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What is a regular quadrangular pyramid. The main properties of the correct pyramid

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

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Definition

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

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

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

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

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

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

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

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

Basic properties

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

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

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

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

The square is based

Regular quadrangular pyramid - a polyhedron based on a square.

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

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

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

Based on a regular triangle

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

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

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

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

Sections of a polyhedron

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

  • axial;
  • parallel basis.

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

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

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

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

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

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

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

Surface areas

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

There are two types of surface area:

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

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

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

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

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

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

What is a regular pyramid in geometry

Properties of a regular quadrangular pyramid

A three-dimensional figure that often appears in geometric problems is a pyramid. The simplest of all the figures of this class is triangular. In this article, we will analyze in detail the basic formulas and properties of the correct

Geometric representations of the figure

Before proceeding to consider the properties of a regular triangular pyramid, let's take a closer look at which figure in question.

Let's assume that there is an arbitrary triangle in three-dimensional space. We choose any point in this space that does not lie in the plane of the triangle, and connect it to three vertices of the triangle. We got a triangular pyramid.

It consists of 4 sides, all of which are triangles. The points where three faces meet are called vertices. The figure also has four of them. The intersection lines of two faces are edges. The pyramid under consideration has 6 ribs. The figure below shows an example of this figure.

Since the figure is formed by four sides, it is also called a tetrahedron.

Correct pyramid

Above, an arbitrary figure with a triangular base was considered. Now suppose we draw a perpendicular line from the top of the pyramid to its base. This segment is called the height. It is obvious that it is possible to spend 4 different heights for the figure. If the height intersects the triangular base in the geometric center, then such a pyramid is called a straight pyramid.

A straight pyramid whose base is an equilateral triangle is called a regular pyramid. For her, all three triangles that form the lateral surface of the figure are isosceles and equal to each other. A special case of a regular pyramid is the situation when all four sides are equilateral identical triangles.

Consider the properties of a regular triangular pyramid and give the appropriate formulas for calculating its parameters.

Base side, height, lateral edge and apothem

Any two of the listed parameters uniquely determine the other two characteristics. We give formulas that connect the named quantities.

Suppose that the side of the base of a regular triangular pyramid is a. The length of its side edge is equal to b. What will be the height of a regular triangular pyramid and its apothem?

For the height h we get the expression:

This formula follows from the Pythagorean theorem for which are the side edge, the height and 2/3 of the height of the base.

The apothem of a pyramid is the height for any lateral triangle. The length of apotema a b is:

a b \u003d √ (b 2 - a 2 / 4)

From these formulas it can be seen that whatever the side of the base of a triangular regular pyramid and the length of its lateral edge, the apotema will always be more height pyramids.

The two formulas presented contain all four linear characteristics the figure in question. Therefore, from the known two of them, you can find the rest by solving the system from the written equalities.

figure volume

For absolutely any pyramid (including an inclined one), the value of the volume of space bounded by it can be determined by knowing the height of the figure and the area of ​​its base. The corresponding formula looks like:

Applying this expression to the figure in question, we obtain the following formula:

Where the height of a regular triangular pyramid is h and its base side is a.

It is not difficult to obtain a formula for the volume of a tetrahedron, in which all sides are equal to each other and represent equilateral triangles. In this case, the volume of the figure is determined by the formula:

That is, it is uniquely determined by the length of side a.

Surface area

We continue to consider the properties of a triangular regular pyramid. total area of all the faces of a figure is called its surface area. It is convenient to study the latter by considering the corresponding development. The figure below shows what a regular triangular pyramid looks like.

Suppose we know the height h and the side of the base a of the figure. Then the area of ​​its base will be equal to:

Every student can get this expression if he remembers how to find the area of ​​a triangle, and also takes into account that the height of an equilateral triangle is also a bisector and a median.

The area of ​​the lateral surface formed by three identical isosceles triangles is:

S b = 3/2*√(a 2 /12+h 2)*a

This equality follows from the expression of the apotema of the pyramid in terms of the height and length of the base.

The total surface area of ​​the figure is:

S = S o + S b = √3/4*a 2 + 3/2*√(a 2 /12+h 2)*a

Note that for a tetrahedron, in which all four sides are the same equilateral triangles, the area S will be equal to:

Properties of a regular truncated triangular pyramid

If the top of the considered triangular pyramid is cut off by a plane parallel to the base, then the remaining Bottom part will be called a truncated pyramid.

In the case of a triangular base, as a result of the section method described, a new triangle is obtained, which is also equilateral, but has a smaller side length than the base side. A truncated triangular pyramid is shown below.

We see that this figure is already limited by two triangular bases and three isosceles trapezoids.

Suppose that the height of the resulting figure is h, the lengths of the sides of the lower and upper bases are a 1 and a 2, respectively, and the apothem (height of the trapezoid) is equal to a b. Then the surface area of ​​the truncated pyramid can be calculated by the formula:

S = 3/2*(a 1 +a 2)*a b + √3/4*(a 1 2 + a 2 2)

Here the first term is the area of ​​the lateral surface, the second term is the area of ​​the triangular bases.

The volume of the figure is calculated as follows:

V = √3/12*h*(a 1 2 + a 2 2 + a 1 *a 2)

To unambiguously determine the characteristics of a truncated pyramid, it is necessary to know its three parameters, which is demonstrated by the above formulas.

A triangular pyramid is a pyramid based on a triangle. The height of this pyramid is the perpendicular, which is lowered from the top of the pyramid to its bases.

Finding the Height of a Pyramid

How to find the height of a pyramid? Very simple! To find the height of any triangular pyramid, you can use the volume formula: V = (1/3)Sh, where S is the base area, V is the volume of the pyramid, h is its height. From this formula, derive the height formula: to find the height of a triangular pyramid, you need to multiply the volume of the pyramid by 3, and then divide the resulting value by the base area, it will be: h \u003d (3V) / S. Since the base of a triangular pyramid is a triangle, you can use the formula for calculating the area of ​​a triangle. If we know: the area of ​​the triangle S and its side z, then according to the area formula S=(1/2)γh: h = (2S)/γ, where h is the height of the pyramid, γ is the edge of the triangle; the angle between the sides of the triangle and the two sides themselves, then using the following formula: S = (1/2)γφsinQ, where γ, φ are the sides of the triangle, we find the area of ​​the triangle. The value of the sine of the angle Q must be viewed in the table of sines, which is on the Internet. Next, we substitute the area value into the height formula: h = (2S)/γ. If the task requires calculating the height of a triangular pyramid, then the volume of the pyramid is already known.

Regular triangular pyramid

Find the height of a regular triangular pyramid, i.e. a pyramid in which all faces are equilateral triangles, knowing the size of the edge γ. In this case, the edges of the pyramid are the sides of equilateral triangles. The height of a regular triangular pyramid will be: h = γ√(2/3), where γ is the edge of an equilateral triangle, h is the height of the pyramid. If the area of ​​the base (S) is unknown, and only the length of the edge (γ) and the volume (V) of the polyhedron are given, then the necessary variable in the formula from the previous step must be replaced by its equivalent, which is expressed in terms of the length of the edge. The area of ​​a triangle (regular) is equal to 1/4 of the product of the length of the side of this triangle, squared by the square root of 3. We substitute this formula instead of the base area in the previous formula, and we get the following formula: h \u003d 3V4 / (γ 2 √3) = 12V/(γ 2 √3). The volume of a tetrahedron can be expressed in terms of the length of its edge, then all variables can be removed from the formula for calculating the height of a figure and only the side of the triangular face of the figure can be left. The volume of such a pyramid can be calculated by dividing by 12 from the product the length of its face cubed by the square root of 2.

We substitute this expression into the previous formula, we get the following formula for calculating: h = 12(γ 3 √2/12)/(γ 2 √3) = (γ 3 √2)/(γ 2 √3) = γ√(2 /3) = (1/3)γ√6. Also, a regular triangular prism can be inscribed in a sphere, and knowing only the radius of the sphere (R), you can find the very height of the tetrahedron. The edge length of the tetrahedron is: γ = 4R/√6. We replace the variable γ with this expression in the previous formula and obtain the formula: h = (1/3)√6(4R)/√6 = (4R)/3. The same formula can be obtained by knowing the radius (R) of a circle inscribed in a tetrahedron. In this case, the length of the edge of the triangle will be equal to 12 ratios between square root of 6 and radius. We substitute this expression into the previous formula and have: h = (1/3)γ√6 = (1/3)√6(12R)/√6 = 4R.

How to find the height of a regular quadrangular pyramid

To answer the question of how to find the length of the height of the pyramid, you need to know what a regular pyramid is. A quadrangular pyramid is a pyramid based on a quadrilateral. If in the conditions of the problem we have: the volume (V) and the area of ​​\u200b\u200bthe base (S) of the pyramid, then the formula for calculating the height of the polyhedron (h) will be as follows - divide the volume multiplied by 3 by the area S: h \u003d (3V) / S. With a square base of a pyramid with known: given volume (V) and side length γ, replace the area (S) in the previous formula with the square of the side length: S = γ 2 ; H = 3V/γ 2 . The height of the regular pyramid h = SO passes just through the center of the circle, which is circumscribed near the base. Since the base of this pyramid is a square, the point O is the point of intersection of the diagonals AD and BC. We have: OC = (1/2)BC = (1/2)AB√6. Further, we find in a right triangle SOC (according to the Pythagorean theorem): SO = √(SC 2 -OC 2). Now you know how to find the height of a regular pyramid.

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

  1. 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.
  2. Geometry. Grade 10-11: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill.
  3. 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.
  1. Internet portal "Yaklass" ()
  2. Internet portal "Festival of Pedagogical Ideas "First of September" ()
  3. Internet portal "Slideshare.net" ()

Homework

  1. Can a regular polygon be the base of an irregular pyramid?
  2. Prove that non-intersecting edges of a regular pyramid are perpendicular.
  3. 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.
  4. RAVS is a regular triangular pyramid. Construct the linear angle of the dihedral angle at the base of the pyramid.

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

Target: examining the pyramid geometric body, to explain the perfection of its form.

Tasks:

1. Give a mathematical definition of a 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 the unique shape of the pyramid be explained mathematically?

3. What explains the geometric wonders of the pyramid?

4. What explains the perfection of the shape of the pyramid?

Definition of a pyramid.

PYRAMID (from Greek pyramis, genus n. 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 corners of the base, pyramids are triangular, quadrangular, etc.

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

It is possible that Greek word"pyramid" comes from the Egyptian expression per-em-us, i.e., from a term that meant the height of the pyramid. The 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. Butuzova and others, we learned that: A polyhedron composed of n-gon A1A2A3 ... An and n triangles RA1A2, RA2A3, ..., RAnA1 is called a pyramid. The polygon A1A2A3 ... An is the base of the pyramid, and the triangles RA1A2, RA2A3, ..., PAnA1 are the lateral faces of the pyramid, P is the top of the pyramid, the segments RA1, RA2, ..., RAn are the lateral edges.

However, such a definition of the 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 solid figure bounded by planes that converge from one plane to one point.

But this definition has been criticized already in antiquity. So Heron suggested the following definition pyramids: "This 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 studied these definitions and found the definition of Adrien Marie Legendre, who in 1794 in his work “Elements of Geometry” defines the pyramid as follows: “Pyramid is a bodily figure formed by triangles converging at one point and ending on different sides of a flat base.”

It seems to us that last definition gives a clear idea of ​​\u200b\u200bthe pyramid, since it talks about 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.”

Pyramid as a geometric body.

That. A pyramid is a polyhedron, one of whose faces (base) is a polygon, the other faces (sides) are triangles that have one common vertex (the top 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 right pyramid, at the base of which is a regular polygon and truncated pyramid.

In the figure - the pyramid PABCD, ABCD - its base, PO - height.

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

Sfull = Sside + Sbase, where Sside is the sum of the areas of the side faces.

pyramid volume is found according to the formula:

V=1/3Sbase h, where Sosn. - base area h- height.
The axis of a regular pyramid is a straight line containing its height.
Apothem ST - the height of the side face of a regular pyramid.

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

1) side edges 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="287" height="151">

The bases of the truncated pyramid are similar polygons ABCD and A`B`C`D`, side faces are trapezoids.

Height truncated pyramid - the distance between the bases.

Truncated volume pyramid is found by the formula:

V=1/3 h(S + https://pandia.ru/text/78/390/images/image019_2.png" align="left" width="91" height="96"> The lateral surface area of ​​a regular truncated pyramid is expressed as follows: Sside. = ½(P+P') h, where P and P’ are the perimeters of the bases, h- the height of the side face (the apothem of a regular truncated by feasts

Sections of the pyramid.

Sections of the pyramid by planes passing through its top 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 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 pyramid section and designate it;

construct a straight line passing through given point and the resulting intersection point;

· Repeat these steps for the next faces.

, which corresponds to the ratio of the legs of a right triangle 4:3. This ratio of the legs corresponds to the well-known right triangle with sides 3:4:5, 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 what is born from both.

For a triangle 3:4:5, the equality is true: 32 + 42 = 52, which expresses the Pythagorean theorem. Isn't this the theorem they wanted to perpetuate Egyptian priests, building a pyramid based on a 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 Egyptian pyramids sought to impress distant descendants with the depth of their knowledge, and they achieved this by choosing as the "main geometric idea" for the pyramid of Cheops - "golden" right triangle, and for the Khafre pyramid - the "sacred" or "Egyptian" triangle.

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

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

Algebraic finding of the Golden section of a segment AB = a reduces to solving the equation a: x = x: (a - x), whence x is approximately equal to 0.62a. The x ratio can be expressed as fractions 2/3, 3/5, 5/8, 8/13, 13/21…= 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 connected, DE \u003d BE is postponed, and, finally, AC \u003d AD, then the equality AB is fulfilled: CB = 2: 3.

golden ratio often used in works of art, architecture, 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 have a width to length ratio close to 0.618. Considering the arrangement of leaves on a common stem of plants, one can notice that between every two pairs of leaves, the third is located in the place of the Golden Ratio (slides). Each of us “wears” the Golden Ratio with us “in our hands” - this is the ratio of the phalanges of the fingers.

Thanks to the discovery of several mathematical papyri, Egyptologists have learned something about the ancient Egyptian systems of calculus and measures. The tasks contained in them were solved by scribes. One of the most famous is the Rhind Mathematical Papyrus. By studying these puzzles, Egyptologists learned how the ancient Egyptians coped with various quantities that arose in the calculation of measures of weight, length and volume, in which fractions were often used, as well as how they dealt with angles.

The ancient Egyptians used a method of 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 slope gradient was expressed as a ratio of an integer, called "seked". In 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 plane of the base, measured by a nth number of horizontal units per vertical unit of elevation. Thus, this unit of measure is equivalent to our modern cotangent of the angle of inclination. Therefore, the Egyptian word "seked" is related to our modern word"gradient"".

The numerical key to the pyramids lies in the ratio of their height to the base. AT in practical terms- this is the easiest way to make templates necessary for constantly checking 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, hence the differences in the angles of inclination for each pyramid. But there could be another reason. Perhaps they all wanted to embody different symbolic associations hidden in different proportions. However, the angle of Khafre's pyramid (based on the triangle (3:4:5) appears in the three problems presented by the pyramids in the Rhind 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's say that the length of the hypotenuse 5 was never mentioned. But mathematical problems concerning the pyramids are always solved on the basis of the seked angle - the ratio of the height to the 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 no doubt known to the ancient Egyptians. It is possible that these ratios for each pyramid were chosen arbitrarily. However, this contradicts the importance attached to numerical symbolism in all types of Egyptian visual arts. It is very likely that such relationships were of significant importance, since they expressed specific religious ideas. In other words, the whole complex of Giza was subject to a coherent design, designed to reflect some kind of divine theme. This would explain why the designers chose different angles for the three pyramids.

In The Secret of Orion, Bauval and Gilbert presented convincing evidence of the connection of the pyramids of Giza with the constellation of Orion, in particular with the stars of Orion's Belt. The same 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 occupied by Great Pyramid of Pharaoh Cheops (Khufu). 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).

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

Most researchers agree that the length of the side of the base of the pyramid, for example, GF is equal to L\u003d 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.

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 pyramid of Cheops is truncated. Its upper platform today has a size of approximately 10 ´ 10 m, and a century ago it was equal to 6 ´ 6 m. It is obvious that the top of the pyramid was dismantled, and it does not correspond to the original one.

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

What was the original height of the pyramid? This height can be recreated if you find the basic "geometric idea" of the pyramid.


Figure 2.

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

And here the researchers were in for a big surprise!.png" width="25" height="24">= 1.272. Comparing this value with the tg value 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 minute of arc, then the value a will become equal to 1.272, that is, it will coincide with the value of . It should be noted that in 1840 G. Wise repeated his measurements and clarified that the value of the angle a=51°50".

These measurements have led researchers to the following very interesting hypothesis: the triangle ASV of the pyramid of Cheops was based on the relation AC / CB = = 1,272!

Consider now a right triangle ABC, in which the ratio of legs AC / CB= (Fig.2). If now the lengths of the sides of the rectangle ABC denote by x, y, z, and also take into account that the ratio y/x= , then, in accordance with the Pythagorean theorem, the length z can be calculated by the formula:

If accept x = 1, y= https://pandia.ru/text/78/390/images/image027_1.png" width="143" height="27">


Figure 3"Golden" right triangle.

A right triangle in which the sides are related as t: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 per unit, that is: CB= 1. But then the length of the side of the base of the pyramid GF= 2, and the area of ​​the base EFGH will be equal to SEFGH = 4.

Let us now calculate the area of ​​the side face of the Cheops pyramid SD. Because the height AB triangle AEF is equal to t, then the area of ​​the side face will be equal to SD = t. Then the total area of ​​all four side faces of the pyramid will be equal to 4 t, and the ratio of the total external area of ​​the pyramid to the base area will be equal to the golden ratio! That's what it is - the main geometric secret of the pyramid of Cheops!

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

As a rule, they are obtained in search of some "constant", in particular, the number "pi" (Ludolf number), equal to 3.14159...; bases of natural logarithms "e" (Napier's number) equal to 2.71828...; the number "F", the number of the "golden section", equal, for example, 0.618 ... etc..

You can name, for example: 1) Property of Herodotus: (Height) 2 \u003d 0.5 st. main x Apothem; 2) Property of V. Price: Height: 0.5 st. osn \u003d Square root of "Ф"; 3) Property of M. Eist: Perimeter of the base: 2 Height = "Pi"; in a different interpretation - 2 tbsp. main : Height = "Pi"; 4) G. Reber's property: Radius of the inscribed circle: 0.5 st. main = "F"; 5) Property of K. Kleppish: (St. main.) 2: 2 (st. main. x Apothem) \u003d (st. main. W. Apothem) \u003d 2 (st. main. x Apothem) : ((2 st. main X Apothem) + (st. main) 2). Etc. You can come up with a lot of such properties, especially if you connect two adjacent pyramids. For example, as "Properties of A. Arefiev" it can be mentioned that the difference between the volumes of the pyramid of Cheops and the pyramid of Khafre is equal to twice the volume of the pyramid of Menkaure...

Many interesting positions, in particular, about the construction of pyramids according to the "golden section" are described in the books of D. Hambidge "Dynamic Symmetry in Architecture" and M. Geek "Aesthetics of Proportion in Nature and Art". Recall that the "golden section" is the division of the segment in such a ratio, when part A is as many times greater 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" == 1.618. .. The use of the "golden section" is indicated not only in individual pyramids, but in the entire pyramid complex in Giza.

The most curious thing, however, is that the same pyramid of Cheops simply "cannot" accommodate so many miraculous properties. Taking a certain property one by one, you can "adjust" it, but all at once they do not fit - they do not coincide, they contradict each other. Therefore, if, for example, when checking all properties, one and the same side of the base of the pyramid (233 m) is initially taken, 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 those of 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. A "miracle" should be considered only something obviously impossible for the ancient Egyptians. This, in particular, includes "cosmic" miracles, in which the measurements of the pyramid of Cheops or the pyramid complex in 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" relations.

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 you use the "Egyptian elbow" invented by him, then the side of the pyramid will correspond to "the most accurate duration solar year, expressed to the nearest billionth of a 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 the height of 146.6 m is usually taken, Smith took it as 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.

Last curious statement:

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

So, despite the skepticism, let's note the well-known harmony of the construction of statements: 1) the height of the pyramid, as a line "going 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 earth's 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 bee language, analyzed by Karl von Frisch. However, we refrain from commenting on this for now.

SHAPE OF THE PYRAMIDS

The famous tetrahedral shape of the pyramids did not appear immediately. The Scythians made burials in the form of earthen hills - barrows. The Egyptians built "hills" of stone - pyramids. This happened for the first time after the unification of Upper and Lower Egypt, in the 28th century BC, when the founder of the III dynasty, Pharaoh Djoser (Zoser), faced the task of strengthening the unity of the country.

And here, according to historians, the "new concept of deification" of the tsar played an important role in strengthening the central power. Although the royal burials were distinguished by greater splendor, they did not differ in principle from the tombs of 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 a small building of large stone blocks was then placed - "mastaba" (in Arabic - "bench"). On the site of the mastaba of his predecessor, Sanakht, Pharaoh Djoser erected the first pyramid. It was stepped and was a visible transitional stage from one architectural form to another, from a mastaba to a pyramid.

In this way, the pharaoh was "raised" by the sage and architect Imhotep, who was later considered a magician and identified by the Greeks with the god Asclepius. It was as if six mastabas were erected in a row. Moreover, the first pyramid occupied an area of ​​1125 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, as it were.

This situation did not satisfy the architect, and on the top platform of a huge flat mastaba, Imhotep placed three more, 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 more familiar tetrahedral pyramids. Why, however, not triangular or, say, octagonal? An indirect answer is given by the fact that almost all the pyramids are perfectly oriented to the four cardinal points, and therefore have four sides. In addition, 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 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 Hambidge, Geek and others.

What is the advantage of the angle of the semioctahedron? According to the descriptions of archaeologists and historians, some pyramids collapsed under their own weight. What was needed was a "durability angle", an angle that was the most energetically reliable. Purely empirically, this angle can be taken from the vertex angle in a pile of crumbling dry sand. But to get accurate data, you need to use the model. Taking four firmly fixed balls, you need to put the fifth one on them and measure the angles of inclination. However, here you can make a mistake, therefore, a theoretical calculation helps out: you should connect the centers of the balls 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 a regular semi-octahedron.

However, why do many pyramids, gravitating towards a similar form, nevertheless do not retain it? Probably the pyramids are getting old. Contrary to the famous saying:

"Everything in the world is afraid of time, and time is afraid of the pyramids", the buildings of the pyramids must age, they can and should take place not only the processes of external weathering, but also the processes of internal "shrinkage", 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 mutilated?” asks V. Zamarovsky. “The usual references to the destructive effects of time and “the use of stone for other buildings” do not fit here.

After all, most of its blocks and facing slabs have remained in place to this day, in ruins at its foot. "As we will see, a number of provisions make one think even about the fact that famous pyramid Cheops also "shrunken". 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 diamond and gold crystals. Characteristically a large number of"intersecting" signs for such concepts as Pharaoh, Sun, Gold, Diamond. Everywhere - noble, brilliant (brilliant), great, flawless and so on. The similarities are not accidental.

The solar cult, as you know, was an important part of the religion. ancient egypt. "No matter how we translate the name of the greatest of the pyramids, - it is noted in one of the modern manuals - "Sky Khufu" or "Sky Khufu", it meant that the king is the sun. If Khufu, in the brilliance of his power, imagined himself a 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 was symbolized by almost all peoples as "solar metal", gold. "Big disk of bright gold" - so the Egyptians called our daylight. The Egyptians knew gold very well, they knew its native forms, where gold crystals can appear in the form of octahedrons.

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

Currently, the main supplier of diamonds is South Africa, but West 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 paleovisit hypothesis pin many hopes (see below). Diamonds could not be the reason for the contacts of the ancient Egyptians with this region. However, one way or another, but, it is possible that it was precisely by copying the octahedrons of diamond and gold crystals that the ancient Egyptians deified thereby "indestructible" like diamond and "brilliant" like gold pharaohs, the sons of the Sun, comparable only to the most wonderful creations nature.

Conclusion:

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

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

BIBLIOGRAPHY

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

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

Geometry grade 10-11, M: "Enlightenment", 2000

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

Internet resources

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

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

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

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