Here are collected basic information about the pyramids and related formulas and concepts. All of them are studied with a tutor in mathematics in preparation for the exam.

Consider a plane, a polygon lying in it and a point S not lying in it. Connect S to all vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called lateral edges. The polygon is called the base, and the point S is called the top of the pyramid. Depending on the number n, the pyramid is called triangular (n=3), quadrangular (n=4), pentagonal (n=5) and so on. Alternative name for the triangular pyramid - tetrahedron. The height of a pyramid is the perpendicular drawn from its apex to the base plane.

A pyramid is called correct if a regular polygon, and the base of the height of the pyramid (the base of the perpendicular) is its center.

Tutor's comment:
Do not confuse the concept of "regular pyramid" and "regular tetrahedron". In a regular pyramid, the side edges are not necessarily equal to the edges of the base, but in a regular tetrahedron, all 6 edges of the edges are equal. This is his definition. It is easy to prove that the equality implies that the center P of the polygon with a height base, so a regular tetrahedron is a regular pyramid.

What is an apothem?
The apothem of a pyramid is the height of its side face. If the pyramid is regular, then all its apothems are equal. The reverse is not true.

Mathematics tutor about his terminology: work with pyramids is 80% built through two types of triangles:
1) Containing apothem SK and height SP
2) Containing the lateral edge SA and its projection PA

To simplify references to these triangles, it is more convenient for a math tutor to name the first of them apothemic, and second costal. Unfortunately, you will not find this terminology in any of the textbooks, and the teacher has to introduce it unilaterally.

Pyramid volume formula:
1) , where is the area of the base of the pyramid, and is the height of the pyramid
2) , where is the radius of the inscribed sphere, and is the area full surface pyramids.
3) , where MN is the distance of any two crossing edges, and is the area of the parallelogram formed by the midpoints of the four remaining edges.

Pyramid Height Base Property:

Point P (see figure) coincides with the center of the inscribed circle at the base of the pyramid if one of the following conditions is met:
1) All apothems are equal
2) All side faces are equally inclined towards the base
3) All apothems are equally inclined to the height of the pyramid
4) The height of the pyramid is equally inclined to all side faces

Math tutor's commentary: note that all items are united by one common property: one way or another, side faces participate everywhere (apothems are their elements). Therefore, the tutor can offer a less precise, but more convenient formulation for memorization: the point P coincides with the center of the inscribed circle, the base of the pyramid, if there is any equal information about its lateral faces. To prove it, it suffices to show that all apothemical triangles are equal.

The point P coincides with the center of the circumscribed circle near the base of the pyramid, if one of the three conditions is true:
1) All side edges are equal
2) All side ribs are equally inclined towards the base
3) All side ribs are equally inclined to the height

Definition

Pyramid is a polyhedron composed of a polygon $A_1A_2...A_n$ and $n$ triangles with a common vertex $P$ (not lying in the plane of the polygon) and opposite sides coinciding with the sides of the polygon.
Designation: $PA_1A_2...A_n$ .
Example: pentagonal pyramid $PA_1A_2A_3A_4A_5$ .

Triangles $PA_1A_2, \ PA_2A_3$ etc. called side faces pyramids, segments $PA_1, PA_2$, etc. - side ribs, polygon $A_1A_2A_3A_4A_5$ – basis, point $P$ – summit.

Height Pyramids are a perpendicular dropped from the top of the pyramid to the plane of the base.

A pyramid with a triangle at its base is called tetrahedron.

The pyramid is called correct, if its base is a regular polygon and one of the following conditions is met:

$(a)$ side edges of the pyramid are equal;

$(b)$ the height of the pyramid passes through the center of the circumscribed circle near the base;

$(c)$ side ribs are inclined to the base plane at the same angle.

$(d)$ side faces are inclined to the base plane at the same angle.

regular tetrahedron is a triangular pyramid, all the faces of which are equal equilateral triangles.

Theorem

The conditions $(a), (b), (c), (d)$ are equivalent.

Proof

Draw the height of the pyramid $PH$ . Let $\alpha$ be the plane of the base of the pyramid.

1) Let us prove that $(a)$ implies $(b)$ . Let $PA_1=PA_2=PA_3=...=PA_n$ .

Because $PH\perp \alpha$ , then $PH$ is perpendicular to any line lying in this plane, so the triangles are right-angled. So these triangles are equal in common leg $PH$ and hypotenuse $PA_1=PA_2=PA_3=...=PA_n$ . So $A_1H=A_2H=...=A_nH$ . This means that the points $A_1, A_2, ..., A_n$ are at the same distance from the point $H$ , therefore, they lie on the same circle with radius $A_1H$ . This circle, by definition, is circumscribed about the polygon $A_1A_2...A_n$ .

2) Let us prove that $(b)$ implies $(c)$ .

$PA_1H, PA_2H, PA_3H,..., PA_nH$ rectangular and equal in two legs. Hence, their angles are also equal, therefore, $\angle PA_1H=\angle PA_2H=...=\angle PA_nH$.

3) Let us prove that $(c)$ implies $(a)$ .

Similar to the first point, triangles $PA_1H, PA_2H, PA_3H,..., PA_nH$ rectangular and along the leg and sharp corner. This means that their hypotenuses are also equal, that is, $PA_1=PA_2=PA_3=...=PA_n$ .

4) Let us prove that $(b)$ implies $(d)$ .

Because in a regular polygon, the centers of the circumscribed and inscribed circles coincide (generally speaking, this point is called the center of a regular polygon), then $H$ is the center of the inscribed circle. Let's draw perpendiculars from the point $H$ to the sides of the base: $HK_1, HK_2$, etc. These are the radii of the inscribed circle (by definition). Then, according to the TTP, ($PH$ is a perpendicular to the plane, $HK_1, HK_2$, etc. are projections perpendicular to the sides) oblique $PK_1, PK_2$, etc. perpendicular to the sides $A_1A_2, A_2A_3$, etc. respectively. So, by definition $\angle PK_1H, \angle PK_2H$ equal to the angles between the side faces and the base. Because triangles $PK_1H, PK_2H, ...$ are equal (as right-angled on two legs), then the angles $\angle PK_1H, \angle PK_2H, ...$ are equal.

5) Let us prove that $(d)$ implies $(b)$ .

Similarly to the fourth point, the triangles $PK_1H, PK_2H, ...$ are equal (as rectangular along the leg and acute angle), which means that the segments $HK_1=HK_2=...=HK_n$ are equal. Hence, by definition, $H$ is the center of a circle inscribed in the base. But since for regular polygons, the centers of the inscribed and circumscribed circles coincide, then $H$ is the center of the circumscribed circle. Chtd.

Consequence

The side faces of a regular pyramid are equal isosceles triangles.

Definition

The height of the side face of a regular pyramid, drawn from its top, is called apothema.
The apothems of all lateral faces of a regular pyramid are equal to each other and are also medians and bisectors.

Important Notes

1. The height of a regular triangular pyramid falls to the intersection point of the heights (or bisectors, or medians) of the base (the base is a regular triangle).

2. The height of a regular quadrangular pyramid falls to the point of intersection of the diagonals of the base (the base is a square).

3. The height of a regular hexagonal pyramid falls to the point of intersection of the diagonals of the base (the base is a regular hexagon).

4. The height of the pyramid is perpendicular to any straight line lying at the base.

Definition

The pyramid is called rectangular if one of its lateral edges is perpendicular to the plane of the base.

Important Notes

1. Do rectangular pyramid the edge perpendicular to the base is the height of the pyramid. That is, $SR$ is the height.

2. Because $SR$ perpendicular to any line from the base, then $\triangle SRM, \triangle SRP$ are right triangles.

3. Triangles $\triangle SRN, \triangle SRK$ are also rectangular.
That is, any triangle formed by this edge and the diagonal coming out of the vertex of this edge, which lies at the base, will be right-angled.

\[(\Large(\text(Volume and surface area of the pyramid)))\]

Theorem

The volume of a pyramid is equal to one third of the product of the area of the base and the height of the pyramid: \

Consequences

Let $a$ be the side of the base, $h$ be the height of the pyramid.

1. The volume of a regular triangular pyramid is $V_(\text(right triangle pyr.))=\dfrac(\sqrt3)(12)a^2h$,

2. The volume of a regular quadrangular pyramid is $V_(\text(right.four.pyre.))=\dfrac13a^2h$.

3. The volume of a regular hexagonal pyramid is $V_(\text(right.hex.pyr.))=\dfrac(\sqrt3)(2)a^2h$.

4. The volume of a regular tetrahedron is $V_(\text(right tetra.))=\dfrac(\sqrt3)(12)a^3$.

Theorem

The area of the lateral surface of a regular pyramid is equal to half the product of the perimeter of the base and the apothem.

\[(\Large(\text(Truncated pyramid)))\]

Definition

Consider an arbitrary pyramid $PA_1A_2A_3...A_n$ . Let us draw a plane parallel to the base of the pyramid through a certain point lying on the side edge of the pyramid. This plane will divide the pyramid into two polyhedra, one of which is a pyramid ($PB_1B_2...B_n$ ), and the other is called truncated pyramid($A_1A_2...A_nB_1B_2...B_n$ ).

The truncated pyramid has two bases - polygons $A_1A_2...A_n$ and $B_1B_2...B_n$ , which are similar to each other.

The height of a truncated pyramid is a perpendicular drawn from some point of the upper base to the plane of the lower base.

Important Notes

1. All side faces of a truncated pyramid are trapezoids.

2. The segment connecting the centers of the bases of a regular truncated pyramid (that is, a pyramid obtained by a section of a regular pyramid) is the height.

Pyramid Concept

Definition 1

A geometric figure formed by a polygon and a point that does not lie in the plane containing this polygon, connected to all the vertices of the polygon, is called a pyramid (Fig. 1).

The polygon from which the pyramid is composed is called the base of the pyramid, the triangles obtained by connecting with the point are the side faces of the pyramid, the sides of the triangles are the sides of the pyramid, and the point common to all triangles is the top of the pyramid.

Types of pyramids

Depending on the number of corners at the base of the pyramid, it can be called triangular, quadrangular, and so on (Fig. 2).

Figure 2.

Another type of pyramid is a regular pyramid.

Let us introduce and prove the property of a regular pyramid.

Theorem 1

All side faces of a regular pyramid are isosceles triangles that are equal to each other.

Proof.

Consider a regular $n-$gonal pyramid with vertex $S$ of height $h=SO$. Let's describe a circle around the base (Fig. 4).

Figure 4

Consider triangle $SOA$. By the Pythagorean theorem, we get

Obviously, any side edge will be defined in this way. Therefore, all side edges are equal to each other, that is, all side faces are isosceles triangles. Let us prove that they are equal to each other. Since the base is a regular polygon, the bases of all side faces are equal to each other. Consequently, all side faces are equal according to the III sign of equality of triangles.

The theorem has been proven.

We now introduce the following definition related to the concept of a regular pyramid.

Definition 3

The apothem of a regular pyramid is the height of its side face.

Obviously, by Theorem 1, all apothems are equal.

Theorem 2

The lateral surface area of a regular pyramid is defined as the product of the semi-perimeter of the base and the apothem.

Proof.

Let us denote the side of the base of the $n-$coal pyramid as $a$, and the apothem as $d$. Therefore, the area of the side face is equal to

Since, by Theorem 1, all sides are equal, then

The theorem has been proven.

Another type of pyramid is the truncated pyramid.

Definition 4

If a plane parallel to its base is drawn through an ordinary pyramid, then the figure formed between this plane and the plane of the base is called a truncated pyramid (Fig. 5).

Figure 5. Truncated pyramid

The lateral faces of the truncated pyramid are trapezoids.

Theorem 3

The area of the lateral surface of a regular truncated pyramid is defined as the product of the sum of the semiperimeters of the bases and the apothem.

Proof.

Let us denote the sides of the bases of the $n-$coal pyramid by $a\ and\ b$, respectively, and the apothem by $d$. Therefore, the area of the side face is equal to

Since all sides are equal, then

The theorem has been proven.

Task example

Example 1

Find the area of the lateral surface of a truncated triangular pyramid if it is obtained from a regular pyramid with base side 4 and apothem 5 by cutting off by a plane passing through the midline of the lateral faces.

Solution.

According to the median line theorem, we obtain that the upper base of the truncated pyramid is equal to $4\cdot \frac(1)(2)=2$, and the apothem is equal to $5\cdot \frac(1)(2)=2.5$.

Then, by Theorem 3, we obtain

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Introduction

When we meet the word "pyramid", then 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 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.).

That's why 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.

Item 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 that confirms the use of pyramid properties in various fields 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 "monumental structure with 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:

	A polyhedron whose base is a polygon and the other faces are triangles that have a common vertex	Explanatory dictionary of the Russian language, ed. D. N. Ushakova
	A body bounded by equal triangles, composed of vertices at one point and forming a square with their bases	Explanatory Dictionary of V.I.Dal
	A polyhedron whose base is a polygon and the remaining faces are triangles with a common vertex	Explanatory Dictionary, ed. S. I. Ozhegova and N. Yu. Shvedova
	A polyhedron whose base is a polygon and whose side faces are triangles that have a common vertex	T. F. Efremov. New explanatory and derivational dictionary of the Russian language.
	A polyhedron, one face of which is a polygon, and the other faces are triangles having a common vertex	Dictionary of foreign words
	A geometric body whose base is a polygon and whose sides are as many triangles as the base has sides whose vertices converge to one point.	Dictionary of foreign words of the Russian language
	A polyhedron, one face of which is some kind of flat polygon, and all other faces are triangles, the bases of which are the sides of the base of the triangle, and the vertices converge at one point	F. Brockhaus, I.A. Efron. encyclopedic Dictionary
	A polyhedron whose base is a polygon and the remaining faces are triangles that have a common vertex	Modern Dictionary
	A polyhedron, one of whose faces is a polygon and the other faces are triangles with a common vertex	Mathematical encyclopedic Dictionary

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 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 intersected by the plane A'B'C'D' parallel to the base, then side ribs and the height is 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 - 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 (the 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 science 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 the pyramid is a monumental structure with geometric shape pyramids (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. Behind 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 right triangle ABC, in which the ratio of 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, exactly equal to area each 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 square root from the ratio of the golden section, 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 the globe. If we connect the Mexican pyramids and the Egyptian ones, then we will see two equal triangle. 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. Distance from Kailash to North Pole equals 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 images of three-dimensional objects are recorded and then restored using a laser, in 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 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 various kinds 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 these data it is possible to determine possible place failure or problem. For more complete information you need to turn 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 Pyramide- specific event 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 in ascending order, explaining such a construction by the fact that a person cannot experience needs. high level while in need of 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 consumed 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 meant for something more than those practical ways its uses, which are now open.

Conclusion

We constantly meet with 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 considered in 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.

Van der Waerden, Barthel Leendert. Awakening Science. Mathematics ancient egypt, Babylon and Greece. [Text] / B. L. Van der Waerden - KomKniga, 2007

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

The 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: Publishing House"Neva"; 2003.

Perelman Ya. I. Entertaining arithmetic. [Text] / Ya. I. Perelman- M .: Tsentrpoligraf, 2017

Reichard G. Pyramids. [Text] / Hans Reichard - M .: Slovo, 1978

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

First level

Pyramid. visual guide (2019)

What is a pyramid?

How does she look?

You see: at the pyramid below (they say " at the base"") some polygon, and all the vertices of this polygon are connected to some point in space (this point is called " vertex»).

This whole structure has side faces, side ribs And base ribs. Once again, let's draw a pyramid along with all these names:

Some pyramids may look very strange, but they are still pyramids.

Here, for example, quite "oblique" pyramid.

And a little more about the names: if there is a triangle at the base of the pyramid, then the pyramid is called triangular;

At the same time, the point where it fell height, is called height base. Note that in the "crooked" pyramids height may even be outside the pyramid. Like this:

And there is nothing terrible in this. It looks like an obtuse triangle.

Correct pyramid.

A lot of complex words? Let's decipher: " At the base - correct"- this is understandable. And now remember that a regular polygon has a center - a point that is the center of and , and .

Well, and the words “the top is projected into the center of the base” mean that the base of the height falls exactly into the center of the base. Look how smooth and cute it looks right pyramid.

Hexagonal: at the base - a regular hexagon, the vertex is projected into the center of the base.

quadrangular: at the base - a square, the top is projected to the intersection point of the diagonals of this square.

triangular: at the base is a regular triangle, the vertex is projected to the intersection point of the heights (they are also medians and bisectors) of this triangle.

Very important properties of a regular pyramid:

IN right pyramid

all side edges are equal.
all side faces are isosceles triangles and all these triangles are equal.

Pyramid Volume

The main formula for the volume of the pyramid:

Where did it come from exactly? This is not so simple, and at first you just need to remember that the pyramid and cone have volume in the formula, but the cylinder does not.

Now let's calculate the volume of the most popular pyramids.

Let the side of the base be equal, and the side edge equal. I need to find and.

This is the area of a right triangle.

Let's remember how to search for this area. We use the area formula:

We have "" - this, and "" - this too, eh.

Now let's find.

According to the Pythagorean theorem for

What does it matter? This is the radius of the circumscribed circle in, because pyramidcorrect and hence the center.

Since - the point of intersection and the median too.

(Pythagorean theorem for)

Substitute in the formula for.

Let's plug everything into the volume formula:

Attention: if you have a regular tetrahedron (i.e.), then the formula is:

Let the side of the base be equal, and the side edge equal.

There is no need to search here; because at the base is a square, and therefore.

Let's find. According to the Pythagorean theorem for

Do we know? Almost. Look:

(we saw this by reviewing).

Substitute in the formula for:

And now we substitute and into the volume formula.

Let the side of the base be equal, and the side edge.

How to find? Look, a hexagon consists of exactly six identical regular triangles. We have already searched for the area of a regular triangle when calculating the volume of a regular triangular pyramid, here we use the found formula.

Now let's find (this).

According to the Pythagorean theorem for

But what does it matter? It's simple because (and everyone else too) is correct.

We substitute:

\displaystyle V=\frac(\sqrt(3))(2)((a)^(2))\sqrt(((b)^(2))-((a)^(2)))

PYRAMID. BRIEFLY ABOUT THE MAIN

A pyramid is a polyhedron that consists of any flat polygon (), a point that does not lie in the plane of the base (top of the pyramid) and all segments connecting the top of the pyramid to the base points (side edges).

A perpendicular dropped from the top of the pyramid to the plane of the base.

Correct pyramid- a pyramid, which has a regular polygon at the base, and the top of the pyramid is projected into the center of the base.

Property of a regular pyramid:

In a regular pyramid, all side edges are equal.
All side faces are isosceles triangles and all these triangles are equal.

Pyramid Height Base Property:

Types of pyramids

Task example

Pyramid. visual guide (2019)

What is a pyramid?

Correct pyramid.

Pyramid Volume

The main formula for the volume of the pyramid:

PYRAMID. BRIEFLY ABOUT THE MAIN

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