It happens that there is a tree in your backyard or in the forest nearby that is quite suitable for harvesting, or you are simply interested in its height. And many in this case will ask the question: how to determine the height of a tree, but at the same time not perform any manipulations with the tree, and, among other things, not climb on it. In fact, there are several methods that work equally well to measure the height of a tree, and with reasonable accuracy.
The first method is to compare the sizes of the shadows and calculate the proportion. To do this, you need to stand next to a tree in sunny weather to see the shadow of the tree and your shadow, and thus measure the size of the shadows. Knowing your height, you can easily calculate the height of the tree.
For example, your height is 170 centimeters. Your shadow will be approximately 120 centimeters. The shadow of a tree, for example, will be 320 centimeters. This means that you need to solve the following proportion: divide the length of your shadow by your height and equate it to the ratio of dividing the shadow of a tree by an unknown value. Thus, it turns out: 120:170=320:x. In this case, x = 170*320:120 = 453 centimeters. This will be an approximate value, but it is enough for us.
The second method of obtaining the height of a tree can be to measure it by relative size. So, if your friend or relative stands next to a tree, and you take a pencil and move away a certain distance, then look at the pencil at arm’s length. In this case, the pencil should appear the same size as the tree. Then position the pencil in front of your friend. After that, measure your friend's height from a distance using a pencil. Thus, if a pencil has a length of 22 centimeters and a tree of the same size at a distance, then your friend’s height, for example, will be equal to seven centimeters, and his real growth- 168 centimeters. In this case we also get the proportion: 7:168=22:x. Having solved it, we find that the height of the tree is 528 centimeters, which is a little more than five meters.
In addition to these two methods, there are other ways to measure height. Some methods, for example, require knowledge of trigonometry and the use of an angle meter. For example, if you stand a friend at a certain distance from a tree and look from the side, you can mentally draw lines from the ground to the top of the tree through the friend's head. By measuring the resulting angle, you can calculate the height of the tree.
As you can see, calculating the height of a tree does not take much time and does not require complex or dangerous manipulations. To do this, you just need to know how to solve the proportion, as well as how to measure relative values. And the methods used can be used not only to measure the height of trees, but also any other tall objects, for example, buildings, mountains, pillars and so on.
It is necessary on a sunny day to choose an hour when the length of his own shadow will be equal to his height. To use a shadow to solve a problem, you need to know some geometric properties of a triangle, namely the following two:
1) The angles at the base of an isosceles triangle are equal, and vice versa - that the sides lying opposite the equal angles of the triangle are equal to each other;
2) the sum of the angles of any triangle is equal to 180 0 (i.e. two right angles)
On a sunny day, you can use any shade. By measuring the length of the pole (av) and the length of its shadow (vs). Then the required height is calculated from the proportion: AB: av = BC: sun.
Measuring the height of a tree using an isosceles triangle.
Approaching an object (for example, a tree) or moving away from it, 
place a triangle near the eye so that one of its legs is directed vertically, and the other coincides with the line of sight to the top of the tree. The height of the tree will be equal to the distance to the tree (in steps) plus the height to the observer's eyes.
7. Measuring the height of a tree using a puddle.
If there is a puddle not far from the tree, you need to position yourself so that it fits between you and the object, and then use a horizontally placed mirror to find the reflection of the top of the tree in the water (Fig. 4). The height of the tree will be as many times greater than the height of a person, how many times the distance from it to the puddle is greater than the distance from the puddle to the observer.
Measuring the height of a tree using a photograph.
Let's take a photograph that shows the object being measured and the measure. Let's find the ratio of the real length of the measurement to the length of the measurement from the photograph, then multiply the resulting result by the length of the measured object from the photograph? Maybe we'll get a more accurate result.
Measuring the height of a tree by eye (by eye).
Visually - this is the simplest and quick way. The main thing in it is the training of visual memory and the ability to mentally lay down on the ground a well-imagined constant measure (50, 100, 200, 500 meters). Having fixed these standards in memory, it is not difficult to compare with them and estimate distances on the ground.
gist: offer as much as possible more People can estimate the height of a tree by eye by placing a meter ruler vertically next to the tree.
With a balloon
The point: compare the height of the tree with the length of a suitable thread.
Equipment: balloon, filled with helium; long light rope (thread); tape measure or the like meter.
Progress:
1) tie a long thread to the ball and push it gradually upward until the ball reaches the top of the tree
2) make a mark on the thread (for example, a knot).
3) return the ball down, measure the length of the released part of the thread.
Pencil method
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Equipment: pencil (or pen, or any stick), assistant, tape measure.
Progress:
1) stand at such a distance from the tree that you can see it entirely - from the base to the top. Place an assistant near the trunk.
2) extend your hand in front of you with a pencil clenched in your fist. Squint one eye and bring the tip of the stylus to the top of the tree. Now move your thumbnail so that it is under the base of the trunk.
3) rotate your fist 90 degrees so that the pencil is parallel to the ground. In this case, your nail should still remain at the base of the trunk.
4) shout to your assistant to move away from the tree. When he reaches the point where the pencil point is pointing, signal him to stop.
5) measure the distance from the trunk to the place where the assistant froze. It will
equal to the height of the tree.
2.2. Choosing the optimal methods for determining the height of a tree.
We discussed all 11 methods for determining the height of a tree. Among them there are both physical and geometric methods. Selected physical methods applicable to autumn weather conditions:
using a pole (method No. 2.1.), an isosceles triangle (No. 6), a photograph (No. 8), by eye (No. 9), using the “pencil” method (No. 11).
How to measure tree height without cutting it down and climbing to its top? Of course there are many in various ways make such measurements using very simple instruments. But this kind of miracles is very easy to perform, having only knowledge of the principles of geometry.
Without a doubt, the sage Thales, who lived six centuries BC, could have solved the problem of how to measure the height of a tree. It was he who first identified it in Egypt using it. Thales chose the day and hour when the length of his own shadow was equal to his height; at this moment the height of the pyramid must also be equal to the length of the shadow it casts.
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 his height, they encounter level ground at an angle of half a right angle, and therefore the top of the pyramid, the middle of its base and the end of its shadow should form isosceles triangle. After this, the task of how to measure the height of a tree seems childishly simple.
But in our latitudes the sun is low above the horizon, and shadows are equal to the height of the objects casting them only in the afternoon hours summer months. Besides this, this in a simple way measuring the height of a tree is convenient to use on a clear sunny day for measuring alone standing trees, the shadow of which does not merge with the shadow of its neighbors.
Therefore, to measure the height of a tree using any shadow, no matter how long it is, you need to change this method. By measuring your shadow or the shadow of some pole, you can calculate the required height from the proportion:
AB: ab = BC: bс,
those. the height of the tree is as many times greater than your own height (or the height of the pole) as 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 triangles ABC And abc.
In another way to measure the height of a tree, you need a pole, which will have to be stuck vertically into the ground so that the protruding part is exactly equal to your height. The place for the pole must be chosen so that, lying horizontally, as shown in the figure, one can see the top of the tree in a straight line with the top point of the pole. Since the triangle ABC– isosceles and rectangular, then the angle A= 45 0, and therefore AB = BC, i.e. the desired height of the tree.
Such geometric techniques in the problem of measuring the height of a tree can only be performed using light. Try applying them to shadows cast by light street lamp or indoor - they will not come true.
Determining the size of inaccessible objects is easiest done using specialized geodetic equipment. Modern electronic total stations with a reflectorless measurement mode, laser tape measures and altimeters greatly simplify the task, allowing you to measure the height of a tree or the width of a river.
Unfortunately, not everyone can afford to have equipment worth several thousand dollars in reserve in the pantry, and sometimes one has to deal with similar problems at the everyday level. To solve these problems, knowledge gleaned from the series “Applied Geodesy” comes to our aid: “History of the industry”, “Choosing an electronic total station”, “Independent measurements using a tape measure, pegs and ingenuity”, school course geometry, and a little ingenuity (where would we be without it).
Determining the height of an inaccessible object
To determine the location of the future country house or any other building, it is important to know the heights of nearby objects, such as poles or dead trees. This will eliminate the possibility of destruction of your property from a falling object in the event natural disaster or for any other reasons.
Another important point before the start of construction is determining the sagging height of power line wires passing in the area of the site. A construction crane can hit a power line, which will lead to dire consequences. Do not forget about the breakdown voltage - there is a possibility of electric shock even a few meters from a high voltage wire in wet weather.
For the experiment, we will try to determine the height of a 10 kV power transmission line support from the ground to the top insulator using various methods and write the obtained values in the table.
Statistical evaluation method
It is also popularly called the “by eye” method. Its essence is a visual comparison of a known height and an inaccessible one. For convenience, near the object being measured, you install a vertical stick with known height. The “standard” for comparison should be as high as possible. Having moved away to a convenient distance, estimate the height, and write the result in the table. As you understand, one person cannot accurately take “measurements”, so to obtain good result ask your relatives or friends to perform similar actions. How more people participate in “measurements” - the more accurate the result.
Then comes the time for information processing: discard the extreme values (maximum and minimum), and calculate the arithmetic mean from the remaining results. The resulting value will give an idea of the height of the inaccessible object. The error of this method depends on the experience of people and the quality of their spatial orientation.
Evaluation by photo
The rapid development of technology has made it possible to integrate a camera into almost any modern gadget, so selecting equipment for such an experiment will not be difficult. The essence is the same - estimating the height of an inaccessible object, but not by eye, but by calculating the proportion between the photographic image of the standard and its real height.
Near the object to be measured, you place a stick with a known height (we used a geodetic pole), move away to a distance when the top of the object fits into the frame. Ideally, the height of the reference and shooting level should be approximately the same, and the camera itself should be held level. If possible, use a photographic tripod, the height of which should be set using a tape measure.
Download the image to your computer and refresh your memory from the article in our series, in which we introduced the concept of scale. We have received an image whose dimensions are proportional to its real size; we just need to calculate the scale and recalculate the height of the inaccessible object. To do this, you can print a photo for measurements with a ruler or use any image processing program that allows you to measure distances in a photo in centimeters.
This method is more progressive, but requires a computer and a camera, and field conditions this cannot always be ensured.
Ball pen
There is always a writing device in desk, and it will help us in determining the height of an object using the perspective method. Instead of a pen, you can use a pencil, a straight stick or any other similar item. We will also need an assistant and a tape measure.
We move away to such a distance that we can see the entire measurement object. Holding the pen in your fist, extend your straight arm in front of you so that its tip coincides with the top of the object. Pull it out thumb arms to the side parallel to the ground so that you end up with a right angle. Then we turn the brush with a ballpoint pen 90 degrees, as a result, our thumb looks into the ground parallel to the object being measured, and the tip of the pen points to the place where the assistant needs to move.
We projected the height of the object by parallel transfer to the ground. Now it won’t be difficult to measure the resulting distance from the assistant to the pole with a tape measure; it will be equal to the determined height. The method is well suited for field conditions, quite accurate, but requires an assistant.
Shadow measurement
The method used by the ancient Egyptians and Greeks is easily reproduced in modern realities and requires a minimum of labor costs. To determine the height, we need to simultaneously measure the length of the shadow from an object with a known height and the length of the shadow from an inaccessible object.
Measurements should be taken in the evening or morning, when the length of the shadow is maximum. This will eliminate errors, and the result is calculated by drawing up the simplest proportion:
Pole height = person's height * length of the pole's shadow / length of the person's shadow
Mirror method
The angle of incidence is known to be equal to angle reflections. We will use this postulate to calculate the inaccessible height. We place the mirror on the ground approximately as shown in the photo, step aside until the top of the object being measured is reflected in the mirror.
We measure the required distances from the person to the mirror, from the mirror to the pole, and obtain the required height after calculating the proportion.
Pole height = person's height * distance from mirror to pole / distance from person to mirror
Balloon
You can take measurements using the “Winnie the Pooh method” to please your own children. Despite some comedy, he also has the right to life, because... at favorable conditions it gives acceptable results.
To work, we will need a ball filled with inert gas, a rope and a tape measure. Carefully release the ball on the string parallel to the object being measured, when it reaches the top, fix the height of the string, lower the ball and measure the required distance with a tape measure. To obtain a more accurate value, your assistant should move a considerable distance away to more accurately visualize the place when the height of the ball and the object are equal. This method We present it as one of the alternatives, so we will not conduct laboratory tests on it.
Piano in the bushes
We've covered several basic methods for measuring the height of an inaccessible object. I would like to find out which one is the most accurate. The “piano in the bushes” will help us with this - an electronic tacheometer with a special software, which allows you to obtain the height value in the field.
For convenience, we will enter all the values in a table, which will give us the necessary clarity. As can be seen from the table, all methods have a small error, but this is quite enough to estimate the height of an inaccessible object.
The arithmetic mean of the obtained values is 9.47 m, so to obtain an optimal result, the methods must be combined and the obtained values averaged. In case it is necessary high accuracy, you can purchase a pendulum altimeter, which is used by foresters to assess green spaces.
Well, the most accurate result will be obtained when ordering a topographic survey. In the technical specifications it is worth noting special requirement— measuring the heights of trees and other inaccessible objects. As you understand, this will affect the estimated cost of the work, so we will devote the next article in our series to estimates. Incomprehensible terms may hide work processes that you don’t need, and vice versa, some important points unknowingly may fall out of sight. Armed with this knowledge, you will be able to communicate with specialists almost on equal terms, which will ultimately save money.
Vladimir Stefansky, rmnt.ru
Direct methods for determining linear distances
Accurate measurements are made using a measuring tape or steel tape, 10 or 20 meters long. Sometimes, a long cord is used (in the form of a thick wire), on which marks are placed: white - every 2 m and red - every 10 m, with pins fixed at the ends (steel pins or wooden stakes). It is important that the measuring devices do not stretch and are accurately measured and adjusted to the standard.
When measuring fields and taking measurements along winding contours on the ground, the field surveying compass-meter “Kovalyok” (“two-meter”, old name -), in the form of the letter A, is still used. This is a folding wooden fork, with a constant leg opening equal to 2 meters .
During topographic survey work, they keep a measurement log compiled in a standard form, where the numbers of standing points and the results of current measurements are immediately entered. Additionally, they draw up, by hand, an outline (schematic drawing of the film being taken, in this moment, locality).
Approximate, rough measurements with low accuracy are made by pedometer - in pairs of your steps (equal to approximately your height, minus 10-20 centimeters, depending on the pace of walking, the degree of roughness of the terrain and the angle of inclination earth's surface). The counting results are sequentially entered, recorded in a notebook, in the form of a data table for further recalculation of distances covered and route segments into meters.
Remote visual methods for determining distances
Remote-visual methods of measuring lengths - they are used in cases where there is an insurmountable barrier or obstacle (river, swamp, lake, deep ravine, mountain gorge), but direct visibility sufficient for making measurements is maintained.
The width of the river can be determined geometrically by eye, by constructing two equal right triangles. Having selected on the opposite bank (in the direction perpendicular to the channel) some noticeable object “A” (tree, large stone, etc.), located at the very edge of the water, drive a peg “B” opposite it (Figure 1). Along the shore, perpendicular to line AB, measure with a tape measure or in steps, for example 20m, and drive in a peg “C”. On the continuation of line BC, at a distance also equal to 20 m, another peg “D” is driven in. From peg "D" in the direction DE, perpendicular (directions are set by spreading your arms to the sides and bringing them together with your palms, straight in front of you or using a cross-shaped ecker) to the line DV, you need to go from the river until peg "C" will be in line with object "A". Since triangles ABC and EDS are absolutely and completely equal, the width of the river will be equal to the distance DE minus VK (the interval to the water's edge). If the shoulders DS and SV are not equal (it is not possible to walk along the shore; they interfere dense thickets), then AB = DE*BC/CD
The width of the river can be determined without leaving the water by constructing a rectangular isosceles triangle ADV on the ground (Fig. 2). Having constructed a right angle at point “A”, they move away in the direction AC to such a point “D”, from which object “B” will be detected at an angle of 45° (in this case, AB = AD). To lay out the corners, a homemade cross-shaped ecker is used (in the form of a square sheet of paper with corners bent upward or, mounted on a stand, a flat wooden cross with four pins driven in square), with the help of which angles of 45° and 90° from the chassis are built lines (main highway). At point “A”, for better visibility when placing poles on the target, a clearly visible “layout” is placed (for example, a white sheet of paper is attached, facing towards point “D”).
Express method, without installing an ecker on a tripod - hold two crossed straight branches of the same length horizontally at eye level so that one branch is parallel to the river flow and directed to point “A” (look with one eye closed). Then, the forty-five-angle line passing through the ends of the branches is looked at by closing the other eye and slightly tilting the head. You can also sight using a compass scale or dial wristwatch(you can use a measuring ruler as a guide, placing it with its edge through the center of the dial).
Having the opportunity to carry out triangulation on the ground (measure with a protractor or on the dial of a compass) and (in field conditions, this can be done without a calculator and precise ones, using a protractor, ruler and compass), you can sight at any angle, and then calculate using the formula:
AB = AD * tg ADV.
If the angle is 45 degrees, then tg(45°)=1 and, accordingly, AB=AD
tg(64°) = 2 and AB=AD*2
tg(72°) = 3 and AB=AD*3
Fig.2
The width of the river can be determined quite accurately using the direct notch method (Fig. 3). To do this, a noticeable object “C” is selected on the opposite bank, and along the bank on which the researcher is located, a base AB is laid and its length is measured. From points “A” and “B” make notches to point “C”, i.e. measure angles CAB and ABC. By constructing triangle ABC using a measuring ruler, we can obtain the basis accepted AB scale the desired width of the river.
In the same way, the width of the river can be determined without directly measuring the angles CAB and ABC, using graphic notches on a tablet. It is necessary to plot on paper the length of the basis AB on the selected scale, then from the ends of the basis, orienting the planchette while standing on the corner points, draw directions to some visible object “C” on the opposite bank. Then, the width of the river can be determined graphically - on the drawing, recalculating it to its scale.
Fig.3
A very simple and convenient approximate method for determining the width of a river using a blade of grass or thread. Standing on the bank of the river at point “A”, they notice on the opposite bank two noticeable objects (for example, boat B and tree “C”), located near the edge (Fig. 4). Then, taking a blade of grass (thread) by its ends with hands extended in front of you, notice its length “d”, which closes the gap BC between the selected objects (you need to look with one eye). Then, folding the blade of grass in half, they move away from the river until (point “D”) until the gap BC is closed by the blade of grass. The distance AD traveled will be equal to the width of the river.
Fig.4
There is also the fastest, but very approximate way to determine the width of the river - close your right eye and point the thumb of your horizontally extended hand raised upward (Fig. 5) in the direction of a noticeable object “A” on the opposite bank. Then, changing open eye(this is how a stereoscopic effect appears in the form of a stereo pair of images from two various points observations), they notice that the finger seemed to bounce sideways from the observed object to point “B”. By eyeballing the distance AB, in meters (assuming approximately the height or width of objects), and multiplying it by 10, you get the approximate width of the river. In such measurements, a person acts as a stereophotogrammetric device.
Fig.5
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