# How to calculate the height of a tree

## Height measurements | BC BigTree

#### Equipment needed

You 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!

**The 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 measure height

Height is the hardest measurement to take accurately, especially for larger trees. Measurements become more reliable the greater the distance you are from the tree (the distance you are away from the tree must be greater than the total tree height). In dense forests it can be challenging to get a clear view of the tree top. The slope of the ground can also make measurement difficult. Trees that are leaning significantly should be measured with the lean to the right or left, not with the lean toward or away from you. In challenging forest situations we recommend making more than one attempt to measure height. If possible try and remeasure from a different view point, and always double check your measurements.

#### 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.

**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.

**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.

**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 estimate the height of a tree?

There are a number of easy ways to get a rough estimate of the height of a tree.

You can obtain a rough guess of a tree's height by:

**comparing with a measurable object**nearby the trunk, eg. a pole or a house of which you know or can measure the height and by looking from a distance how many times that objects fits the tree. You could also do this on a photograph that was preferably taken from a distance as large as possible (to have the smallest perspective distortion of the tree possible) with the largest zoomfactor your camera has (to have the least lens distortion possible).- a more correct method is based on
**goniometry**: you start to walk away from the base of the trunk until you see the tree's top from an angle of 45 (which you can check using your arm).The height of the tree is then the distance from the tree to where you're standing (eg. 80 ft) + your eye height (the distance from the ground to your eyes, eg. 5 ft).

The idea is that if there's an angle of 45 (the angle between your line of sight and the ground) in a right-angled triangle (a triangle with an angle of 90, the corner tree-ground), then both small sides have an equal length. This means that the height of the tree then equals the distance tree-observer. But since your eyes are not on the ground, you need to add your own eye height as an extra (see image).

Provided you train yourself a couple of times in making steps of eg. 3 ft, you can relatively quickly determine the height of something. If you make your steps correctly (or use a tape to measure your distance from the tree) and if you're sufficiently able to determine an angle of 45 (or make you use of an inclinometer or tilt meter), then such a height guess can be quite good theoretically. But keep in mind that height measurements should be looked at very critically.

If you are not sure your guess of the 45 angle is any good, you can do it like this too: take a stick of about 1 to 2 ft and keep it vertically with a straight, horizontal arm. Walk away from the tree until the top of the tree corresponds with the tip of the stick, from your point of view. Then turn the stick horizontally and remember with which spot (as far away from you as the tree is) the tip of stick corresponds. The height of the tree equals the distance from the tree to that spot.

These methods assume the tree is not growing on a slope and the top of the tree can be determined (by surrounding trees or by the round shape of the crown the actual top of the tree can remain hidden for the observer). Also, 'distance to the tree' is not entirely correct: it should be 'distance to the orthogonal projection of the tree's top on the ground'. The difference between the 'distance to the tree' and that projection is half the diameter of the tree near the ground.

- another estimation useful for standalone trees standing on flat ground on sunny days is to
**measure the tree's shadow length on the ground**. If you measure the shadow's length of an object with known length (e.g. yourself), this would allow a rough estimation of the tree's size as well. See this calculator. - a third and the most accurate way consists of
**climbing to the top of the tree**and doing a direct tape drop. Of course this is something that should only be doing taking extreme precautions. This method is rarely used. On the photo on the right you can see Ronny Schreurs climbing a giant sequoia in military area Massy in Houthalen-Helchteren (Belgium) to determine its height.Video material of the tape drop of Hyperion, the tallest tree in the world.

Professional arborists measure trees' height by using a inclinometer like the Suunto clinometer or the Forester Vertex, by which a number of distances
and angles are measured. In more recent years handy laser equipment is used and is rapidly replacing the traditional inclinometers.
Using analoguous formulas from goniometry (like mentioned above) the height can be calculated.

A tape is used much more infrequently, except for specific trees like record trees or felled trees.

Above (left) you see the Nikon Forestry 550, a professional laser based range finder that can be used to measure the height of trees accurately and quickly. On the right see a climber working his way to the top of the tallest known beech tree in the Sonian forest to measure its height using a tape.

More on tree measuring:

- How to exactly measure tree height
- How to measure tree girth
- How to measure tree volume

## Determination of tree height and river width without special instruments

Determination of tree height and river width without special instruments - WoodWool Landscaping, tree care

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Special tools are available for determining different values. But, as practice shows, extreme tourists - tourists and travelers, as a rule, do not put altimeters in their backpacks. Although, it would seem, they really need these devices. For example, in order to quickly turn a coastal tree into a bridge across a fast river (unless, of course, we are talking about a protected area or someone's plot).

But this is the essence of extreme hobbies, that they allow you to enjoy your own victories - over laziness, routine, intellectual dependence on technical devices invented by someone. Everyone can feel like an experienced tracker or scout. One has only to want it and try to abstract from stereotypes. In particular, it is possible to determine whether the height of a tree is sufficient so that it, having fallen, could block the river, using items that are always at hand.

Contents

### How to find out the height of a tree

To determine the size of high-rise objects in "unsettled" conditions, you can use non-standard use of geometry rules or apply age-old techniques. By the way, using the shadow method, the "father of philosophy" Thales of Miletus (7th-6th centuries BC) calculated the height of the Egyptian pyramid.

Do not give up the simplest devices - rulers or tape measures. Theoretically, you can do without them, but the error will be too large.

So, here are some ways to determine the height of a tree:

- "Falling tree". Find a straight stick and, holding it vertically, move away from the tree. Move away until the visual dimensions (height) of the tree and the stick match. Now tilt the "instrument" to the left or right by 90 °, simulating the situation of a falling plant object. Remember the point on the ground where the top of the stick will be when it visually “falls”. It remains to measure the distance from it to the base of the tree. This is the value to be determined.

- Line of sight. In addition to the previous "measuring device", the help of an "assistant" will be required. He must place the stick at an arbitrary distance from the tree. You need to take a “lying” position face up in such a way that you can see the crown of the object above the top edge of the stick. Now you need to collect all the data in the formula: H \u003d (h * b) / a (H is the value we are trying to find out, h is the vertical size of the stick, the distance from your top to its base is a, to the foot of the tree is b).

- Shadow-based measurements. The proportionality between the object and the shadow cast by it is maintained throughout the solar day. Accordingly, knowing the ratio of the height of the stick (h) and the length of its shadow (u), you can calculate the size of the tree. We use the formula: H = (h * U) / u (U is the length of the shadow of the object of interest to us).

- Using a balloon (preferably filled with helium) on a long string. We release the thread until the ball reaches the top of the tree. After that, we return it to the ground and measure the length of the released string (thread).

- With a pencil (small stick). You will also need an assistant. Move away from the tree at such a distance that it will be comparable to the size of a pencil that you hold in your hand extended parallel to the ground. The silhouette of the tree should merge with the outline of the pencil (the top points strictly match). We fix the point of the base of the tree on the pencil with the thumb. After that, we turn the pencil in a clamped outstretched hand to the ground (by 90 °), left or right. Visually fix the point where the tip of the stylus intersects with the surface line. We ask the assistant to measure the distance from it to the base of the tree. The desired result has been obtained.

### Determination of the width of the river

At least you will need a tape measure, an indispensable stick (maybe more than one). The situation will help to simplify the elementary device for measuring angles.

#### Proven methods:

- Stand sideways to the river. Take a stick and, on an outstretched hand, turn it so that visually it resembles a bridge over a river in the place where you plan to cross. Its top should intersect with the contour of the opposite bank. Now you need to take the stick (without changing its position) at the point where you can see the border between the water and the bank on your side (the width of the river). Without moving your hand to the side, turn the stick 180°. Remember the landmark to which its top will point. The distance from it to the river will be equal to the width of the stream.
- We find an approximate object on the opposite bank. Standing in front of him, we stick a stick. We take a few steps from it at an angle of 90 ° (along the river), mark the point with another stick. We do not stop the movement, we repeat the number of steps passed and set the third landmark. We turn 90 ° (back to the water) and move without changing direction until the central (second) stick and the landmark on the opposite bank are on the same line (coincide). The length of the distance traveled from the third stick is equal to the width of the river.
- We visually fix a convenient object on the other bank at the place of the future crossing. We stand opposite him (then this place will become a reference point). Then we begin to move along the river until we stand on a line that is at an angle of 45 ° to the object on the opposite bank. The path that we have traveled from the reference point is equal to the width of the stream.

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## Tree height determination

- Authors
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Pakhomova E.V. 1

1MBOU secondary school No. 70 in Lipetsk Municipal budgetary educational institution secondary school No. 70 in Lipetsk

Bueva L.P. 1

1MBOU secondary school No. 70, Lipetsk Municipal budgetary educational institution secondary school No. 70, Lipetsk

##### The author of the work was awarded a diploma of the winner of the III degree

Diploma of a school studentCertificate of the head

The text of the work is placed without images and formulas.

The full version of the work is available in the "Job Files" tab in PDF format

Introduction

In the modern world, we determine the value of various quantities (length, mass, temperature) using various instruments and tools.

In their professional activities, builders and architects use complex expensive devices to determine the height of an object: electronic and laser altimeters. [3]

At the lessons of geometry in the 8th grade, the topic is studied: "Similar triangles." The properties of such triangles can be used to carry out various measurements on the ground. Two tasks were considered: determining the height of an object and the distance to an inaccessible point.

To determine the height of an object, the authors of the textbook "Geometry 7-9" L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev, E. G. Poznyak, I. I. Yudina suggest using a pole with a rotating bar. P.64. Practical applications of similar triangles. Determining the height of an object. To measure the height of a telegraph pole, you need to put a pole with a rotating bar at some distance from the pole and direct the bar to the top of the pole. A point is marked on the surface of the earth at which a straight line passing through the bar and the top point of the column intersects with the surface of the earth. The resulting right-angled triangles are similar according to the first sign of similarity of triangles. From the similarity of triangles, we find the height of the pole, having previously measured the distance from the pole to the point of intersection of the straight line with the surface of the earth, the distance from the point of intersection of the straight line with the surface of the earth to the base of the pole and the length of the pole.[1]

To prepare for the exam in mathematics, the OGE FIPI Mathematics Open Bank of Tasks is used. Problem 132764. A man 1.7 m tall stands at a distance of 8 steps from the pole on which the lantern hangs. The shadow of a person is 4 steps. At what height (in meters) does the lantern hang? Problem 134914. A person, whose height is 1. 8 m, is standing at a distance of 16 m from a street lamp. In this case, the length of the shadow of a person is 9 m. Determine the height of the lantern (in meters). [4]

In problem No. 581 "Geometry 7-9» L. S. Atanasyan to measure the height of a tree, it is proposed to use a mirror. A beam of light, reflected from a mirror, enters the human eye.[1]

Educational and methodological guide for preparing for the OGE 2019, edited by F. F. Lysenko. Option 6. Task. A pine tree casts a shadow 3 m long. Find its height if a person 1.6 m tall, standing near a pine tree, casts a shadow 0.4 m. Option 9. Task. The height of an apartment building is 12 m. Find the height of a tree growing 3 meters from it, if the shadows from the house and from the tree reach exactly to the bush. The distance between the bush and the tree is 200 cm.[2]

A tall beautiful birch grows near the house. (Picture 1) Hypothesis: “Is it possible to measure the height of this birch using the theoretical material of the textbook and tasks from the open bank of the OGE FIPI?” So the theme of the work was chosen.

Figure 1. Birch growing near the house.

Object of study: the process of using the material of the problem book and improvised means to measure the height of a tree.

Subject of research: similar triangles and ways to measure the height of a tree.

The purpose of my work: to show the relationship between theory and practice on the example of measuring the height of a tree without special instruments.

Before performing the practical part of the work, topics in mathematics were repeated: "Similarity", "Proportions", "Isosceles triangle", "Proportional segments" and additional literature was studied.

The topic is quite relevant. Measurements are made without special devices. Anyone can repeat them in practice. Problems of this type can be found on the main state exam in mathematics.

Assigned tasks:

Examine and analyze the literature that discusses various ways to determine the height of a tree using improvised means.

Carry out the necessary calculations and measurements in practice.

Document the results of measurements and draw conclusions.

2. Description of how to measure the height of a tree

2.1. Method "Shadow"

Equipment: tape measure, assistant.

1 way. The shadow of a tree and a person are on the same line.

A tree and a person are perpendicular to the earth, and the rays of the sun fall at the same angles on the earth. Similar triangles are formed, the sides of which are proportional.

Figure 2. Measuring the height of a tree with a shadow.

The ratio of the height of a tree to the height of a person is equal to the ratio of the length of the tree's shadow to the length of the person's shadow.

Human height - 169 cm, tree shadow length - 945 cm, human shadow length 272 cm.

To calculate the height of a tree, you need to multiply the height of a person by the length of the tree's shadow and divide the result by the length of the person's shadow.

X:169=945:272. X=587. It turned out approximately 5.9 m.

2 way. A person stands so that he does not fall into the shadow of a tree.

A tree and a person are perpendicular to the earth, and the rays of the sun fall at the same angles on the earth. Similar triangles are formed, the sides of which are proportional.

The ratio of the height of a tree to the height of a person is equal to the ratio of the length of the tree's shadow to the length of the person's shadow.

Human height - 169 cm, tree shadow length -245 cm, human shadow length 70 cm.

X:169=245:70; x=592. It turned out approximately 5.9 m.

2. 2. "Photography" method

The height of a tree is as many times the height of a person as the height of the tree in the photograph is greater than the height of the person in the photograph.

Equipment: camera, tape measure, person and assistant.

The person must stand close to the tree so that the person and the tree can be seen in full growth in the photo. (Picture 3)

Figure 3. Measuring the height of a tree using a photograph.

In the photo, we measure the height of the tree - 12 cm, the height of a person - 3 cm.

To calculate the height of a tree, you need to multiply the height of a person -169cm by the height of the tree in the photo and divide the result by the height of the person in the photo. Approximately 6.7m.

2.3. Balloon Method

You need to compare the height of the tree with the length of the thread that is tied to the balloon.

Equipment: balloon filled with helium, reel, tape measure.

The ball must be tied to a thread and released until the ball rises to the top of the tree. Then make a mark on the thread, lower the ball and measure the length of the thread.

Approximately 6.8 m. (Figure 4)

Figure 4. Measuring the height of a tree with a ball.

3.Conclusions and evaluation of the results.

Considered and put into practice a method of measuring the height of a tree with the help of improvised means: "Shadow". This method is found in tasks for preparing for the OGE. In additional literature, such methods of measuring the height of a tree as "Pencil", "Isosceles triangle", "Eye gauge", "Pole" are considered. Methods "Photography" and "Balloon" were applied to this tree to compare the results of measurements and calculations. The methods are based on the properties of similar triangles and on determining the length of a segment.

The most accessible of these methods is the "Balloon" method. In this way, you can measure the height of a tree without an assistant. This method requires calm weather.

The most economical method is the "Shadow" method. You need one roll. For the "Balloon" method, it was necessary to buy a balloon, fill it with helium. For the “Photography” method, you need a good camera, photographer, photos need to be developed.

The “Shadow” method turned out to be the most difficult to perform. Measurements should be taken on a clear sunny day. Several unsuccessful attempts were made. The sun does not stand still. The length of the shadow changes quickly. There is a rope on the ground. One end of the rope should be at the tree trunk, and the other should match the end of the tree's shadow. You need to stand on the rope so that the shadow of a person coincides with the shadow of a tree. And only after that carry out measurements of distances. When the shadow of a person and a tree are not on the same line, the assistant needs to measure distances very quickly.

Measuring the height of a tree with a mirror. The method is described in the school textbook in the problem. This method turned out to be more complicated than the three described. With the help of a mirror taken for measurement, it turned out to be impossible to find such a position in which the reflection of the top of the tree was visible in the mirror. In the textbook, the problem says: "A small mirror lying on the ground. " [1] After several unsuccessful attempts at measurements, it was concluded that the mirror should be large, at least 30cm. It is difficult to catch the reflection of the top of a tree in the mirror.

The method is affordable, but very time consuming.

The measurement results are different: 5.9 m, 5.9 m, 6.7 m, 6.8 m. It can only be stated that the height of the tree is about 7 m. To calculate the error of each of the methods, you need to know the exact height of the tree. This requires an altimeter.

What is the most accurate way? To do this, it is enough to carry out the same measurements, but with an object of known height (or with an object whose height can be measured with a tape measure, for example, a person and a meter ruler). After carrying out the experiments (Man and meter ruler), the conclusion was made: the shadow is a more accurate method of measuring the height of an object with improvised means than photography. (Table 1)

Method name | Settlements | Received ruler length |

Shadow method | 169:x=70:42 | 101.4 cm |

Photo method | 169:x=3:1.7 | 95.8 cm |

Table 1. Experiments "Man and Ruler"

When performing the work, the material studied in mathematics lessons on the topic “Similarity of triangles”, “Proportional segments” was applied to a practical problem. The study of this topic allows you to be more competent in the exam in mathematics.

The problem is solved in different ways.

In the course of the work, experience was gained in independently carrying out measurements and calculations, setting up an experiment, correcting the work on the structure, searching and studying additional literature.

Conclusion

All three methods of measuring the height of a tree "Shadow", "Photo" and "Balloon" can be used if necessary in real life. For example, if you need to cut down an old tree near the house for safety reasons, so that when it falls, it does not touch buildings or power lines. Any person can carry out the considered methods for measuring the height of a tree. The work clearly shows that geometry is not only a school subject, but a science that finds application in life.

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