How to measure the height of a tree

How to measure the height of a tree without cutting it and climbing its top? Of course, there are many different ways to produce such measurements using very simple instruments. But, such miracles are very simply performed, knowing only the knowledge of the principles of geometry.

How to measure the height of a tree without cutting it and climbing its top?

Without a doubt, the task of measuring the height of a tree could be solved by the sage Thales, who lived six centuries before our era. It was he who first identified the height of the pyramid in Egypt, using its shadow. Thales chose the day and the hour when the length of his own shadow was equal to his height; at this moment, the height of the pyramid should also be equal to the length of the shadow cast by it.

Knowing that the angles at the base of an isosceles triangle are equal, and that the sum of the angles of any triangle is equal to two right angles, Thales concluded that when his own shadow is equal to its growth, the sun's rays meet the even ground at an angle of half the straight, and hence the vertex The pyramids, the middle of its base and the end of its shadow should form an isosceles triangle. After that, the task of measuring the height of a tree is childlike.

But in our latitudes the sun is low above the horizon, and the shadows are equal to the height of the objects throwing them only in the midday hours of the summer months. In addition, this simple way to measure the height of a tree is convenient to use on a clear sunny day to measure lonely standing trees, the shadow of which does not merge with the shadow of neighboring trees.

Therefore, to measure the height of a tree using any shadow, no matter how long it is, you need to change this method. Measuring your shadow or the shadow of some pole, you can calculate the desired height from the proportion:

Therefore, to measure the height of a tree using any shadow, no matter how long it is, you need to change this method

AB : ab = BC : bc,

those. the height of the tree is as many times greater than your own height (or the height of the pole), how many times the shadow of the tree is longer than your shadow (or the shadow of the pole). This follows, of course, from the geometric similarity of the triangles ABC and abc.

In another way how to measure the height of a tree, you need a pole that will have to be pushed vertically into the ground so that the protruding part just equals your height. Place for the pole should be chosen so that, lying horizontally, as shown in the figure, you could see the top of the tree on one straight line with the top point of the pole. Since the triangle Abc is isosceles and rectangular, the angle A = 450, and hence AB = BC, i.e. the desired height of the tree.

Such geometric methods in the task of measuring the height of a tree can only be performed using the sun's light. Try to apply them to the shadows cast by the light of a street lamp or room lamp - they will not be justified.

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