# Calculate Height of a Tree - Geometry

by thender
Tags: geometry, height, tree
 P: 29 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.
 HW Helper P: 3,540 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).
P: 788
 Quote by thender 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.
One of these might be useful for your demonstration:
http://en.wikipedia.org/wiki/Sextant