or more exactly to the midpoint of the base of the tree – can you see how to make allowance for that?

Using the Measurements to Find the Height of the Tree

You can calculate the height of the tree using a scale diagram on a piece of paper, following these steps:

  • At a point, C, near the right end of the line, measure an angle equal to the angle, θ, from your experiment:
  • Choosing a suitable scale, measure the distance x equivalent to the distance d you measured in your experiment. Mark a point B:
  • Draw the perpendicular at B to meet the sloping line at A:
  • Measure the height, y, of A above B:
  • Using the scale from step 3, convert y back to find the height, h, of the tree.

Wait!

  • Did you notice anything wrong?
  • Where was your eye when you measured the angle θ?
  • Can you think of a way to compensate for that error?

Another Way

There is an alternative way to calculate the height of the tree using trigonometry. If you know how to do this, then go ahead.

  • How accurate will your answer be?

Can you list below some things about this experiment that tell you that your answer is not very accurate?

Any more?

Try to find out how a real theodolite is made and why it gives much more accurate measurements.

Copyright © 2020 MathsIsFun.com

Tree-mendous Measuring

Give the students the two similar right-angled paper triangles. Similar means that they are enlarged or reduced copies of one another. Discuss what features of shapes helped them spot that the triangles were similar. Make a table of variant and invariant features of similar triangles, i.e., features that change and those that do not change as the triangle is enlarged.

How to Use Trigonometry to Measure the Height of a Tree

Most students won’t know that the ratios of the side lengths within and between similar triangles remain invariant under enlargement. This concept is fundamental to understanding trigonometry.

For the pair of similar triangles, first label the hypotenuse, then mark an angle and label the sides that are opposite and adjacent to that angle. Discuss the words adjacent (meaning next to), opposite (meaning across from), and hypotenuse (the side that is opposite the right angle).

How to Use Trigonometry to Measure the Height of a Tree

Get the students to measure the side lengths of the triangles in millimetres. Stress that accuracy is important. Firstly, get the students to compare the ratios of the matching sides in the different triangles, i.e., opposite side of A : opposite side of B, adjacent side of A : adjacent side of B, and hypotenuse of A : hypotenuse of B.

For example:

How to Use Trigonometry to Measure the Height of a Tree

The ratios 60:80, 104:139, and 120:160 contain the operator 1.33, which maps a side length from triangle A onto its matching side from triangle B. This is the scale factor.

Is there an easy way to measure the height of a tree?

Use a Rock:
(For quick results skip to NOTE)

I had one that was more for fun than anything else. You can use a rock (Assuming that the height of the tree is within throwing distance, you can choose a small rock) and use the 1 dimensional kinematic equations from classical mechanics.

That might sound kind of scary, but really just throw the rock as hard as you can and start counting seconds, and when you see it reach the top of the tree, and record this time.

You can then use the equations:
How to Use Trigonometry to Measure the Height of a Tree

Where delta x is the height of the tree, v0 is initial velocity, vf is final velocity (Which is 0 since it stops at the height of the tree, or approximately the height of the tree), and t is the time you measured to get to the top, which you have from counting. You have two equations and two unknowns (delta x and v0), you can solve this by substitution, elimination, or by matrix algebra (RREF) to get the height of the tree). NOTE: the height of the tree will be in meters, you know, the better format 😀

NOTE: That's the complicated way, the simple way, after derivations is reduced to:
How to Use Trigonometry to Measure the Height of a Tree

We see that the result is easy to compute in your head, it's:
How to Use Trigonometry to Measure the Height of a Tree
So, if the rock takes 2 seconds to stop in the air, it went a height of approximately: 5 * 2^2 = 20m

How to Measure the Height of a Tree

There are many reasons to measure the height of a tree. You may need to know what it could hit if it falls, how high up the fruit is, or may simply want to establish bragging rights. Whatever the reason, measuring the height of a tree is not difficult. Below are four methods that work in most situations. All methods require that you are fairly level with base of the tree.

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45° Triangle Method – Easy

This method requires a square piece of paper or card and a way to measure distance from the tree.

How to Use Trigonometry to Measure the Height of a TreeMeasuring the height of a tree using a 45 degree angle.

  1. Fold the paper/card square in half to make a 45° right angle triangle.
  2. Hold the triangle up to your eye and look along the longest side at the top of the tree.
  3. Move backwards/forwards until your eye lines up with the top of the tree and the two shorter sides run parallel with the ground and tree trunk.
  4. Measure the distance from where you stand in step 3 to the base of the tree. The distance is equal to the height of the tree from the height of your eye (see diagram above for clarification).
  5. Add the measured distance/height to your eye height. Remember you calculated the tree height from your eye level and so you need to add your height for total tree height.

A limitation is when the tree has a spreading canopy, you might be unable to see the top with a 45 degree angle. In that case you would need to use the ‘full version’ of this technique, the trigonometry method.

Trigonometry Method – Medium

This is the ‘full version’ of the 45° triangle method above. With an angle of 45°, the math is easy and height equals distance. When you use an angle other than 45° (when you are forced to stand closer or further away) the math is a little more involved. For this method you will need clinometer and a way to measure your distance from the base of the tree.

How to Use Trigonometry to Measure the Height of a TreeMeasuring the height of a tree using trigonometry.

  1. Measure the angle between the top of the tree and the ground from your eye. Ideally do this using a clinometer. Stand up straight on level ground. You need to be far enough back that you can easily see the top of tree. Put the clinometer to your eye and line it up with the top of the tree while holding the lever (lets the indicator swing freely). Release the lever to trap the indicator and then read off the indicated angle. If you don’t have a clinometer you can just mark the angle on a peice of card and measure using a protractor.
  2. Now that you have the angle, measure the distance between the tree and where you were standing in step one. Do this with a measuring tape or wheel for a more accurate result.
  3. Use the Tangent rule to calculate height of the tree (above eye level). tan(angle) = opposite/adjacent Where opposite is the height of the tree and adjacent is the distance between you and the tree. This is rearranged to:

    opposite = tan(angle) x adjacent

    or more simply

    height = tan(angle) x distance

  4. Add the height of your eyes to the calculated height of the tree. This gives you the total height of the tree. Remember you measured from your eye height and so you have to include how high your eyes are. See the diagram above for clarification.
  • For example, if I stood 20 m from a tree and the angle between the tree’s top and horizontal from my eye was 30 degrees, I would calculate the height as follows:
  • height = tan(angle) x distanceheight = tan(30) x 20
  • height = 11.55 m
  • I am 1.75 m tall at my eyes and so

total height = 11.55 m + 1.75 mtotal height = 13.3 m

Pencil Method – Easy

An easy method that can be done on the fly. I find this especially useful when I want a quick method for seeing if anything is within the fall zone of a tree. Note the danger zone when felling trees is generally considered to be a radius of two times the height of the tree.

  1. Hold a pencil up in your outstretched arm. You can also use a small, straight stick for this method.
  2. Stand far enough back that the tree height fits within the length of the pencil.
  3. Hold the base of the pencil in line with the base of the tree and mark the top of the tree on the pencil (you can just use your thumb on the other hand).
  4. Turn the pencil sideways as though the tree had fallen over. Remember (or have somebody mark) where the ‘top of tree’ mark (or thumb) is in relation to the tree along the ground. The distance from this point on the ground to the base of the tree is roughly equal to the tree’s height.

Smartphone Method – Very Easy

How to Use Trig to Calculate the Height of Things

••• scientific calculator image by Stanisa Martinovic from Fotolia.com

Updated April 24, 2017

By Mark Kennan

When you see a tall object, such as a tree or a flagpole, you may wonder how tall the object is but not have any way to reach the top to measure the height. Instead, you can use trigonometry to calculate the height of the object.

The tangent function, abbreviated “tan” on most calculators, is the ratio between the opposite and adjacent sides of a right triangle.

If you know, or can measure the distance from the object to where you are, you can calculate the height of the object.

    Measure the distance from the object you want to calculate the height of to where you are standing.

    Use the protractor to estimate the angle formed by the line parallel to the ground at your eye level and the line from the top of the object to your eyes.

    Use your calculator to find the tangent of the angle from step two. For example, if the angle from step two was 35 degrees, you would get approximately 0.700.

    Multiply your distance from the object by the result from step three. For example, if you were 20 feet from the object, you would multiply 20 by 0.700 to get about 14 feet.

    Measure the distance from the ground to your eyeball and add the result to the result from step four to calculate the height of the object. For example, if you measure five feet from the ground to your eyeballs, you would add five to 14 to find the total height of the object equals 19 feet.

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About the Author

Mark Kennan is a writer based in the Kansas City area, specializing in personal finance and business topics. He has been writing since 2009 and has been published by “Quicken,” “TurboTax,” and “The Motley Fool.”

How to Measure the Height of a Tree

  1. 1

    Use this method as an alternative to the shadow method. While this method is less accurate, you can use it when the shadow method won't work, such as on an overcast day. Also, if you have a tape measure with you, you can avoid having to do math. Otherwise, you'll need to find a tape measure later and do some simple multiplication problems.[6]

  2. 2

    Stand far enough from the tree so you can view the whole tree—top to bottom—without moving your head. For the most accurate measurement, you should stand so that you are on a piece of ground that is about level with the ground at the tree’s base, not higher or lower. Your view of the tree should be as unobstructed as possible.

  3. 3

    Hold a pencil at arm's length. You can use any small, straight object, such as a paint stick or ruler. Hold it in one hand and stretch your arm out so that the pencil is at arm’s length in front of you (between you and the tree).

  4. 4

    Close one eye and adjust the pencil up or down so that you can sight the very top of the tree at the top of the pencil. This is easiest if you turn the pencil so that the sharpened point is pointing straight up. The tip of the pencil should thus just cover the top of the tree in your line of sight as you look at the tree “through” the pencil.

  5. 5

    Move your thumb up or down the pencil so that the tip of your thumbnail is aligned with the tree’s base.

    While holding the pencil in position so that the tip is aligned with the tree’s top (as in step 3), move your thumb to the point on the pencil that covers the point (again, as you look “through” the pencil with one eye) where the tree meets the ground. Now the pencil is “covering” the entire height of the tree, from the base to the tip.

  6. 6

    Rotate your arm so that the pencil is horizontal (parallel to the ground). Keep your arm held straight out at the same distance, and make sure your thumbnail is still aligned with the tree’s base. The thumb should be even with the center of the bottom of the tree.

  7. 7

    Have your friend move so that you can sight him or her “through” the point of your pencil. That is, your friend’s feet should be aligned with the pencil’s tip.

    He or she should be about the same distance from you as the tree is, not further away or closer toward you.

    Since, depending on the height of the tree, you may need to be some distance away from your friend, consider using hand signals (with the hand that is not holding the pencil) to tell him or her to go farther, come closer, or move to the left or right.

  8. 8

    If you have a tape measure with you, measure the distance between your friend and the tree. Have your friend remain in the place or mark the spot with a stick or rock. Then use a measuring tape to measure the distance between that spot and the base of the tree. The distance between your friend and the tree is the height of the tree.

  9. 9

    If you don't have a tape measure with you, mark the height of your friend and the height of the tree on the pencil.

    Scratch or draw a mark on the pencil where your thumbnail is; this is how long the tree appears from your perspective.

    Use the same method as before to arrange the pencil so it covers your friend, with the tip at your friend's head and your thumbnail at his or her feet. Make a second mark at this position of your thumbnail.

  10. 10

    Find the answer once you have access to a tape measure. You'll need to measure the length of each mark and the height of your friend, but you can do this after you go home, without having to return to the tree. Scale the difference in lengths on the pencil up to your friend's height.

    For instance, if the mark showing your friend's height is 2 inches (5 cm) from the tip and the mark for the tree's height is 7 inches (17.5 cm), then the tree is 3.5 times as tall as your friend, since 7 inches / 2 inches = 3.5 (17.5 cm / 5 cm = 3.5). If your friend is 6 feet (180 cm) tall, the tree is 6 x 3.

    5 = 21 feet tall (180 cm x 3.5 = 630 cm).

    • Note: If you do have a tape measure with you when you're near the tree, you do not need to do any calculations. Read the step above for “if you have a tape measure” carefully.

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