# Calculate Height of a Tree - Geometry

1. ### thender

39
Hi,

I visited my Mother and Stepfather recently, and admired the tall trees around their house.

We estimated them to be around 180-200 feet tall.

I told my Stepfather that the *Actual* height could of course be calculated.

I said there are three angles and three lengths for any triangle, and if you have any three of them, you can calculate the others. (I said a bit less than that, but that was my idea).

He said the Pythagorean theorem is the only way to calculate the height of the tree, and it is not practical because we can't measure the distance from the apex to the observer.

I said I just need to stand at a fixed distance D, from the tree, and measure the angle A made by a straight line from that point to the apex, relative to a straight line from that point to the base of the tree. The tree grows straight, so its angle to the ground is 90*.

He told me I was a fool. And I told him I'd prove him wrong later.

From a logical standpoint, if two angles are fixed and one length is known, the other two lengths MUST intersect at only one point. If they can only have one intersection, they can only have one length.

I really doubt it's impossible to calculate it, but it might take a little more than A^2 + B^2 = C^2.

2. ### Mentallic

3,783
What you're doing here is trigonometry. To try convince him that it works, tell him that if you move far enough away from the tree such that the angle from the ground to the top of the tree is 45o (right in between 0 and 90) then the distance you are from the tree would be equal to the height of the tree. This should make intuitive sense at least.

If going that far is not feasible, then you just need to use the formula

$$\tan\theta = \frac{O}{A}$$

Re-arranging,

$$O=A\tan\theta$$

where O is the opposite length of a right-angled triangle, and A is the adjacent length. Hence O is the height of the tree, and A is the distance from the tree.

And yes, you're correct. Knowing 3 pieces of information about the triangle (except all 3 angles, because knowing 2 angles automatically gives you the 3rd, so it's only 2 pieces of info) gives you enough to figure out everything about the triangle. Right triangles are a more simple case, but there are generalizations for any triangle.

But also, don't be surprised if you find an answer for the height of the tree but in reality it turns out to be wrong, or someone else does the measurements and calculations and finds a different value. You will certainly have errors in your work (and by errors I mean uncertainties in your measured values and hopefully not mathematical errors). To minimize the errors, try to go as far away from the tree as possible such that the angle is not too close to 90o.

For example, if whatever angle measuring tool you're using can only realistically give you a value that is within 1o of the actual angle (so if it were actually 67o but your tool gave you between 66-68) then if we imagine that the height of the tree is precisely 200ft, if you moved 20 ft away and measured the angle, the precise angle would be about 84.29o. If we take the lower extreme of the angle measured by your tool and say it gave you 83.29o then your calculations would give the height of the tree to be 170ft, and the upper error of 85.29o will give 243ft. Meanwhile, if you moved 100ft away, then the actual angle would be 63.43o and the lower error of 62.43o => 191.5ft and 64.43 => 209ft.
Notice that the uncertainty is drastically minimized by going further out and working with angles that are further from 90o (and don't go so far that they approach 0o as well).

Last edited: Jul 13, 2014
1 person likes this.
3. ### olivermsun

969
One of these might be useful for your demonstration:
http://en.wikipedia.org/wiki/Sextant

4. ### davenn

4,354
There's an even easier way without trig.

being able to show you stepfather 2 differ4ent methods may help cement the reality

Step 1

Plant a straight rod in the ground so that it stands vertically. Measure the height of the rod sticking out of the ground and the length of the shadow cast by the rod.

Step 2

Divide the length of the rod by the length of its shadow. This is the current shadow ratio or shadow factor. For example, if the rod is 21 inches tall and its shadow is 12.5 inches long, then the shadow ratio is 21/12.5 = 1.68.

Step 3

Measure the total length of the tree's shadow in the direction of the sun's rays. The total length is measured from the center of the tree's base to the tip of the shadow. Since you cannot place one end of the measuring tape at the center of the tree, you will have to add the tree's radius to the visible shadow length in order to get the total length.

For example, suppose a tree has a diameter of 3 feet and the length of the shadow it casts is 177.5 feet measuring from the base of the tree to the tip of the shadow. Since the tree has a radius of 1.5 feet, the total shadow length is 177.5 + 1.5 = 179 feet.

Step 4

Multiply the tree's total shadow length by the shadow factor you calculated in Step 2. This is the tree's height.

Using the same example as above, if a tree has a total shadow length of 179 feet and the shadow factor is 1.68, then the height of the tree is 179*1.68 = 300.72 feet, or about 300 feet 9 inches.

cheers
Dave

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5. ### thender

39
Thank you very much to everyone, doing the math might be easier than changing the man's mind!

I like how Mentallic was clever enough to foresee the actual practical ramifications of the method, and I might submit that in order to help bring the estimate closer to reality, a number of samples at different lengths could be taken, and either averaged or compared, sort of statistically.

And I greatly enjoy Davenn's approach which takes advantage of the natural geometry of the sun. An untapped resource!

I also remembered the navigation by stars using Sextants I think, but couldn't remember the name or any details.

Thanks to everyone for your help.

6. ### larrybud

26
The shadow method might not work depending on the shape of the tree and the angle of the sun, as you may not be able to tell where the top of the tree's shadow ends!

### Staff: Mentor

Perhaps a simpler method...

Take a square piece of paper or cardboard, and fold it along its diagonal to form a 45 degree angle. Start at the base of the tree and pace away from it, turning occasionally to sight the top of the tree. To sight it, sit down on the ground (to minimize error), hold the paper so one flat side is parallel to the ground and you are looking up along the diagonal edge. Repeat this enough times to find the place where you are able to sight the top of the tree along the 45 degree line of your paper. You are now as far away from the tree as it is tall.

You can improve the accuracy of this using a level and a mirror, but you should be able to get pretty close with the basic procedure.

8. ### mp3car

73
Someone may have mentioned this, i apologize if so, I only skimmed the responses. You can also use the principle of similar triangles, which you learn in geometry rather than trigonometry (like tangent function).

You pick a point and measure how far you are from the tree along the ground, suppose 100'. You can then do one of two things, I'll start with the simpler method. Lie down and look up at the top of the tree. Then have someone else hold something like a broom stick on the ground and have them get a distance away from you such that the very end of the broom stick aligns with the top of the tree. Measure the distance between you and the broom stick, suppose 5'. Then you take the 100' distance from the tree (the base of the triangle) and divide by the 5' from you to the broomstick, giving you 20. Then multiply the height of the broomstick by 20, and that's the height of the tree. I am typing this in a big hurry, so someone else please check my logic! Thanks!! :D

9. ### micromass

20,054
Staff Emeritus
10. ### infinity149

2
Mentallic, that's beautiful, simplicity defined!

Thender, get a right triangle from somewhere such that the length of the base is equal to the hight. Maybe cut one out of cardboard. Hold it in front of your face such that the base is parallel to the ground and the hight is facing toward the tree. Sight up the length of the hypotenuse and adjust your distance to the tree such that the tip just covers the top of the tree. Now measure how far you are from the tree and that is how tall the tree is. Of coarse, the accuracy of this approach depends on how level you hold the device. The base must be held perfectly parallel to the ground.