Correct wording pyramid and examples. Pyramid and truncated pyramid

We are well aware of the great Egyptian pyramids, everyone can imagine what they look like. This representation will help us understand the features of such a geometric figure as a pyramid.

A pyramid is a polyhedron consisting of a flat polygon - the base of the pyramid, a point that does not lie in the plane of the base - the top of the pyramid and all segments connecting the top with the points of the base. The segments that connect the top of the pyramid with the top of the base are called lateral edges. On fig. 1 shows the pyramid SABCD. Quadrilateral ABCD is the base of the pyramid, point S is the top of the pyramid, segments SA, SB, SC and SD are the edges of the pyramid.

The height of the pyramid is the perpendicular dropped from the top of the pyramid to the plane of the base. On fig. 1 SO is the height of the pyramid.

A pyramid is called n-gonal if its base is an n-gon. Figure 1 shows a quadrangular pyramid. A triangular pyramid is called a tetrahedron.

A pyramid is called regular if its base is a regular polygon, and the base of the height coincides with the center of this polygon. The lateral edges of a regular pyramid are equal, and, therefore, the lateral faces are isosceles triangles. In a regular pyramid, the height of the side face drawn from the top of the pyramid is called apothem.

The pyramid has a number of properties.

All diagonals of a pyramid belong to its faces.

If all side edges are equal, then:

• near the base of the pyramid, a circle can be described, and the top of the pyramid is projected into its center;
• the side edges form equal angles with the base plane, and, conversely, if the side edges form equal angles with the base plane, or if a circle can be circumscribed near the base of the pyramid, and the top of the pyramid is projected into its center, then all side edges of the pyramid are equal.

If the side faces are inclined to the base plane at one angle, then:

• a circle can be inscribed at the base of the pyramid, and the top of the pyramid is projected into its center;
• the heights of the side faces are equal;
• the area of ​​the lateral surface is equal to half the product of the perimeter of the base and the height of the lateral face.

Consider the formulas for finding the volume, surface area of ​​the pyramid.

The volume of the pyramid can be calculated using the following formula:

where S is the area of ​​the base and h is the height.

To find the total surface area of ​​a pyramid, use the formula:

S p \u003d S b + S o,

where S p is the total surface area, S b is the side surface area, S o is the base area.

A truncated pyramid is a polyhedron enclosed between the base of the pyramid and a cutting plane parallel to its base. The faces of the truncated pyramid, lying in parallel planes, are called the bases of the truncated pyramid, the remaining faces are called the side faces. The bases of a truncated pyramid are similar polygons, the side faces are trapezoids. A truncated pyramid that is obtained from a regular pyramid is called a regular truncated pyramid. The side faces of a regular truncated trapezoid are equal isosceles trapezoids, their heights are called apothems.

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Students come across the concept of a pyramid long before studying geometry. Blame the famous great Egyptian wonders of the world. Therefore, starting the study of this wonderful polyhedron, most students already clearly imagine it. All of the above sights are in the correct shape. What right pyramid, and what properties it has and will be discussed further.

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Definition

There are many definitions of a pyramid. Since ancient times, it has been very popular.

For example, Euclid defined it as a solid figure, consisting of planes, which, starting from one, converge at a certain point.

Heron provided a more precise formulation. He insisted that it was a figure that has a base and planes in the form of triangles, converging at one point.

Based on the modern interpretation, the pyramid is presented as a spatial polyhedron, consisting of a certain k-gon and k flat triangular figures that have one common point.

Let's take a closer look, What elements does it consist of?

• k-gon is considered the basis of the figure;
• 3-angled figures protrude as the sides of the side part;
• the upper part, from which the side elements originate, is called the top;
• all segments connecting the vertex are called edges;
• if a straight line is lowered from the top to the plane of the figure at an angle of 90 degrees, then its part enclosed in the inner space is the height of the pyramid;
• in any side element to the side of our polyhedron, you can draw a perpendicular, called apothem.

The number of edges is calculated using the formula 2*k, where k is the number of sides of the k-gon. How many faces a polyhedron like a pyramid has can be determined by the expression k + 1.

Important! A regular-shaped pyramid is a stereometric figure whose base plane is a k-gon with equal sides.

Basic properties

Correct pyramid has many properties that are unique to her. Let's list them:

1. The base is a figure of the correct form.
2. The edges of the pyramid, limiting the side elements, have equal numerical values.
3. The side elements are isosceles triangles.
4. The base of the height of the figure falls into the center of the polygon, while it is simultaneously the central point of the inscribed and described.
5. All side ribs are inclined to the base plane at the same angle.
6. All side surfaces have the same angle of inclination with respect to the base.

Thanks to all the listed properties, the performance of element calculations is greatly simplified. Based on the above properties, we pay attention to two signs:

1. In the case when the polygon fits into a circle, the side faces will have equal angles with the base.
2. When describing a circle around a polygon, all the edges of the pyramid emanating from the vertex will have the same length and equal angles with the base.

The square is based

Regular quadrangular pyramid - a polyhedron based on a square.

It has four side faces, which are isosceles in appearance.

On a plane, a square is depicted, but they are based on all the properties of a regular quadrilateral.

For example, if it is necessary to connect the side of a square with its diagonal, then the following formula is used: the diagonal is equal to the product of the side of the square and the square root of two.

Based on a regular triangle

A regular triangular pyramid is a polyhedron whose base is a regular 3-gon.

If the base is a regular triangle, and the side edges are equal to the edges of the base, then such a figure called a tetrahedron.

All faces of a tetrahedron are equilateral 3-gons. In this case, you need to know some points and not waste time on them when calculating:

• the angle of inclination of the ribs to any base is 60 degrees;
• the value of all internal faces is also 60 degrees;
• any face can act as a base;
• drawn inside the figure are equal elements.

Sections of a polyhedron

In any polyhedron there are several types of sections plane. Often in a school geometry course they work with two:

• axial;
• parallel basis.

An axial section is obtained by intersecting a polyhedron with a plane that passes through the vertex, side edges and axis. In this case, the axis is the height drawn from the vertex. The cutting plane is limited by the lines of intersection with all faces, resulting in a triangle.

Attention! In a regular pyramid, the axial section is an isosceles triangle.

If the cutting plane runs parallel to the base, then the result is the second option. In this case, we have in the context of a figure similar to the base.

For example, if the base is a square, then the section parallel to the base will also be a square, only of a smaller size.

When solving problems under this condition, signs and properties of similarity of figures are used, based on the Thales theorem. First of all, it is necessary to determine the coefficient of similarity.

If the plane is drawn parallel to the base, and it cuts off the upper part of the polyhedron, then a regular truncated pyramid is obtained in the lower part. Then the bases of the truncated polyhedron are said to be similar polygons. In this case, the side faces are isosceles trapezoids. The axial section is also isosceles.

In order to determine the height of a truncated polyhedron, it is necessary to draw the height in an axial section, that is, in a trapezoid.

Surface areas

The main geometric problems that have to be solved in the school geometry course are finding the surface area and volume of a pyramid.

There are two types of surface area:

• area of ​​side elements;
• the entire surface area.

From the title itself it is clear what it is about. The side surface includes only the side elements. From this it follows that to find it, you simply need to add up the areas of the lateral planes, that is, the areas of isosceles 3-gons. Let's try to derive the formula for the area of ​​the side elements:

1. The area of ​​an isosceles 3-gon is Str=1/2(aL), where a is the side of the base, L is the apothem.
2. The number of side planes depends on the type of the k-gon at the base. For example, a regular quadrangular pyramid has four lateral planes. Therefore, it is necessary to add up the areas of four figures Sside=1/2(aL)+1/2(aL)+1/2(aL)+1/2(aL)=1/2*4a*L. The expression is simplified in this way because the value 4a=POS, where POS is the perimeter of the base. And the expression 1/2 * Rosn is its semi-perimeter.
3. So, we conclude that the area of ​​​​the side elements of a regular pyramid is equal to the product of the semi-perimeter of the base and the apothem: Sside \u003d Rosn * L.

The area of ​​the full surface of the pyramid consists of the sum of the areas of the lateral planes and the base: Sp.p. = Sside + Sbase.

As for the area of ​​\u200b\u200bthe base, here the formula is used according to the type of polygon.

Volume of a regular pyramid is equal to the product of the base plane area and the height divided by three: V=1/3*Sbase*H, where H is the height of the polyhedron.

What is a regular pyramid in geometry

Properties of a regular quadrangular pyramid

When preparing for the exam in mathematics, students have to systematize their knowledge of algebra and geometry. I would like to combine all known information, for example, how to calculate the area of ​​a pyramid. Moreover, starting from the base and side faces to the entire surface area. If the situation is clear with the side faces, since they are triangles, then the base is always different.

What to do when finding the area of ​​the base of the pyramid?

It can be absolutely any figure: from an arbitrary triangle to an n-gon. And this base, in addition to the difference in the number of angles, can be a regular figure or an incorrect one. In the USE tasks of interest to schoolchildren, there are only tasks with the correct figures at the base. Therefore, we will only talk about them.

right triangle

That is equilateral. One in which all sides are equal and denoted by the letter "a". In this case, the area of ​​\u200b\u200bthe base of the pyramid is calculated by the formula:

S = (a 2 * √3) / 4.

Square

The formula for calculating its area is the simplest, here "a" is the side again:

Arbitrary regular n-gon

The side of a polygon has the same designation. For the number of corners, the Latin letter n is used.

S = (n * a 2) / (4 * tg (180º/n)).

How to proceed when calculating the lateral and total surface area?

Since the base is a regular figure, all the faces of the pyramid are equal. Moreover, each of them is an isosceles triangle, since the side edges are equal. Then, in order to calculate the lateral area of ​​\u200b\u200bthe pyramid, you need a formula consisting of the sum of identical monomials. The number of terms is determined by the number of sides of the base.

The area of ​​an isosceles triangle is calculated by the formula in which half the product of the base is multiplied by the height. This height in the pyramid is called apothem. Its designation is "A". The general formula for lateral surface area is:

S \u003d ½ P * A, where P is the perimeter of the base of the pyramid.

There are situations when the sides of the base are not known, but the side edges (c) and the flat angle at its vertex (α) are given. Then it is supposed to use such a formula to calculate the lateral area of ​​\u200b\u200bthe pyramid:

S = n/2 * in 2 sin α .

Task #1

Condition. Find the total area of ​​the pyramid if its base lies with a side of 4 cm, and the apothem has a value of √3 cm.

Decision. You need to start by calculating the perimeter of the base. Since this is a regular triangle, then P \u003d 3 * 4 \u003d 12 cm. Since the apothem is known, you can immediately calculate the area of ​​\u200b\u200bthe entire lateral surface: ½ * 12 * √3 = 6√3 cm 2.

For a triangle at the base, the following area value will be obtained: (4 2 * √3) / 4 \u003d 4√3 cm 2.

To determine the entire area, you will need to add the two resulting values: 6√3 + 4√3 = 10√3 cm 2.

Answer. 10√3 cm2.

Task #2

Condition. There is a regular quadrangular pyramid. The length of the side of the base is 7 mm, the side edge is 16 mm. You need to know its surface area.

Decision. Since the polyhedron is quadrangular and regular, then its base is a square. Having learned the areas of the base and side faces, it will be possible to calculate the area of ​​\u200b\u200bthe pyramid. The formula for the square is given above. And at the side faces, all sides of the triangle are known. Therefore, you can use Heron's formula to calculate their areas.

The first calculations are simple and lead to this number: 49 mm 2. For the second value, you will need to calculate the semi-perimeter: (7 + 16 * 2): 2 = 19.5 mm. Now you can calculate the area of ​​an isosceles triangle: √ (19.5 * (19.5-7) * (19.5-16) 2) = √2985.9375 = 54.644 mm 2. There are only four such triangles, so when calculating the final number, you will need to multiply it by 4.

It turns out: 49 + 4 * 54.644 \u003d 267.576 mm 2.

Answer. The desired value is 267.576 mm 2.

Task #3

Condition. For a regular quadrangular pyramid, you need to calculate the area. In it, the side of the square is 6 cm and the height is 4 cm.

Decision. The easiest way is to use the formula with the product of the perimeter and the apothem. The first value is easy to find. The second is a little more difficult.

We'll have to remember the Pythagorean theorem and consider It is formed by the height of the pyramid and the apothem, which is the hypotenuse. The second leg is equal to half the side of the square, since the height of the polyhedron falls into its middle.

The desired apothem (the hypotenuse of a right triangle) is √(3 2 + 4 2) = 5 (cm).

Now you can calculate the desired value: ½ * (4 * 6) * 5 + 6 2 \u003d 96 (cm 2).

Answer. 96 cm2.

Task #4

Condition. The correct side of its base is 22 mm, the side ribs are 61 mm. What is the area of ​​the lateral surface of this polyhedron?

Decision. The reasoning in it is the same as described in problem No. 2. Only there was given a pyramid with a square at the base, and now it is a hexagon.

First of all, the area of ​​\u200b\u200bthe base is calculated using the above formula: (6 * 22 2) / (4 * tg (180º / 6)) \u003d 726 / (tg30º) \u003d 726√3 cm 2.

Now you need to find out the semi-perimeter of an isosceles triangle, which is a lateral face. (22 + 61 * 2): 2 = 72 cm. It remains to calculate the area of ​​\u200b\u200beach such triangle using the Heron formula, and then multiply it by six and add it to the one that turned out for the base.

Calculations using the Heron formula: √ (72 * (72-22) * (72-61) 2) \u003d √ 435600 \u003d 660 cm 2. Calculations that will give the lateral surface area: 660 * 6 \u003d 3960 cm 2. It remains to add them up to find out the entire surface: 5217.47≈5217 cm 2.

Answer. Base - 726√3 cm 2, side surface - 3960 cm 2, entire area - 5217 cm 2.

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Introduction

When we meet the word "pyramid", then the associative memory takes us to Egypt. If we talk about the early monuments of architecture, then it can be argued that their number is at least several hundred. An Arab writer of the 13th century said: "Everything in the world is afraid of time, and time is afraid of the pyramids." The pyramids are the only miracle of the seven wonders of the world that has survived to our time, to the era of computer technology. However, researchers have not yet been able to find clues to all their mysteries. The more we learn about the pyramids, the more questions we have. Pyramids are of interest to historians, physicists, biologists, physicians, philosophers, etc. They are of great interest and encourage a deeper study of their properties, both from mathematical and other points of view (historical, geographical, etc.).

So purpose Our study was the study of the properties of the pyramid from different points of view. As intermediate goals, we have identified: consideration of the properties of the pyramid from the point of view of mathematics, the study of hypotheses about the existence of secrets and mysteries of the pyramid, as well as the possibilities of its application.

object study in this paper is a pyramid.

Thing research: features and properties of the pyramid.

Tasks research:

To study scientific - popular literature on the research topic.

Consider the pyramid as a geometric body.

Determine the properties and features of the pyramid.

Find material confirming the application of the properties of the pyramid in various fields of science and technology.

Methods research: analysis, synthesis, analogy, mental modeling.

Expected result of the work should be structured information about the pyramid, its properties and applications.

Stages of project preparation:

Determining the theme of the project, goals and objectives.

Studying and collecting material.

Drawing up a project plan.

Formulation of the expected result of the activity on the project, including the assimilation of new material, the formation of knowledge, skills and abilities in the subject activity.

Formulation of research results.

Reflection

Pyramid as a geometric body

Consider the origins of the word and term " pyramid". It is immediately worth noting that the "pyramid" or " pyramid"(English), " pyramide"(French, Spanish and Slavic languages), pyramide(German) is a Western term with its origins in ancient Greece. In ancient Greek πύραμίς ("P iramis"and many others. h. Πύραμίδες « pyramides"") has several meanings. The ancient Greeks called pyramis» a wheat cake that resembled the shape of Egyptian structures. Later, the word came to mean "a monumental structure with a square area at the base and with sloping sides meeting at the top. The etymological dictionary indicates that the Greek "pyramis" comes from the Egyptian " pimar". The first written interpretation of the word "pyramid" found in Europe in 1555 and means: "one of the types of ancient buildings of kings." After the discovery of the pyramids in Mexico and with the development of science in the 18th century, the pyramid became not just an ancient monument of architecture, but also a regular geometric figure with four symmetrical sides (1716). The beginning of the geometry of the pyramid was laid in ancient Egypt and Babylon, but it was actively developed in ancient Greece. The first to establish what the volume of the pyramid is equal to was Democritus, and Eudoxus of Cnidus proved it.

The first definition belongs to the ancient Greek mathematician, the author of theoretical treatises on mathematics that have come down to us, Euclid. In the XII volume of his "Beginnings" he defines the pyramid as a bodily figure, bounded by planes that from one plane (base) converge at one point (top). But this definition has been criticized already in antiquity. So Heron proposed the following definition of a pyramid: "This is a figure bounded by triangles converging at one point and the base of which is a polygon."

There is a definition of the French mathematician Adrien Marie Legendre, who in 1794 in his work “Elements of Geometry” defines the pyramid as follows: “A pyramid is a bodily figure formed by triangles converging at one point and ending on different sides of a flat base.”

Modern dictionaries interpret the term "pyramid" as follows:

Analyzing the definitions of the pyramid, we can conclude that all sources have similar formulations:

A pyramid is a polyhedron whose base is a polygon, and the remaining faces are triangles that have a common vertex. According to the number of corners of the base, pyramids are triangular, quadrangular, etc.

The polygon A 1 A 2 A 3 ... An is the base of the pyramid, and the triangles RA 1 A 2, RA 2 A 3, ..., PAnA 1 are the side faces of the pyramid, P is the top of the pyramid, the segments RA 1, RA 2, ..., PAn - side ribs.

The perpendicular drawn from the top of the pyramid to the plane of the base is called h pyramids.

In addition to an arbitrary pyramid, there is a regular pyramid, at the base of which there is a regular polygon and a truncated pyramid.

area The total surface of a pyramid is the sum of the areas of all its faces. Sfull = S side + S main, where S side is the sum of the areas of the side faces.

Volume pyramid is found by the formula: V=1/3S main.h, where S main. - base area, h - height.

To pyramid properties relate:

When all lateral edges are of the same size, then it is easy to describe a circle near the base of the pyramid, while the top of the pyramid will be projected into the center of this circle; side ribs form the same angles with the base plane; in addition, the converse is also true, i.e. when the side edges form equal angles with the base plane, or when a circle can be described near the base of the pyramid and the top of the pyramid will be projected into the center of this circle, then all the side edges of the pyramid have the same size.

When the side faces have an angle of inclination to the plane of the base of the same value, then it is easy to describe a circle near the base of the pyramid, while the top of the pyramid will be projected into the center of this circle; the heights of the side faces are of equal length; the area of ​​the lateral surface is equal to half the product of the perimeter of the base and the height of the lateral face.

The pyramid is called correct, if its base is a regular polygon, and the vertex is projected into the center of the base. The side faces of a regular pyramid are equal, isosceles triangles (Fig. 2a). axis A regular pyramid is called a straight line containing its height. Apothem - the height of the side face of a regular pyramid, drawn from its top.

Square side face of a regular pyramid is expressed as follows: Sside. \u003d 1 / 2P h, where P is the perimeter of the base, h is the height of the side face (the apothem of a regular pyramid). If the pyramid is crossed by a plane A'B'C'D' parallel to the base, then the side edges and height are divided by this plane into proportional parts; in section, a polygon A'B'C'D' is obtained, similar to the base; the areas of the section and the base are related as the squares of their distances from the top.

Truncated pyramid is obtained by cutting off from the pyramid its upper part by a plane parallel to the base (Fig. 2b). The bases of the truncated pyramid are similar polygons ABCD and A`B`C`D`, the side faces are trapezoids. The height of a truncated pyramid is the distance between the bases. The volume of a truncated pyramid is found by the formula: V=1/3 h (S + + S’), where S and S’ are the areas of the bases ABCD and A’B’C’D’, h is the height.

The bases of a regular truncated n-gonal pyramid are regular n-gons. The area of ​​the lateral surface of a regular truncated pyramid is expressed as follows: Sside. \u003d ½ (P + P ') h, where P and P' are the perimeters of the bases, h is the height of the side face (the apothem of a regular truncated pyramid)

Sections of the pyramid by planes passing through its top are triangles. A section passing through two non-neighboring side edges of a pyramid is called a diagonal section. If the section passes through a point on the side edge and the side of the base, then this side will be its trace on the plane of the base of the pyramid. A section passing through a point lying on the face of the pyramid and a given trace of the section on the plane of the base, then the construction should be carried out as follows: find the intersection point of the plane of the given face and the trace of the section of the pyramid and designate it; build a straight line passing through a given point and the resulting intersection point; Repeat these steps for the next faces.

Rectangular pyramid - it is a pyramid in which one of the side edges is perpendicular to the base. In this case, this edge will be the height of the pyramid (Fig. 2c).

Regular triangular pyramid- This is a pyramid, the base of which is a regular triangle, and the top is projected into the center of the base. A special case of a regular triangular pyramid is tetrahedron. (Fig. 2a)

Let's consider the theorems connecting the pyramid with other geometric bodies.

Sphere

A sphere can be described near the pyramid when at the base of the pyramid lies a polygon around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the point of intersection of the planes passing through the midpoints of the edges of the pyramid perpendicular to them. It follows from this theorem that a sphere can be described both about any triangular and about any regular pyramid; A sphere can be inscribed in a pyramid when the bisector planes of the internal dihedral angles of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.

Cone

A cone is called inscribed in a pyramid if their vertices coincide and its base is inscribed in the base of the pyramid. Moreover, it is possible to inscribe a cone into a pyramid only when the apothems of the pyramid are equal to each other (a necessary and sufficient condition); A cone is called inscribed near the pyramid when their vertices coincide and its base is inscribed near the base of the pyramid. Moreover, it is possible to describe the cone near the pyramid only when all the side edges of the pyramid are equal to each other (a necessary and sufficient condition); The heights of such cones and pyramids are equal to each other.

Cylinder

A cylinder is called inscribed in a pyramid if one of its bases coincides with a circle inscribed in the section of the pyramid by a plane parallel to the base, and the other base belongs to the base of the pyramid. A cylinder is called inscribed near the pyramid if the top of the pyramid belongs to one of its bases, and its other base is inscribed near the base of the pyramid. Moreover, it is possible to describe a cylinder near the pyramid only when there is an inscribed polygon at the base of the pyramid (a necessary and sufficient condition).

Very often in their research, scientists use the properties of the pyramid with the proportions of the Golden Ratio. We will consider how the golden section ratios were used when building the pyramids in the next paragraph, and here we will dwell on the definition of the golden section.

The mathematical encyclopedic dictionary gives the following definition golden section- this is the division of the segment AB into two parts in such a way that most of its AC is the average proportional between the entire segment AB and its smaller part CB.

The algebraic finding of the Golden section of the segment AB = a is reduced to solving the equation a: x = x: (a-x), whence x is approximately equal to 0.62a. The ratio x can be expressed as fractions n/n+1= 0,618, where n is the Fibonacci number numbered n.

The golden ratio is often used in works of art, architecture, and is found in nature. Vivid examples are the sculpture of Apollo Belvedere, the Parthenon. During the construction of the Parthenon, the ratio of the height of the building to its length was used and this ratio is 0.618. Objects around us also provide examples of the Golden Ratio, for example, the bindings of many books also have a width to length ratio close to 0.618.

Thus, having studied popular scientific literature on the research problem, we came to the conclusion that a pyramid is a polyhedron, the base of which is a polygon, and the remaining faces are triangles with a common vertex. We examined the elements and properties of the pyramid, its types and correlation with the proportions of the Golden Section.

2. Features of the pyramid

So in the Big Encyclopedic Dictionary it is written that a pyramid is a monumental structure that has the geometric shape of a pyramid (sometimes stepped or tower-shaped). The tombs of the ancient Egyptian pharaohs of the 3rd - 2nd millennium BC were called pyramids. e., as well as the pedestals of temples in Central and South America, associated with cosmological cults. Among the grandiose pyramids of Egypt, the Great Pyramid of Pharaoh Cheops occupies a special place. Before proceeding to the analysis of the shape and size of the pyramid of Cheops, we should remember what system of measures the Egyptians used. The Egyptians had three units of length: "cubit" (466 mm), equal to seven "palms" (66.5 mm), which, in turn, was equal to four "fingers" (16.6 mm).

Most researchers agree that the length of the side of the base of the pyramid, for example, GF, is L = 233.16 m. This value corresponds almost exactly to 500 "cubits". Full compliance with 500 "cubits" will be if the length of the "cubit" is considered equal to 0.4663 m.

The height of the pyramid (H) is estimated by researchers differently from 146.6 to 148.2 m. And depending on the accepted height of the pyramid, all the ratios of its geometric elements change. What is the reason for the differences in the estimate of the height of the pyramid? The fact is that the pyramid of Cheops is truncated. Its upper platform today has a size of approximately 10x10 m, and a century ago it was 6x6 m. It is obvious that the top of the pyramid was dismantled, and it does not correspond to the original one. Estimating the height of the pyramid, it is necessary to take into account such a physical factor as the settlement of the structure. For a long time, under the influence of colossal pressure (reaching 500 tons per 1 m 2 of the lower surface), the height of the pyramid decreased compared to its original height. The original height of the pyramid can be recreated if you find the basic geometric idea.

In 1837, the English Colonel G. Wise measured the angle of inclination of the faces of the pyramid: it turned out to be equal to a = 51 ° 51 ". This value is still recognized by most researchers today. The indicated value of the angle corresponds to the tangent (tg a), equal to 1.27306. This value corresponds to the ratio of the height of the pyramid AC to half of its base CB, that is, AC / CB = H / (L / 2) = 2H / L.

And here the researchers were in for a big surprise! The fact is that if we take the square root of the golden ratio, then we get the following result = 1.272. Comparing this value with the value tg a = 1.27306, we see that these values ​​are very close to each other. If we take the angle a \u003d 51 ° 50 ", that is, reduce it by only one arc minute, then the value of a will become equal to 1.272, that is, it will coincide with the value. It should be noted that in 1840 G. Wise repeated his measurements and clarified that the value of the angle a \u003d 51 ° 50 ".

These measurements led the researchers to the following interesting hypothesis: the triangle ASV of the Cheops pyramid was based on the ratio AC / CB = 1.272.

Consider now a right triangle ABC, in which the ratio of the legs AC / CB = . If we now denote the lengths of the sides of the rectangle ABC as x, y, z, and also take into account that the ratio y / x \u003d, then in accordance with the Pythagorean theorem, the length z can be calculated by the formula:

If we accept x = 1, y = , then:

A right triangle in which the sides are related as t::1 is called a "golden" right triangle.

Then, if we take as a basis the hypothesis that the main “geometric idea” of the Cheops pyramid is the “golden” right-angled triangle, then from here it is easy to calculate the “design” height of the Cheops pyramid. It is equal to:

H \u003d (L / 2) / \u003d 148.28 m.

Let us now derive some other relations for the pyramid of Cheops, which follow from the "golden" hypothesis. In particular, we find the ratio of the outer area of ​​the pyramid to the area of ​​its base. To do this, we take the length of the leg CB as a unit, that is: CB = 1. But then the length of the side of the base of the pyramid is GF = 2, and the base area EFGH will be equal to S EFGH = 4.

Let us now calculate the area of ​​the side face of the Cheops pyramid S D . Since the height AB of triangle AEF is equal to t, then the area of ​​the side face will be equal to S D = t. Then the total area of ​​all four side faces of the pyramid will be equal to 4t, and the ratio of the total external area of ​​the pyramid to the area of ​​​​the base will be equal to the golden ratio. This is the main geometric secret of the Cheops pyramid.

And also, during the construction of the Egyptian pyramids, it was found that the square built at the height of the pyramid is exactly equal to the area of ​​\u200b\u200beach of the side triangles. This is confirmed by the latest measurements.

We know that the ratio between the circumference of a circle and its diameter is a constant value, well known to modern mathematicians, schoolchildren - this is the number "Pi" = 3.1416 ... But if we add the four sides of the base of the Cheops pyramid, we get 931.22 m. Dividing this is the number twice the height of the pyramid (2x148.208), we get 3.1416 ..., that is, the number "Pi". Consequently, the pyramid of Cheops is a one-of-a-kind monument, which is the material embodiment of the number "Pi", which plays an important role in mathematics.

Thus, the presence in the size of the pyramid of the golden section - the ratio of the doubled side of the pyramid to its height - is a number very close in value to the number π. This, of course, is also a feature. Although many authors believe that this coincidence is accidental, since the fraction 14/11 is "a good approximation for the square root of the ratio of the golden ratio, and for the ratio of the areas of a square and a circle inscribed in it."

However, it is wrong to speak here only of the Egyptian pyramids. There are not only Egyptian pyramids, there is a whole network of pyramids on Earth. The main monuments (Egyptian and Mexican pyramids, Easter Island and the Stonehenge complex in England) at first glance are randomly scattered around our planet. But if the study includes the Tibetan pyramid complex, then a strict mathematical system of their location on the surface of the Earth appears. Against the backdrop of the Himalayan ridge, a pyramidal formation is clearly distinguished - Mount Kailash. The location of the city of Kailash, the Egyptian and Mexican pyramids is very interesting, namely, if you connect the city of Kailash with the Mexican pyramids, then the line connecting them goes to Easter Island. If you connect the city of Kailash with the Egyptian pyramids, then the line of their connection again goes to Easter Island. Exactly one-fourth of the globe has been outlined. If we connect the Mexican pyramids and the Egyptian ones, then we will see two equal triangles. If you find their area, then their sum is equal to one-fourth of the area of ​​the globe.

An indisputable connection between the complex of Tibetan pyramids was revealed with other structures antiquity - the Egyptian and Mexican pyramids, the colossi of Easter Island and the Stonehenge complex in England. The height of the main pyramid of Tibet - Mount Kailash - is 6714 meters. The distance from Kailash to the North Pole is 6714 kilometers, the distance from Kailash to Stonehenge is 6714 kilometers. If you put aside on the globe from the North Pole these 6714 kilometers, then we will get to the so-called Devil's Tower, which looks like a truncated pyramid. And finally exactly 6714 kilometers from Stonehenge to the Bermuda Triangle.

As a result of these studies, it can be concluded that there is a pyramidal-geographical system on Earth.

Thus, the features are the ratio of the total external area of ​​the pyramid to the area of ​​​​the base will be equal to the golden ratio; the presence in the size of the pyramid of the golden section - the ratio of the double side of the pyramid to its height - is a number very close in value to the number π, i.e. the pyramid of Cheops is a one-of-a-kind monument, which is the material embodiment of the number "Pi"; the existence of a pyramidal-geographical system.

3. Other properties and uses of the pyramid.

Consider the practical application of this geometric figure. For example, hologram. First, let's look at what holography is. Holography - a set of technologies for accurately recording, reproducing and reshaping the wave fields of optical electromagnetic radiation, a special photographic method in which, using a laser, images of three-dimensional objects are recorded and then restored to the highest degree similar to real ones. A hologram is a product of holography, a three-dimensional image created by a laser that reproduces an image of a three-dimensional object. Using a regular truncated tetrahedral pyramid, you can recreate an image - a hologram. A photo file and a regular truncated tetrahedral pyramid from a translucent material are created. A small indent is made from the bottommost pixel and the middle pixel relative to the y-axis. This point will be the midpoint of the side of the square formed by the section. The photo is multiplied, and its copies are located in the same way relative to the other three sides. A pyramid is placed on the square with a section down so that it coincides with the square. The monitor generates a light wave, each of the four identical photographs, being in a plane that is a projection of the face of the pyramid, falls on the face itself. As a result, on each of the four faces we have the same images, and since the material from which the pyramid is made has the property of transparency, the waves seem to be refracted, meeting in the center. As a result, we get the same interference pattern of a standing wave, the central axis, or the axis of rotation of which is the height of a regular truncated pyramid. This method also works with the video image, since the principle of operation remains unchanged.

Considering particular cases, one can see that the pyramid is widely used in everyday life, even in the household. The pyramidal shape is often found, primarily in nature: plants, crystals, the methane molecule has the shape of a regular triangular pyramid - a tetrahedron, the unit cell of a diamond crystal is also a tetrahedron, in the center and four vertices of which are carbon atoms. Pyramids are found at home, children's toys. Buttons, computer keyboards are often similar to a quadrangular truncated pyramid. They can be seen in the form of building elements or architectural structures themselves, as translucent roof structures.

Consider some more examples of the use of the term "pyramid"

Ecological pyramids- these are graphical models (usually in the form of triangles) that reflect the number of individuals (pyramid of numbers), the amount of their biomass (biomass pyramid) or the energy contained in them (energy pyramid) at each trophic level and indicate a decrease in all indicators with an increase in trophic level

Information pyramid. It reflects the hierarchy of different types of information. The provision of information is built according to the following pyramidal scheme: at the top - the main indicators by which you can unambiguously track the pace of the enterprise's movement towards the chosen goal. If something is wrong, then you can go to the middle level of the pyramid - generalized data. They clarify the picture for each indicator individually or in relation to each other. From this data, you can determine the possible location of the failure or problem. For more complete information, you need to refer to the base of the pyramid - a detailed description of the state of all processes in numerical form. This data helps to identify the cause of the problem so that it can be corrected and avoided in the future.

Bloom's taxonomy. Bloom's taxonomy proposes a classification of tasks in the form of a pyramid, set by educators to students, and, accordingly, learning goals. She divides educational goals into three areas: cognitive, affective and psychomotor. Within each individual sphere, in order to move to a higher level, the experience of previous levels, distinguished in this sphere, is necessary.

Financial pyramid- a specific phenomenon of economic development. The name "pyramid" clearly illustrates the situation when people "at the bottom" of the pyramid give money to a small top. At the same time, each new participant pays to increase the possibility of his promotion to the top of the pyramid.

Pyramid of Needs Maslow reflects one of the most popular and well-known theories of motivation - the theory of hierarchy. needs. Maslow distributed the needs as they increase, explaining this construction by the fact that a person cannot experience high-level needs while he needs more primitive things. As the lower needs are satisfied, the needs of a higher level become more and more urgent, but this does not mean at all that the place of the previous need is occupied by a new one only when the former is fully satisfied.

Another example of the use of the term "pyramid" is food pyramid - a schematic representation of the principles of healthy eating developed by nutritionists. Foods at the bottom of the pyramid should be eaten as often as possible, while foods at the top of the pyramid should be avoided or eaten in limited quantities.

Thus, all of the above shows the variety of uses of the pyramid in our lives. Perhaps the pyramid has a much higher purpose, and is intended for something more than the practical uses that are now open.

Conclusion

We constantly meet pyramids in our life - these are ancient Egyptian pyramids and toys that children play with; objects of architecture and design, natural crystals; viruses that can only be seen with an electron microscope. Over the many millennia of its existence, the pyramids have become a kind of symbol that personifies the desire of man to reach the pinnacle of knowledge.

In the course of the study, we determined that the pyramids are a fairly common phenomenon throughout the globe.

We studied popular science literature on the topic of research, examined various interpretations of the term "pyramid", determined that in the geometric sense, a pyramid is a polyhedron, the base of which is a polygon, and the remaining faces are triangles with a common vertex. We studied the types of pyramids (regular, truncated, rectangular), elements (apothem, side faces, side edges, top, height, base, diagonal section) and the properties of geometric pyramids with equal side edges and when the side faces are tilted to the base plane at one angle. Considered the theorems connecting the pyramid with other geometric bodies (sphere, cone, cylinder).

The features of the pyramid are:

the ratio of the total external area of ​​the pyramid to the area of ​​​​the base will be equal to the golden ratio;

the presence in the size of the pyramid of the golden section - the ratio of the double side of the pyramid to its height - is a number very close in value to the number π, i.e. the pyramid of Cheops is a one-of-a-kind monument, which is the material embodiment of the number "Pi";

the existence of a pyramidal-geographical system.

We studied the modern application of this geometric figure. We examined how the pyramid and the hologram are connected, drew attention to the fact that the pyramidal form is most often found in nature (plants, crystals, methane molecules, the structure of the diamond lattice, etc.). Throughout the study, we met with material confirming the use of the properties of the pyramid in various fields of science and technology, in the everyday life of people, in the analysis of information, in the economy, and in many other areas. And they came to the conclusion that perhaps the pyramids have a much higher purpose, and are intended for something more than the practical uses for them that are now open.

Bibliography.

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Voloshinov A. V. Mathematics and Art. [Text] / A.V. Voloshinov - Moscow: "Enlightenment" 2000.

World History (encyclopedia for children). [Text] / - M .: “Avanta +”, 1993.

hologram . [Electronic resource] - https://hi-news.ru/tag/hologramma - article on the Internet

Geometry [Text]: Proc. 10 - 11 cells. for educational institutions L. S. Atanasyan, V. F. Butuzov and others - 22nd edition. - M.: Enlightenment, 2013

Coppens F. New era of pyramids. [Text] / F. Coppens - Smolensk: Rusich, 2010

Mathematical Encyclopedic Dictionary. [Text] / A. M. Prokhorov and others - M .: Soviet Encyclopedia, 1988.

Muldashev E.R. The world system of pyramids and monuments of antiquity saved us from the end of the world, but ... [Text] / E.R. Muldashev - M .: "AiF-Print"; M.: "OLMA-PRESS"; St. Petersburg: Neva Publishing House; 2003.

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Terra Lexicon. Illustrated encyclopedic dictionary. [Text] / - M.: TERRA, 1998.

Tompkins P. Secrets of the Great Pyramid of Cheops. [Text]/ Peter Tompkins. - M.: "Tsentropoligraf", 2008

Uvarov V. The magical properties of the pyramids. [Text] / V. Uvarov - Lenizdat, 2006.

Sharygin I.F. Geometry grade 10-11. [Text] / I.F. Sharygin:. - M: "Enlightenment", 2000

Yakovenko M. The key to understanding the pyramid. [Electronic resource] - http://world-pyramids.com/russia/pyramid.html - article on the Internet

In the case of the pyramids, as is often the case, experience precedes scientific reasoning. Currently, there are many recorded phenomena and phenomena regarding the properties of the pyramids. Science is not yet able to explain them all (at least this information is not widely available). However, the lack of a scientific explanation does not prevent individuals and groups from using some of the well-known properties of the pyramids for the benefit of their lives. So what are these properties?

The energies inside the pyramid change the internal structure of the objects in it. The following phenomena have been noted:

● mummification (dehydration and sterilization),

● regeneration of damaged tissues,

● structurization of water (does not freeze at negative temperatures),

● food products placed at the level of 1/3 of the height from the base improve their palatability and increase their shelf life several times (in Bulgaria, pyramidal vegetable stores have been used for many years),

● germination of seeds improves (if seeds are kept before planting in a pyramid at the level of 1/3 of the height from the base for 10-15 days, germination and yield increase by about 2 times).

● blunt razor blades and knives located in the pyramid at the level of 1/3 of the height from the base are sharpened within 24 hours (the patented discovery of Karel Drbal - Karel Drbal).

● If a generator is placed in a large pyramid at a height of 1/3 to 1/2, then the pyramid will be able to generate electrical energy. (The generator is made from aluminum sheetsor copper. The assembled generator is connected to an alkaline battery. In the manufacture of such a generator, it should be borne in mind that the more plates it has, the more voltage it can give, and the larger the surface of the plates, the more current the generator can produce).

Effects on the human body (as a result of drinking water and food aged in the pyramid, or resting in the large or above the small pyramids):

● Relieves stress at the physical and mental levels.

● There is a noticeable effect on the parasympathetic nervous system (the pulse and pressure indicators decrease, stabilize).

● Gives a general healing effect, enhances immunity, vitality.

● Improved blood counts (increase in hemoglobin, decrease in ESR, decrease in leukocytes).

● The pain syndrome is reduced.

● Increases efficiency, improves sleep.

● Decreases susceptibility to stress.

The pyramid affects its environment:

● reduces the level of radiation;

● changes the ionization level from positive to negative;

● reflects the flow of electromagnetic radiation of technical and natural origin

● neutralize harmful radiation of pathogenic zones. With the help of the pyramids, by selecting their height and relative position, it is possible to neutralize or reduce to a safe value for humans, the dangerous influence of geopathic zones, both natural and man-made. It has been established what pyramids defined a lot size able to cancel the effect geopathogenic points of low and medium strength, since the pyramidal field shifts the linesHartman-Curry , modifying and muting them ();

● improves the ecological situation: water bodies, air, etc. are cleaned;

Features of the shape, size and material of pyramids with the properties described above:

The pyramid must be regular (the base is a regular polygon (with equal sides), and the vertex is projected into the center of the base).

Depending on the geometric parameters, material and dimensions of the pyramids, their properties will vary to varying degrees. Polyhedra with the ratio of the lengths of the edges of the Cheops pyramid are very effective: simplified - if the side of the square at the base of the pyramid is equal to one, then the height is 0.63, and the side edge is about 0.95.

The inner space of the pyramids at the level from 1/3 to 2/3 of the height of the pyramid from the base (the Bovi-Drbal zone) has the maximum energy potential.

With doubling the height of the pyramid, the activity of its action increases significantly. This is evidenced by the experiments of A.E. Hunger. A distinctive feature of the Hunger pyramids is that in them the proportion of the golden section is applied to the ratio of the diameters of neighboring balls, sequentially inscribed in a regular tetrahedral pyramid. When this condition is met, the ratio of the height of the pyramid to the side of the square lying at its base is 2.05817 ..., and the angle between the faces of the pyramid is 27.3 °:

The pyramid shows its properties provided that the sides of its base are clearly oriented to the cardinal points (with an accuracy of 2-3 degrees).

When manufacturing a pyramid from dielectric materials, the use of metal joints (using nails, self-tapping screws, staples, etc.) is unacceptable. Adhesive bonding is preferred.

Experiments of thermal physicist A.I. Veinik show that pyramids can be monolithic or hollow, made, for example, from paper, cardboard, plastic, metal, etc. You can also do without faces at all, it is enough to reproduce only edges from wire or rods.

The force of the influence of the pyramids increases with time and tends to accumulate (if the pyramid is not moved).

● personalpyramid

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