Pyramid concept

Definition 1

A geometric figure formed by a polygon and a point not lying in the plane containing this polygon, connected to all 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 to 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 angles at the base of the pyramid, it can be called triangular, quadrangular, and so on (Fig. 2).

Figure 2.

Another type of pyramid is the regular pyramid.

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

Theorem 1

All lateral faces of a regular pyramid are isosceles triangles, which are equal to each other.

Evidence.

Consider a regular $ n- $ coal pyramid with top $ S $ and height $ h \u003d SO $. Let us describe a circle around the base (Fig. 4).

Figure 4.

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

Obviously, this will define any lateral edge. Therefore, all side edges are equal to each other, that is, all side edges 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 criterion of equality of triangles.

The theorem is proved.

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 lateral edge.

Obviously, by Theorem One, all apothems are equal to each other.

Theorem 2

The lateral surface area of \u200b\u200ba regular pyramid is defined as the product of the base half-perimeter and the apothem.

Evidence.

Let's denote the side of the base of the $ n- $ coal pyramid by $ a $, and the apothem by $ d $. Therefore, the area of \u200b\u200bthe side face is

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

The theorem is proved.

Another type of pyramid is a truncated pyramid.

Definition 4

If a plane is drawn through an ordinary pyramid parallel to its base, 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 side faces of the truncated pyramid are trapezoids.

Theorem 3

The lateral surface area of \u200b\u200ba regular truncated pyramid is defined as the product of the sum of the base semiperimeters and the apothem.

Evidence.

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 \u200b\u200bthe side face is

Since all sides are equal, then

The theorem is proved.

Example task

Example 1

Find the area of \u200b\u200bthe 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 with a plane passing through the midline of the lateral faces.

Decision.

By the middle line theorem, we get that the upper base of the truncated pyramid is $ 4 \\ cdot \\ frac (1) (2) \u003d 2 $, and the apothem is $ 5 \\ cdot \\ frac (1) (2) \u003d 2.5 $.

Then, by Theorem 3, we obtain

Here you can find basic information about the pyramids and related formulas and concepts. All of them are studied with a mathematics tutor 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 the vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called side ribs. The polygon is called the base and point S is called the top of the pyramid. Depending on the number n, the pyramid is called triangular (n \u003d 3), quadrangular (n \u003d 4), ptyagonal (n \u003d 5), and so on. An alternative name for the triangular pyramid is tetrahedron... The height of the pyramid is called the perpendicular, lowered from its top to the plane of the base.

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

Tutor comment:
Do not confuse the concept of "regular pyramid" and "correct 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 the coincidence of the center P of the polygon with the base of the height, so a regular tetrahedron is a regular pyramid.

What is Apothema?
The apothem of a pyramid is the height of its lateral face. If the pyramid is correct, then all its apothems are equal. The converse is not true.

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

To make it easier to refer to these triangles, it is more convenient for a math tutor to call the first one apothemic, and second costal... Unfortunately, you will not find this terminology in any of the textbooks, and the teacher has to enter it unilaterally.

The formula for the volume of a pyramid:
1) , where is the area of \u200b\u200bthe base of the pyramid, and is the height of the pyramid
2), where is the radius of the inscribed sphere, and is the total surface area of \u200b\u200bthe pyramid.
3) , where MN is the distance of any two crossing edges, and is the area of \u200b\u200bthe 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 Commentary: note that all points have one common property: one way or another, side faces are involved everywhere (apothems are their elements). Therefore, the tutor may offer a less accurate, but more convenient for memorization formulation: the point P coincides with the center of the inscribed circle at the base of the pyramid, if there is any equal information about its side faces. To prove it, it suffices to show that all apothemic triangles are equal.

Point P coincides with the center of a circle described near the base of the pyramid, if one of 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 height

Students are faced with the concept of a pyramid long before the study of geometry. This is due to the famous great Egyptian wonders of the world. Therefore, starting the study of this wonderful polyhedron, most students already clearly imagine it. All of the aforementioned attractions have the correct shape. What correct pyramid, and what properties it has and will be discussed further.

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Definition

There are many definitions of a pyramid. Since ancient times, it has enjoyed great popularity.

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

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

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

Let's figure it out in more detail, what elements does it consist of:

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

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

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

Basic properties

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

1. The base is a figure of regular shape.
2. The edges of the pyramid that bound the side members have equal numeric values.
3. Lateral elements are isosceles triangles.
4. The base of the figure's height falls into the center of the polygon, while at the same time it is the center point of the inscribed and described.
5. All side ribs are inclined to the plane of the base at the same angle.
6. All side surfaces have the same angle of inclination with respect to the base.

All of these properties make it much easier to perform member calculations. Based on the above properties, we draw attention to two signs:

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

It is based on a square

Regular quadrangular pyramid - a polyhedron based on a square.

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

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

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

It is based on a regular triangle

A regular triangular pyramid is a polyhedron with a regular 3-gon at its base.

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

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

• the angle of inclination of the ribs to any base is 60 degrees;
• the size of all inner edges is also 60 degrees;
• any facet can act as a base;
• drawn inside the shape are equal elements.

Polyhedron sections

In any polyhedron, there are several types of sectionplane. Often in the school geometry course, two are worked:

• axial;
• parallel basis.

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

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

If the cutting plane runs parallel to the base, then the result is the second option. In this case, we have a cross-sectional figure similar to the base.

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

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

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

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

Surface areas

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

There are two types of surface area values:

• the area of \u200b\u200bthe side elements;
• the area of \u200b\u200bthe entire surface.

From the name itself it is clear what it is about. The side surface only includes side elements. From this it follows that to find it, you just need to add up the areas of the lateral planes, that is, the areas of isosceles 3-gons. Let's try to derive the formula for the area of \u200b\u200bthe side elements:

1. The area of \u200b\u200ban isosceles 3-gon is equal to Str \u003d 1/2 (aL), where a is the side of the base, L is the apothem.
2. The number of side planes depends on the type of the k-th gon at the base. For example, a regular quadrangular pyramid has four side planes. Therefore, it is necessary to add the areas of the four figures S side \u003d 1/2 (aL) +1/2 (aL) +1/2 (aL) +1/2 (aL) \u003d 1/2 * 4а * L. The expression is simplified in this way because the value 4a \u003d Rosn, where Rosn is the perimeter of the base. And the expression 1/2 * Rosn is its semiperimeter.
3. So, we conclude that the area of \u200b\u200bthe side elements of a regular pyramid is equal to the product of the base half-perimeter by the apothem: Sbok \u003d Rosn * L.

The total surface area of \u200b\u200bthe pyramid consists of the sum of the areas of the lateral planes and the base: Sp.p. \u003d Sside + Sbase.

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

The volume of a regular pyramidis equal to the product of the area of \u200b\u200bthe base plane by the height, divided by three: V \u003d 1/3 * Sbase * H, where H is the height of the polyhedron.

What is a correct pyramid in geometry

Properties of a regular quadrangular pyramid

Hypothesis: we believe that the perfection of the shape of the pyramid is due to the mathematical laws embedded in its shape.

Purpose:having studied the pyramid as a geometric body, to explain the perfection of its shape.

Tasks:

1. Give a mathematical definition of the pyramid.

2. Study the pyramid as a geometric body.

3. Understand what mathematical knowledge the Egyptians laid in their pyramids.

Private questions:

1. What is a pyramid as a geometric body?

2. How can you explain the uniqueness of the shape of the pyramid from a mathematical point of view?

3. What explains the geometric wonders of the pyramid?

4. What explains the perfection of the pyramid shape?

Definition of the pyramid.

PYRAMID (from the Greek pyramis, genus pyramidos) - a polyhedron, the base of which is a polygon, and the remaining faces are triangles with a common vertex (figure). According to the number of angles of the base, pyramids are distinguished triangular, quadrangular, etc.

PYRAMID - a monumental structure with a geometric pyramid shape (sometimes also stepped or tower-like). The pyramids are called the giant tombs of the ancient Egyptian pharaohs of the 3rd - 2nd millennium BC. e., as well as ancient American pedestals of temples (in Mexico, Guatemala, Honduras, Peru) associated with cosmological cults.

It is possible that the Greek word "pyramid" comes from the Egyptian expression per-em-us, that is, from the term meaning the height of the pyramid. Prominent Russian Egyptologist V. Struve believed that the Greek “puram… j” comes from the ancient Egyptian “p” -mr ”.

From the history. Having studied the material in the textbook "Geometry" by the authors of Atanasyan. Butuzov and others, we learned that: A polyhedron composed of n - gon A1A2A3 ... An and n triangles PA1A2, PA2A3, ..., PnA1 is called a pyramid. Polygon A1A2A3… An is the base of the pyramid, and triangles PA1A2, PA2A3,…, PANA1 are the side faces of the pyramid, P is the top of the pyramid, the segments PA1, PA2,…, PAN are the side edges.

However, this definition of a pyramid did not always exist. For example, the ancient Greek mathematician, the author of theoretical treatises on mathematics that have come down to us, Euclid, defines a pyramid as a bodily figure bounded by planes that converge from one plane to one point.

But this definition was criticized already in ancient times. So Heron proposed the following definition of a pyramid: "It is a figure bounded by triangles converging at one point and the base of which is a polygon."

Our group, comparing these definitions, came to the conclusion that they do not have a clear formulation of the concept of “foundation”.

We investigated these definitions and found the definition of Adrien Marie Legendre, who in 1794 in his work "Elements of Geometry" defines the pyramid as follows: "A pyramid is a solid figure formed by triangles converging at one point and ending on different sides of a flat base."

It seems to us that the last definition gives a clear idea of \u200b\u200bthe pyramid, since it refers to the fact that the base is flat. Another definition of a pyramid appeared in a 19th century textbook: "a pyramid is a solid angle intersected by a plane."

The pyramid as a geometric body.

T. about. A pyramid is a polyhedron, one of the faces of which (base) is a polygon, the other faces (side) are triangles that have one common vertex (apex of the pyramid).

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

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

The figure shows the pyramid PABCD, ABCD is its base, PO is the height.

Full surface area pyramid is called the sum of the areas of all its faces.

S full \u003d S side + S main,where S side - the sum of the areas of the side faces.

The volume of the pyramid is found by the formula:

V \u003d 1 / 3Sb. h, where Sosn. - base area, h - height.

The axis of a regular pyramid is called a straight line containing its height.
Apothem ST - the height of the side face of the regular pyramid.

The area of \u200b\u200bthe side face of a regular pyramid is expressed as follows: S side. \u003d 1 / 2P h, where P is the perimeter of the base, h - the height of the side face (apothem of the regular pyramid). If the pyramid is crossed by the plane A'B'C'D 'parallel to the base, then:

1) lateral ribs and height are divided by this plane into proportional parts;

2) in the section, a polygon A'B'C'D 'is obtained, similar to the base;

https://pandia.ru/text/78/390/images/image017_1.png "width \u003d" 287 "height \u003d" 151 "\u003e

Truncated pyramid bases - similar polygons ABCD and A`B`C`D`, side faces - trapezoid.

Height truncated pyramid - the distance between the bases.

Truncated volume pyramid is found by the formula:

V \u003d 1/3 h (S + https://pandia.ru/text/78/390/images/image019_2.png "align \u003d" left "width \u003d" 91 "height \u003d" 96 "\u003e The lateral surface area of \u200b\u200ba regular truncated pyramid is expressed as follows: S side. \u003d ½ (P + P ') h, where P and P 'are the perimeters of the bases, h- the height of the side face (apothem of the correct truncated

Sections of the pyramid.

The sections of the pyramid by planes passing through its apex are triangles.

The section passing through two non-adjacent lateral edges of the pyramid is called 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 edge of the pyramid, and a given trace of the section on the base plane, then the construction should be carried out as follows:

· Find the point of intersection 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 following faces.

, which corresponds to the ratio of the legs of a right-angled triangle 4: 3. This aspect ratio corresponds to the well-known 3: 4: 5 right-angled triangle, which is called the "perfect", "sacred" or "Egyptian" triangle. According to historians, the "Egyptian" triangle was given a magical meaning. Plutarch wrote that the Egyptians compared the nature of the universe to a "sacred" triangle; they symbolically likened the vertical leg to the husband, the base to the wife, and the hypotenuse to that which is born of both.

For a triangle 3: 4: 5, the equality is true: 32 + 42 \u003d 52, which expresses the Pythagorean theorem. Was it not this theorem that the Egyptian priests wanted to perpetuate by erecting a pyramid on the basis of the triangle 3: 4: 5? It is difficult to find a better example to illustrate the Pythagorean theorem, which was known to the Egyptians long before its discovery by Pythagoras.

Thus, the ingenious creators of the Egyptian pyramids sought to amaze distant descendants with the depth of their knowledge, and they achieved this by choosing the "golden" right triangle for the Cheops pyramid, and the "sacred" or "Egyptian" one for the Khafre pyramid triangle.

Very often in their research, scientists use the properties of pyramids with the proportions of the Golden Section.

In the mathematical encyclopedic dictionary, the following definition of the Golden Section is given - this is harmonic division, division in the extreme and average ratio - dividing 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.

Algebraic Finding of the Golden Ratio of a Segment AB \u003d a is reduced to solving the equation a: x \u003d x: (a - x), whence x is approximately equal to 0.62a. The ratio x can be expressed in fractions 2/3, 3/5, 5/8, 8/13, 13/21 ... \u003d 0.618, where 2, 3, 5, 8, 13, 21 are Fibonacci numbers.

The geometric construction of the Golden Section of the segment AB is carried out as follows: at point B, the perpendicular to AB is restored, the segment BE \u003d 1/2 AB is laid on it, A and E are laid, DE \u003d BE and, finally, AC \u003d HELL, then the equality AB is fulfilled: SV \u003d 2: 3.

The golden ratio is often used in works of art, architecture, and occurs in nature. Notable 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. The objects around us also provide examples of the Golden Ratio, for example, the bindings of many books have a ratio of width to length close to 0.618. Considering the arrangement of leaves on the common stem of plants, it can be seen that between every two pairs of leaves, the third is located in the place of the Golden Section (slides). Each of us “carries” the Golden Ratio with us “in our hands” - this is the ratio of the phalanges of the fingers.

Through the discovery of several mathematical papyri, Egyptologists have learned a thing or two about ancient Egyptian systems of numbers and measures. The tasks contained in them were solved by scribes. One of the most famous is the Rindi Mathematical Papyrus. By studying these puzzles, Egyptologists learned how the ancient Egyptians dealt with the varying quantities that came up when calculating the measures of weight, length, and volume, in which fractions were often used, and how they dealt with angles.

The ancient Egyptians used a method for calculating angles based on the ratio of the height to the base of a right triangle. They expressed any angle in the language of the gradient. The gradient of the slope was expressed by an integer ratio called "seced". In his book Mathematics in the Time of the Pharaohs, Richard Pillins explains: “The seked of a regular pyramid is the inclination of any of the four triangular faces to the base plane, measured by an nth number of horizontal units per vertical unit of lift. Thus, this unit is equivalent to our modern tilt cotangent. Hence, the Egyptian word "seked" is akin to our modern word "gradient."

The numerical key to the pyramids lies in the ratio of their height to the base. In practical terms, this is the easiest way to make the templates needed to constantly check the correct angle of inclination throughout the construction of the pyramid.

Egyptologists would be happy to convince us that each pharaoh was eager to express his individuality, which is why the different angles of inclination for each pyramid. But there could be another reason. Perhaps they all wished to embody different symbolic associations, hidden in different proportions. However, the angle of Khafre's pyramid (based on a triangle (3: 4: 5) appears in the three problems represented by the pyramids in the Rindi Mathematical Papyrus). So this attitude was well known to the ancient Egyptians.

To be fair to Egyptologists who claim that the ancient Egyptians did not know the 3: 4: 5 triangle, let us say that the length of the hypotenuse 5 was never mentioned. But mathematical problems related to pyramids are always solved based on the angle seked - the ratio of height to base. Since the length of the hypotenuse was never mentioned, it was concluded that the Egyptians never calculated the length of the third side.

The height to base ratios used in the pyramids of Giza were undoubtedly known to the ancient Egyptians. It is possible that these relationships were chosen arbitrarily for each pyramid. However, this contradicts the importance given to numerical symbolism in all forms of Egyptian visual arts. It is very likely that such relationships were significant because they expressed specific religious ideas. In other words, the entire Giza complex was subordinate to a coherent plan designed to reflect a certain divine theme. This would explain why the designers chose different angles for the three pyramids.

In The Mystery of Orion, Bauval and Gilbert presented convincing evidence of the connection of the pyramids of Giza with the constellation Orion, in particular with the stars of Orion's Belt. This constellation is present in the myth of Isis and Osiris, and there is reason to consider each pyramid as an image of one of the three main deities - Osiris, Isis and Horus.

MIRACLES "GEOMETRIC".

Among the grandiose pyramids of Egypt, a special place is Great Pyramid of Pharaoh Cheops (Khufu)... Before proceeding to the analysis of the shape and size of the Cheops pyramid, one should recall 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, equal to four "fingers" (16.6 mm).

Let's analyze the dimensions of the Cheops pyramid (Fig. 2), following the reasoning given in the wonderful book of the Ukrainian scientist Nikolai Vasyutinsky "The Golden Proportion" (1990).

Most researchers agree that the length of the side of the base of the pyramid, for example, Gf equals L \u003d 233.16 m. This value corresponds to almost exactly 500 "cubits". Full compliance with 500 "cubits" will be if the length of the "cubit" is considered equal to 0.4663 m.

Pyramid height ( 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, strictly speaking, the Cheops pyramid is truncated. Its top platform today is about 10 ´ 10 m, and a century ago it was 6 ´ 6 m. Obviously, the top of the pyramid was taken apart, and it does not correspond to the original one.

When evaluating the height of the pyramid, it is necessary to take into account such a physical factor as the "draft" of the structure. For a long time, under the influence of colossal pressure (reaching 500 tons per 1 m2 of the lower surface), the height of the pyramid has decreased compared to its original height.

What was the initial height of the pyramid? This height can be recreated by finding the basic "geometric idea" of the pyramid.

Figure 2.

In 1837, English Colonel G. Weisz measured the angle of inclination of the pyramid's faces: it turned out to be equal a \u003d 51 ° 51 ". This value is still recognized by the majority of researchers. 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 AS to half of its base CB (Fig. 2), that is AC / CB = H / (L / 2) = 2H / L.

And here the researchers were in for a big surprise! .Png "width \u003d" 25 "height \u003d" 24 "\u003e \u003d 1.272. Comparing this value with the value of tg a \u003d 1.27306, we see that these values \u200b\u200bare very close to each other. If we take the angle a \u003d 51 ° 50 ", that is, to reduce it by only one arc minute, then the value a will become equal to 1.272, that is, coincide with the value. It should be noted that in 1840 G. Weis repeated his measurements and specified that the value of the angle a \u003d 51 ° 50 ".

These measurements led the researchers to the following very interesting hypothesis: the basis of the triangle ACB of the Cheops pyramid was the ratio AC / CB = = 1,272!

Consider now a right-angled triangle ABC, in which the ratio of the legs AC / CB \u003d (Fig. 2). If now the lengths of the sides of the rectangle ABC denote through 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 you accept x = 1, y \u003d https://pandia.ru/text/78/390/images/image027_1.png "width \u003d" 143 "height \u003d" 27 "\u003e

Figure 3. "Golden" right-angled triangle.

A right-angled triangle in which the sides are related as t : golden "right-angled triangle.

Then, if we take as a basis the hypothesis that the main "geometrical 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 deduce some other relations for the Cheops pyramid arising from the "golden" hypothesis. In particular, we find the ratio of the outer area of \u200b\u200bthe pyramid to the area of \u200b\u200bits base. To do this, take the length of the leg CB per unit, that is: CB \u003d 1. But then the length of the side of the base of the pyramid Gf \u003d 2, and the base area EFGH will be equal SEFGH = 4.

Let us now calculate the area of \u200b\u200bthe side face of the Cheops pyramid SD... Since the height AB triangle AEF equals t, then the area of \u200b\u200bthe side face will be SD = t... Then the total area of \u200b\u200ball four side faces of the pyramid will be 4 t, and the ratio of the total outer area of \u200b\u200bthe pyramid to the area of \u200b\u200bthe base will be equal to the golden ratio! That's what it is - the main geometric mystery of the Cheops pyramid!

The group of "geometric miracles" of the Cheops pyramid includes the real and contrived properties of the relationship between different dimensions in the pyramid.

As a rule, they are obtained in search of some "constants", in particular, the number "pi" (Ludolph's number), equal to 3.14159 ...; base of natural logarithms "e" (Napier's number) equal to 2.71828 ...; the number "F", the number of the "golden ratio", equal, for example, 0.618 ... and so on.

You can name, for example: 1) Property of Herodotus: (Height) 2 \u003d 0.5 tbsp. main x Apothem; 2) Property of V. Price: Height: 0.5 st. osn \u003d Square root of "Ф"; 3) Property of M. Eyst: Base perimeter: 2 Height \u003d "Pi"; in a different interpretation - 2 tbsp. main : Height \u003d "Pi"; 4) Property of G. Ribs: Inscribed circle radius: 0.5 st. main \u003d "F"; 5) Property of K. Kleppisch: (Art. Main.) 2: 2 (art. Main. X Apothem) \u003d (art. Main. U. Apothem) \u003d 2 (art. Main. X Apothem): ((2 art. base X Apothem) + (st. base) 2). Etc. You can think of a lot of such properties, especially if you connect two neighboring pyramids. For example, as "Properties of A. Arefiev" it is possible to mention that the difference between the volumes of the Cheops pyramid and the Khafre pyramid is equal to the doubled volume of the Mikerin pyramid ...

Many interesting provisions, in particular, on the construction of pyramids according to the "golden ratio" are set forth in the books by D. Hambidge "Dynamic symmetry in architecture" and M. Geek "Aesthetics of proportion in nature and art". Recall that the "golden ratio" is the division of a segment in such a ratio when part A is as many times larger than part B, how many times A is less than the entire segment A + B. The ratio A / B is equal to the number "Ф" \u003d\u003d 1.618. .. The use of the "golden ratio" is indicated not only in individual pyramids, but also in the entire complex of pyramids in Giza.

The most curious thing, however, is that one and the same pyramid of Cheops simply "cannot" contain so many wonderful properties. Taking a certain property one by one, it can be "adjusted", but all at once they do not fit - they do not coincide, they contradict each other. Therefore, if, for example, when checking all properties, we initially take the same side of the pyramid base (233 m), then the heights of pyramids with different properties will also be different. In other words, there is a certain "family" of pyramids, outwardly similar to Cheops, but corresponding to different properties. Note that there is nothing particularly miraculous in the "geometric" properties - much arises purely automatically, from the properties of the figure itself. Only something clearly impossible for the ancient Egyptians should be considered a "miracle". This, in particular, includes "cosmic" miracles, in which the measurements of the Cheops pyramid or the pyramid complex at Giza are compared with some astronomical measurements and "even" numbers are indicated: a million times, a billion times less, and so on. Let's consider some "cosmic" relationships.

One of the statements is this: "If we divide the side of the base of the pyramid by the exact length of the year, we get exactly 10-millionth of the earth's axis." Calculate: divide 233 by 365, we get 0.638. The radius of the Earth is 6378 km.

Another statement is actually the opposite of the previous one. F. Noetling pointed out that if we use the "Egyptian elbow" invented by him, then the side of the pyramid will correspond to "the most exact duration of a solar year, expressed with an accuracy of one billionth day" - 365.540.903.777.

P. Smith's statement: "The height of the pyramid is exactly one billionth of the distance from the Earth to the Sun." Although an altitude of 146.6 m is usually taken, Smith took it 148.2 m. According to modern radar measurements, the semi-major axis of the earth's orbit is 149.597.870 + 1.6 km. This is the average distance from the Earth to the Sun, but at perihelion it is 5,000,000 kilometers less than at aphelion.

One last curious statement:

"How to explain that the masses of the pyramids of Cheops, Khafre and Mykerinus relate to each other, like the masses of the planets Earth, Venus, Mars?" Let's calculate. The masses of the three pyramids are as follows: Khafre - 0.835; Cheops - 1,000; Mikerin - 0.0915. The ratio of the masses of the three planets: Venus - 0.815; Land - 1,000; Mars - 0.108.

So, in spite of the skepticism, let us note the well-known harmony of the construction of statements: 1) the height of the pyramid, as a line "extending into space" - corresponds to the distance from the Earth to the Sun; 2) the side of the base of the pyramid closest to "the substrate", that is, to the Earth, is responsible for the earth's radius and earthly circulation; 3) the volumes of the pyramid (read - masses) correspond to the ratio of the masses of the planets closest to the Earth. A similar "cipher" can be traced, for example, in the bee language analyzed by Karl von Frisch. However, we will refrain from commenting on this for now.

PYRAMID SHAPE

The famous four-sided shape of the pyramids did not appear immediately. The Scythians made burials in the form of earthen hills - mounds. The Egyptians set up "hills" of stone - pyramids. This happened for the first time after the unification of Upper and Lower Egypt, in the XXVIII century BC, when the founder of the III dynasty, Pharaoh Djoser (Zoser), was faced with the task of strengthening the unity of the country.

And here, according to historians, an important role in strengthening the central government was played by the "new concept of deification" of the king. Although the royal burials were distinguished by greater splendor, they, in principle, did not differ from the tombs of the court nobles, they were the same structures - mastabas. Above the chamber with the sarcophagus containing the mummy, a rectangular hill of small stones was poured, where then a small building of large stone blocks was erected - "mastaba" (in Arabic - "bench"). In place of the mastab of his predecessor, Sanakht, Pharaoh Djoser erected the first pyramid. It was stepwise and was a visible transitional stage from one architectural form to another, from a mastaba to a pyramid.

In this way, the sage and architect Imhotep, who was later considered a magician and identified by the Greeks with the god Asclepius, "elevated" the pharaoh. It was as if six mastabas were erected in a row. Moreover, the first pyramid occupied an area of \u200b\u200b1125 x 115 meters, with an estimated height of 66 meters (according to Egyptian measures - 1000 "palms"). At first, the architect planned to build a mastaba, but not oblong, but square in plan. Later, it was expanded, but since the extension was made lower, two steps were formed.

This situation did not satisfy the architect, and Imhotep placed three more on the upper platform of a huge flat mastaba, gradually decreasing towards the top. The tomb was under the pyramid.

Several more stepped pyramids are known, but later the builders moved on to building the more familiar tetrahedral pyramids for us. Why, however, not triangular or, say, octahedral? An indirect answer is given by the fact that almost all pyramids are perfectly oriented along the four cardinal points, therefore they have four sides. Moreover, the pyramid was a "house", a shell of a quadrangular burial chamber.

But what caused the angle of inclination of the faces? In the book "The principle of proportions" a whole chapter is devoted to this: "What could determine the angles of inclination of the pyramids." In particular, it is indicated that "the image to which the great pyramids of the Old Kingdom gravitate is a triangle with a right angle at the top.

In space, it is a semi-octahedron: a pyramid in which the edges and sides of the base are equal, the faces are equilateral triangles. "Certain considerations are given on this subject in the books of Hambage, Geek and others.

What is the advantage of the angle of the semi-octahedron? According to the descriptions of archaeologists and historians, some of the pyramids collapsed under their own weight. What was needed was a "longevity angle", the angle most energetically reliable. Purely empirically, this angle can be taken from the vertex angle in a heap of crumbling dry sand. But to get accurate data, you need to use a model. Taking four firmly fixed balls, you need to put the fifth on them and measure the angles of inclination. However, here you can be mistaken, therefore, a theoretical calculation helps out: the centers of the balls should be connected with lines (mentally). At the base, you get a square with a side equal to twice the radius. The square will be just the base of the pyramid, the length of the edges of which will also be equal to twice the radius.

Thus, a dense packing of balls of the 1: 4 type will give us the correct semi-octahedron.

However, why do many pyramids, gravitating towards a similar shape, nevertheless not retain it? The pyramids are probably aging. Contrary to the famous saying:

"Everything in the world is afraid of time, and time is afraid of pyramids", the buildings of the pyramids should grow old, not only external weathering processes can and should take place in them, but also internal "shrinkage" processes, from which the pyramids may become lower. Shrinkage is also possible because, as found out by the works of D. Davidovits, the ancient Egyptians used the technology of making blocks from lime chips, in other words, from "concrete". It is these processes that could explain the reason for the destruction of the Medum pyramid, located 50 km south of Cairo. It is 4600 years old, the dimensions of the base are 146 x 146 m, the height is 118 m. “Why is it so disfigured?” Asks V. Zamarovsky. “Usual references to the destructive influence of time and“ the use of stone for other buildings ”are not suitable here.

After all, most of its blocks and facing slabs have remained in place to this day, in the ruins at its foot. "As we will see, a number of provisions even make one think about the fact that the famous pyramid of Cheops has also" dried up. "In any case, in all ancient images, the pyramids are pointed ...

The shape of the pyramids could also be generated by imitation: some natural patterns, "miraculous perfection", say, some crystals in the form of an octahedron.

Such crystals could be crystals of diamond and gold. A large number of "intersecting" signs are characteristic for concepts such as Pharaoh, Sun, Gold, Diamond. Everywhere - noble, shining (brilliant), great, flawless and so on. The similarities are not accidental.

The solar cult is known to have been an important part of the religion of Ancient Egypt. "No matter how we translate the name of the greatest of the pyramids," says one of the modern manuals - "Khufu's Heaven" or "Khufu Heavenly", it meant that the king is the sun. " If Khufu, in the splendor of his power, imagines himself to be the second sun, then his son Jedef-Ra became the first of the Egyptian kings who began to call himself "the son of Ra", that is, the son of the Sun. The sun in almost all peoples was symbolized by the "solar metal", gold. "Great disk of bright gold" - this is how the Egyptians called our daylight. The Egyptians knew gold perfectly, they knew its native forms, where gold crystals can appear in the form of octahedrons.

As a "sample of forms" the "sun stone" - diamond is also interesting here. The name of the diamond came from the Arab world, "almas" is the hardest, hardest, indestructible. The ancient Egyptians knew diamond and its properties quite well. According to some authors, they even used bronze pipes with diamond cutters for drilling.

South Africa is currently the main supplier of diamonds, but Western Africa is also rich in diamonds. The territory of the Republic of Mali is even called the "Diamond Land" there. Meanwhile, it is on the territory of Mali that the Dogon live, with whom the supporters of the Paleovisite hypothesis pin many hopes (see below). Diamonds could not serve as a reason for the contacts of the ancient Egyptians with this land. However, one way or another, it is possible that it was precisely by copying the octahedrons of diamond and gold crystals that the ancient Egyptians deified thereby "indestructible" like a diamond and "brilliant" like gold pharaohs, the sons of the Sun, comparable only with the most wonderful creations of nature.

Output:

Having studied the pyramid as a geometric body, having become acquainted with its elements and properties, we were convinced of the validity of the opinion about the beauty of the pyramid shape.

As a result of our research, we came to the conclusion that the Egyptians, having collected the most valuable mathematical knowledge, embodied it in the pyramid. Therefore, the pyramid is truly the most perfect creation of nature and man.

BIBLIOGRAPHY

"Geometry: Textbook. for 7 - 9 cl. general education. institutions \\, etc. - 9th ed. - M .: Education, 1999

History of mathematics at school, M: "Education", 1982

Geometry Grade 10-11, M: "Education", 2000

Peter Tompkins "Secrets of the Great Pyramid of Cheops", M: "Tsentropoligraf", 2005

Internet resources

http: // veka-i-mig. ***** /

http: // tambov. ***** / vjpusk / vjp025 / rabot / 33 / index2.htm

http: // www. ***** / enc / 54373.html

Pyramid. Truncated pyramid

Pyramid is called a polyhedron, one of whose faces is a polygon ( base ), and all other faces are triangles with a common vertex ( side faces ) (fig. 15). The pyramid is called correct if its base is a regular polygon and the top of the pyramid is projected to the center of the base (Fig. 16). A triangular pyramid in which all edges are equal is called tetrahedron .

Side rib pyramid is the side of the side face that does not belong to the base Height pyramid is called the distance from its apex to the plane of the base. All lateral edges of a regular pyramid are equal to each other, all lateral edges are equal isosceles triangles. The height of the side face of a regular pyramid drawn from the top is called apothem . Diagonal section the section of the pyramid is called a plane passing through two lateral edges that do not belong to one face.

Side surface area pyramid is called the sum of the areas of all side faces. Full surface area called the sum of the areas of all side faces and the base.

Theorems

1. If in a pyramid all lateral edges are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of the circle circumscribed about the base.

2. If in the pyramid all side edges have equal lengths, then the top of the pyramid is projected into the center of the circle circumscribed about the base.

3. If in the pyramid all the faces are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of the circle inscribed in the base.

To calculate the volume of an arbitrary pyramid, the formula is correct:

where V - volume;

S main - base area;

H - the height of the pyramid.

For the correct pyramid, the formulas are correct:

where p - base perimeter;

h a - apothem;

H - height;

S full

S side

S main - base area;

V - the volume of the correct pyramid.

Truncated pyramid called the part of the pyramid, enclosed between the base and the secant plane, parallel to the base of the pyramid (Fig. 17). Regular truncated pyramid is called the part of a regular pyramid, enclosed between the base and the secant plane parallel to the base of the pyramid.

Foundations truncated pyramids - similar polygons. Side faces - trapezoid. Height a truncated pyramid is the distance between its bases. Diagonal a truncated pyramid is called a segment connecting its vertices that do not lie on the same face. Diagonal section a section of a truncated pyramid is called a plane passing through two lateral edges that do not belong to one face.

For a truncated pyramid, the following formulas are valid:

(4)

where S 1 , S 2 - areas of the upper and lower bases;

S full - total surface area;

S side - lateral surface area;

H - height;

V - the volume of the truncated pyramid.

For a correct truncated pyramid, the formula is correct:

where p 1 , p 2 - base perimeters;

h a - the apothem of the regular truncated pyramid.

Example 1. In a regular triangular pyramid, the dihedral angle at the base is 60º. Find the tangent of the angle of inclination of the side edge to the plane of the base.

Decision. Let's make a drawing (fig. 18).

The pyramid is regular, so at the base there is an equilateral triangle and all side faces are equal isosceles triangles. The dihedral angle at the base is the angle of inclination of the side face of the pyramid to the plane of the base. Linear angle is the angle a between two perpendiculars: and i.e. The top of the pyramid is projected in the center of the triangle (the center of the circumcircle and the inscribed circle in the triangle ABC). The angle of inclination of the lateral rib (for example SB) Is the angle between the edge itself and its projection onto the base plane. For rib SB this angle will be the angle SBD... To find the tangent, you need to know the legs SO and OB... Let the length of the segment BD equals 3 and... Dot ABOUT section BD is divided into parts: and From we find SO: From we find:

Answer:

Example 2. Find the volume of a regular truncated quadrangular pyramid if the diagonals of its bases are cm and cm, and the height is 4 cm.

Decision. To find the volume of the truncated pyramid, we will use formula (4). To find the area of \u200b\u200bthe bases, you need to find the sides of the base squares, knowing their diagonals. The sides of the bases are 2 cm and 8 cm, respectively. So the area of \u200b\u200bthe bases and Having substituted all the data in the formula, we calculate the volume of the truncated pyramid:

Answer: 112 cm 3.

Example 3. Find the area of \u200b\u200bthe side face of a regular triangular truncated pyramid, the sides of the bases of which are 10 cm and 4 cm, and the height of the pyramid is 2 cm.

Decision. Let's make a drawing (fig. 19).

The side face of this pyramid is an isosceles trapezoid. To calculate the area of \u200b\u200ba trapezoid, you need to know the base and height. The bases are given by condition, only the height remains unknown. We will find it from where AND 1 E perpendicular from point AND 1 on the plane of the lower base, A 1 D - perpendicular from AND 1 on AS. AND 1 E \u003d 2 cm, since this is the height of the pyramid. To find DE we will make an additional drawing, in which we will depict a top view (fig. 20). Dot ABOUT - projection of the centers of the upper and lower bases. since (see fig. 20) and On the other hand OK Is the radius of the inscribed circle and OM - radius inscribed in a circle:

MK \u003d DE.

By the Pythagorean theorem from

Side face area:

Answer:

Example 4. At the base of the pyramid lies an isosceles trapezoid, the bases of which andand b (a> b). Each side face forms an angle with the base plane of the pyramid equal to j... Find the total surface area of \u200b\u200bthe pyramid.

Decision. Let's make a drawing (fig. 21). Total surface area of \u200b\u200bthe pyramid SABCD equal to the sum of the areas and area of \u200b\u200bthe trapezoid ABCD.

Let us use the statement that if all the faces of the pyramid are equally inclined to the plane of the base, then the vertex is projected to the center of the circle inscribed in the base. Dot ABOUT - vertex projection S at the base of the pyramid. Triangle SOD is the orthogonal projection of the triangle CSD on the plane of the base. By the theorem on the area of \u200b\u200bthe orthogonal projection of a plane figure, we get:

Similarly, it means Thus, the task was reduced to finding the area of \u200b\u200bthe trapezoid ABCD... Draw a trapezoid ABCDseparately (fig. 22). Dot ABOUT - the center of the circle inscribed in the trapezoid.

Since a circle can be inscribed in a trapezoid, either From, by the Pythagorean theorem, we have