What is called the base of the pyramid. Formulas and properties of a regular quadrangular pyramid. Truncated pyramid. Four basic linear parameters

This video tutorial will help users to get an idea about Pyramid theme. Correct pyramid. In this lesson, we will get acquainted with the concept of a pyramid, give it a definition. Consider what a regular pyramid is and what properties it has. Then we prove the theorem on the lateral surface of a regular pyramid.

In this lesson, we will get acquainted with the concept of a pyramid, give it a definition.

Consider a polygon A 1 A 2...A n, which lies in the plane α, and a point P, which does not lie in the plane α (Fig. 1). Let's connect the dot P with peaks A 1, A 2, A 3, … A n. Get n triangles: A 1 A 2 R, A 2 A 3 R etc.

Definition. Polyhedron RA 1 A 2 ... A n, made up of n-gon A 1 A 2...A n and n triangles RA 1 A 2, RA 2 A 3 …RA n A n-1 , called n- coal pyramid. Rice. one.

Rice. one

Consider a quadrangular pyramid PABCD(Fig. 2).

R- the top of the pyramid.

ABCD- the base of the pyramid.

RA- side rib.

AB- base edge.

From a point R drop the perpendicular RN on the ground plane ABCD. The perpendicular drawn is the height of the pyramid.

Rice. 2

The total surface of the pyramid consists of the lateral surface, that is, the area of all lateral faces, and the base area:

S full \u003d S side + S main

A pyramid is called correct if:

its base is a regular polygon;
the segment connecting the top of the pyramid with the center of the base is its height.

Explanation on the example of a regular quadrangular pyramid

Consider a regular quadrangular pyramid PABCD(Fig. 3).

R- the top of the pyramid. base of the pyramid ABCD- a regular quadrilateral, that is, a square. Dot O, the intersection point of the diagonals, is the center of the square. Means, RO is the height of the pyramid.

Rice. 3

Explanation: in the right n-gon, the center of the inscribed circle and the center of the circumscribed circle coincide. This center is called the center of the polygon. Sometimes they say that the top is projected into the center.

The height of the side face of a regular pyramid, drawn from its top, is called apothema and denoted h a.

1. all side edges of a regular pyramid are equal;

2. side faces are equal isosceles triangles.

Let us prove these properties using the example of a regular quadrangular pyramid.

Given: RABCD- regular quadrangular pyramid,

ABCD- square,

RO is the height of the pyramid.

Prove:

1. RA = PB = PC = PD

2.∆ATP = ∆BCP = ∆CDP = ∆DAP See Fig. 4.

Rice. 4

Proof.

RO is the height of the pyramid. That is, straight RO perpendicular to the plane ABC, and hence direct AO, VO, SO and DO lying in it. So the triangles ROA, ROV, ROS, ROD- rectangular.

Consider a square ABCD. It follows from the properties of a square that AO = BO = CO = DO.

Then the right triangles ROA, ROV, ROS, ROD leg RO- general and legs AO, VO, SO and DO equal, so these triangles are equal in two legs. From the equality of triangles follows the equality of segments, RA = PB = PC = PD. Point 1 is proven.

Segments AB and sun are equal because they are sides of the same square, RA = RV = PC. So the triangles AVR and VCR - isosceles and equal on three sides.

Similarly, we get that the triangles ABP, BCP, CDP, DAP are isosceles and equal, which was required to prove in item 2.

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:

For the proof, we choose a regular triangular pyramid.

Given: RAVS is a regular triangular pyramid.

AB = BC = AC.

RO- height.

Prove: . See Fig. 5.

Rice. 5

Proof.

RAVS is a regular triangular pyramid. I.e AB= AC = BC. Let be O- the center of the triangle ABC, then RO is the height of the pyramid. The base of the pyramid is an equilateral triangle. ABC. notice, that .

triangles RAV, RVS, RSA- equal isosceles triangles (by property). A triangular pyramid has three side faces: RAV, RVS, RSA. So, the area of the lateral surface of the pyramid is:

S side = 3S RAB

The theorem has been proven.

The radius of a circle inscribed in the base of a regular quadrangular pyramid is 3 m, the height of the pyramid is 4 m. Find the area of the lateral surface of the pyramid.

Given: regular quadrangular pyramid ABCD,

ABCD- square,

r= 3 m,

RO- the height of the pyramid,

RO= 4 m.

To find: S side. See Fig. 6.

Rice. 6

Decision.

According to the proven theorem, .

Find the side of the base first AB. We know that the radius of a circle inscribed in the base of a regular quadrangular pyramid is 3 m.

Then, m.

Find the perimeter of the square ABCD with a side of 6 m:

Consider a triangle BCD. Let be M- middle side DC. As O- middle BD, then (m).

Triangle DPC- isosceles. M- middle DC. I.e, RM- the median, and hence the height in the triangle DPC. Then RM- apothem of the pyramid.

RO is the height of the pyramid. Then, straight RO perpendicular to the plane ABC, and hence the direct OM lying in it. Let's find an apothem RM from a right triangle ROM.

Now we can find the side surface of the pyramid:

Answer: 60 m2.

The radius of a circle circumscribed near the base of a regular triangular pyramid is m. The lateral surface area is 18 m 2. Find the length of the apothem.

Given: ABCP- regular triangular pyramid,

AB = BC = SA,

R= m,

S side = 18 m 2.

To find: . See Fig. 7.

Rice. 7

Decision.

In a right triangle ABC given the radius of the circumscribed circle. Let's find a side AB this triangle using the sine theorem.

Knowing the side of a regular triangle (m), we find its perimeter.

According to the theorem on the area of the lateral surface of a regular pyramid, where h a- apothem of the pyramid. Then:

Answer: 4 m.

So, we examined what a pyramid is, what a regular pyramid is, we proved the theorem on the lateral surface of a regular pyramid. In the next lesson, we will get acquainted with the truncated pyramid.

Bibliography

Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., Rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
Geometry. Grade 10-11: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill.
Geometry. Grade 10: Textbook for general educational institutions with in-depth and profile study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th ed., stereotype. - M.: Bustard, 008. - 233 p.: ill.

Internet portal "Yaklass" ()
Internet portal "Festival pedagogical ideas"First of September" ()
Internet portal "Slideshare.net" ()

Homework

Can a regular polygon be the base of an irregular pyramid?
Prove that non-intersecting edges of a regular pyramid are perpendicular.
Find the value of the dihedral angle at the side of the base of a regular quadrangular pyramid, if the apothem of the pyramid is equal to the side of its base.
RAVS is a regular triangular pyramid. Construct the linear angle of the dihedral angle at the base of the pyramid.

In this lesson, we will get acquainted with the concept of a pyramid, give it a definition.

Definition. Polyhedron RA 1 A 2 ... A n, made up of n-gon A 1 A 2...A n and n triangles RA 1 A 2, RA 2 A 3 …RA n A n-1 , called n- coal pyramid. Rice. one.

Rice. one

Consider a quadrangular pyramid PABCD(Fig. 2).

R- the top of the pyramid.

ABCD- the base of the pyramid.

RA- side rib.

AB- base edge.

From a point R drop the perpendicular RN on the ground plane ABCD. The perpendicular drawn is the height of the pyramid.

Rice. 2

The total surface of the pyramid consists of the lateral surface, that is, the area of all lateral faces, and the base area:

S full \u003d S side + S main

A pyramid is called correct if:

its base is a regular polygon;
the segment connecting the top of the pyramid with the center of the base is its height.

Explanation on the example of a regular quadrangular pyramid

Consider a regular quadrangular pyramid PABCD(Fig. 3).

Rice. 3

The height of the side face of a regular pyramid, drawn from its top, is called apothema and denoted h a.

1. all side edges of a regular pyramid are equal;

2. side faces are equal isosceles triangles.

Let us prove these properties using the example of a regular quadrangular pyramid.

Given: RABCD- regular quadrangular pyramid,

ABCD- square,

RO is the height of the pyramid.

Prove:

1. RA = PB = PC = PD

2.∆ATP = ∆BCP = ∆CDP = ∆DAP See Fig. 4.

Rice. 4

Proof.

RO is the height of the pyramid. That is, straight RO perpendicular to the plane ABC, and hence direct AO, VO, SO and DO lying in it. So the triangles ROA, ROV, ROS, ROD- rectangular.

Consider a square ABCD. It follows from the properties of a square that AO = BO = CO = DO.

Segments AB and sun are equal because they are sides of the same square, RA = RV = PC. So the triangles AVR and VCR - isosceles and equal on three sides.

Similarly, we get that the triangles ABP, BCP, CDP, DAP are isosceles and equal, which was required to prove in item 2.

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:

For the proof, we choose a regular triangular pyramid.

Given: RAVS is a regular triangular pyramid.

AB = BC = AC.

RO- height.

Prove: . See Fig. 5.

Rice. 5

Proof.

triangles RAV, RVS, RSA- equal isosceles triangles (by property). A triangular pyramid has three side faces: RAV, RVS, RSA. So, the area of the lateral surface of the pyramid is:

S side = 3S RAB

The theorem has been proven.

The radius of a circle inscribed in the base of a regular quadrangular pyramid is 3 m, the height of the pyramid is 4 m. Find the area of the lateral surface of the pyramid.

Given: regular quadrangular pyramid ABCD,

ABCD- square,

r= 3 m,

RO- the height of the pyramid,

RO= 4 m.

To find: S side. See Fig. 6.

Rice. 6

Decision.

According to the proven theorem, .

Find the side of the base first AB. We know that the radius of a circle inscribed in the base of a regular quadrangular pyramid is 3 m.

Then, m.

Find the perimeter of the square ABCD with a side of 6 m:

Consider a triangle BCD. Let be M- middle side DC. As O- middle BD, then (m).

Triangle DPC- isosceles. M- middle DC. I.e, RM- the median, and hence the height in the triangle DPC. Then RM- apothem of the pyramid.

RO is the height of the pyramid. Then, straight RO perpendicular to the plane ABC, and hence the direct OM lying in it. Let's find an apothem RM from a right triangle ROM.

Now we can find the side surface of the pyramid:

Answer: 60 m2.

The radius of a circle circumscribed near the base of a regular triangular pyramid is m. The lateral surface area is 18 m 2. Find the length of the apothem.

Given: ABCP- regular triangular pyramid,

AB = BC = SA,

R= m,

S side = 18 m 2.

To find: . See Fig. 7.

Rice. 7

Decision.

In a right triangle ABC given the radius of the circumscribed circle. Let's find a side AB this triangle using the sine theorem.

Knowing the side of a regular triangle (m), we find its perimeter.

According to the theorem on the area of the lateral surface of a regular pyramid, where h a- apothem of the pyramid. Then:

Answer: 4 m.

So, we examined what a pyramid is, what a regular pyramid is, we proved the theorem on the lateral surface of a regular pyramid. In the next lesson, we will get acquainted with the truncated pyramid.

Bibliography

Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., Rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
Geometry. Grade 10-11: A textbook for general educational institutions / Sharygin I. F. - M .: Bustard, 1999. - 208 p.: ill.
Geometry. Grade 10: Textbook for general educational institutions with in-depth and profile study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th ed., stereotype. - M.: Bustard, 008. - 233 p.: ill.

Internet portal "Yaklass" ()
Internet portal "Festival of Pedagogical Ideas "First of September" ()
Internet portal "Slideshare.net" ()

Homework

Can a regular polygon be the base of an irregular pyramid?
Prove that non-intersecting edges of a regular pyramid are perpendicular.
Find the value of the dihedral angle at the side of the base of a regular quadrangular pyramid, if the apothem of the pyramid is equal to the side of its base.
RAVS is a regular triangular pyramid. Construct the linear angle of the dihedral angle at the base of the pyramid.

The text of the work is placed without images and formulas.
Full version work is available in the "Files of work" tab in PDF format

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 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 goal 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. 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 polyhedron, 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 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 through 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)

Consider the theorems that connect the pyramid with others 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. When evaluating the height of the pyramid, it is necessary to take into account such physical factor, as a draft design. 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 pyramid's faces: it turned out to be equal to a = 51 ° 51 ". This value is still recognized by most researchers today. Specified value 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 rectangular 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 is exactly equal to the area of \u200b\u200beach of the side triangles. This is confirmed by the latest measurements.

We know that the relation between the circumference of a circle and its diameter is constant, well known to modern mathematicians, schoolchildren, is the number "Pi" = 3.1416 ... But if we add up the four sides of the base of the Cheops pyramid, we get 931.22 m. Dividing this number by 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" playing 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 Tibetan pyramid complex is included in the study, then a strict mathematical system their location on the earth's surface. 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 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. 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. Monitor generates light wave, each of the four identical photographs, being in the plane, which is the 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 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 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 - detailed description states 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 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 - schematic representation of the principles 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 much more lofty goal, and is intended for something more than the practical ways of using it that 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 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.

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. world system 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.

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

This lesson provides the definition and properties of a regular triangular pyramid and its special case - a tetrahedron (see below). Links to examples of problem solving are provided at the end of the lesson.

Definition

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.

The figure shows:
ABC- Base pyramids
OS - Height
KS - Apothem
OK - radius of the circle inscribed in the base
AO - radius of a circle circumscribed around the base of a regular triangular pyramid
SKO - the dihedral angle between the base and the face of the pyramid (they are equal in a regular pyramid)

Important. In a regular triangular pyramid, the length of the edge (in the figure AS, BS, CS) may not be equal to the length of the side of the base (in the figure AB, AC, BC). If the length of the edge of a regular triangular pyramid is equal to the length of the side of the base, then such a pyramid is called a tetrahedron (see below).

Properties of a regular triangular pyramid:

lateral edges of a regular pyramid are equal
all side faces of a regular pyramid are isosceles triangles
in a regular triangular pyramid, you can both inscribe and describe a sphere around it
if the centers of the spheres inscribed and circumscribed around a regular triangular pyramid coincide, then the sum of the plane angles at the top of the pyramid is equal to π (180 degrees), and each of them, respectively, is equal to π / 3 (pi divided by 3 or 60 degrees).
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
the top of the pyramid is projected onto the base at the center of a regular equilateral triangle, which is the center of the inscribed circle and the intersection point of the medians

Formulas for a regular triangular pyramid

The formula for the volume of a regular triangular pyramid is:

V is the volume of a regular pyramid with a regular (equilateral) triangle at the base
h - the height of the pyramid
a - the length of the side of the base of the pyramid
R - radius of the circumscribed circle
r - radius of the inscribed circle

Since a regular triangular pyramid is a special case of a regular pyramid, the formulas that are true for a regular pyramid are also true for a regular triangular pyramid - see formulas for a regular pyramid.

Examples of problem solving:

Tetrahedron

A special case of a regular triangular pyramid is tetrahedron.

Tetrahedron is a regular polyhedron (regular triangular pyramid) in which all faces are regular triangles.

At the tetrahedron:

All edges are equal
4 faces, 4 vertices and 6 edges
All dihedral angles at the edges and all trihedral angles at the vertices are equal

Median of a tetrahedron- this is a segment connecting the vertex to the point of intersection of the medians of the opposite face (the medians of an equilateral triangle opposite the vertex)

Bimedian tetrahedron- this is a segment connecting the midpoints of crossing edges (connecting the midpoints of the sides of a triangle, which is one of the faces of a tetrahedron)

Tetrahedron height- this is a segment connecting the vertex with a point of the opposite face and perpendicular to this face (that is, it is the height drawn from any face, also coincides with the center of the circumscribed circle).

Tetrahedron has the following properties:

All medians and bimedians of a tetrahedron intersect at one point
This point divides the medians in a ratio of 3:1, counting from the top
This point bisects the bimedians

Video lesson 2: Pyramid challenge. Pyramid Volume

Video lesson 3: Pyramid challenge. Correct pyramid

Lecture: Pyramid, its base, side edges, height, side surface; triangular pyramid; right pyramid

Pyramid, its properties

Pyramid- This is a three-dimensional body that has a polygon at the base, and all its faces consist of triangles.

A special case of a pyramid is a cone, at the base of which lies a circle.

Consider the main elements of the pyramid:

Apothem is a segment that connects the top of the pyramid with the middle of the lower edge of the side face. In other words, this is the height of the face of the pyramid.

In the figure you can see the triangles ADS, ABS, BCS, CDS. If you look closely at the names, you can see that each triangle has one common letter in its name - S. That is, it means that all side faces (triangles) converge at one point, which is called the top of the pyramid.

The segment OS, which connects the vertex with the point of intersection of the diagonals of the base (in the case of triangles, at the point of intersection of the heights), is called pyramid height.

A diagonal section is a plane that passes through the top of the pyramid, as well as one of the diagonals of the base.

Since the lateral surface of the pyramid consists of triangles, to find the total area of the lateral surface, it is necessary to find the areas of each face and add them. The number and shape of the faces depends on the shape and size of the sides of the polygon that lies at the base.

The only plane in a pyramid that does not have a vertex is called basis pyramids.

In the figure, we see that the base is a parallelogram, however, there can be any arbitrary polygon.

Properties:

Consider the first case of a pyramid, in which it has edges of the same length:

A circle can be described around the base of such a pyramid. If you project the top of such a pyramid, then its projection will be located in the center of the circle.
The angles at the base of the pyramid are the same for each face.
Wherein sufficient condition to the fact that around the base of the pyramid you can describe a circle, and also assume that all the edges different lengths, we can consider the same angles between the base and each edge of the faces.

If you come across a pyramid in which the angles between the side faces and the base are equal, then the following properties are true:

You will be able to describe a circle around the base of the pyramid, the top of which is projected exactly to the center.
If you draw at each side face of the height to the base, then they will be of equal length.
To find the lateral surface area of such a pyramid, it is enough to find the perimeter of the base and multiply it by half the length of the height.
Sbp \u003d 0.5P oc H.
Types of pyramid.
Depending on which polygon lies at the base of the pyramid, they can be triangular, quadrangular, etc. If a regular polygon lies at the base of the pyramid (with equal parties), then such a pyramid will be called regular.

Regular triangular pyramid