Lateral faces of a regular pyramid. Pyramid. Formulas and properties of the pyramid

Students encounter the concept of a pyramid long before studying geometry. The fault lies with the famous great Egyptian wonders of the world. Therefore, when starting to study this wonderful polyhedron, most students already clearly imagine it. All the above-mentioned attractions have the correct shape. What's happened regular pyramid, and what properties it has will be discussed further.

Definition

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

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 this was the figure that has a base and planes in the form of triangles, converging at one point.

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

Let's look at it in more detail, what elements does it consist of:

The k-gon is considered the basis of the figure;
3-gonal shapes protrude as the edges of the side part;
the upper part from which the side elements originate is called the apex;
all segments connecting a vertex are called edges;
if a straight line is lowered from the vertex to the plane of the figure at an angle of 90 degrees, then its part contained in the internal space is the height of the pyramid;
in any lateral element, a perpendicular, called an apothem, can be drawn to the side of our polyhedron.

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 such as a pyramid has can be determined using the expression k+1.

Important! A pyramid of regular shape 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:

The basis is a figure of the correct shape.
The edges of the pyramid that limit the side elements have equal numerical values.
The side elements are isosceles triangles.
The base of the height of the figure falls at the center of the polygon, while it is simultaneously the central point of the inscribed and circumscribed.
All side ribs are inclined to the plane of the base at the same angle.
All side surfaces have the same angle of inclination relative to the base.

Thanks to all of the listed properties, performing element calculations is much simpler. Based on the above properties, we pay attention to two signs:

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

The basis is a square

Regular quadrangular pyramid - a polyhedron whose base is a square.

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

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

For example, if it is necessary to relate 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 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 size of all internal faces is also 60 degrees;
any face can act as a base;
, drawn inside the figure, these are equal elements.

Sections of a polyhedron

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

axial;
parallel to the 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 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 dimensions.

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

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 a 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 the axial section, that is, in the trapezoid.

Surface areas

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

There are two types of surface area values:

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

From the name itself it is clear what we are talking about. The side surface includes only the side elements. It follows from this 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:

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

The area of the total surface of the pyramid consists of the sum of the areas of the side planes and the base: Sp.p. = Sside + Sbas.

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

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

What is a regular pyramid in geometry

Properties of a regular quadrangular pyramid

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 the vertices of the polygon, is called a pyramid (Fig. 1).

The polygon from which the pyramid is made is called the base of the pyramid; the resulting triangles, when connected to a 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 that are equal to each other.

Proof.

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

Figure 4.

Consider the triangle $SOA$. According to the Pythagorean theorem, we get

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

The theorem has been proven.

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

Definition 3

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

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

Theorem 2

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

Proof.

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

Since, according to Theorem 1, all sides are equal, then

The theorem has been proven.

Another type of pyramid is a truncated pyramid.

Definition 4

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

Figure 5. Truncated pyramid

The lateral faces of the truncated pyramid are trapezoids.

Theorem 3

The lateral surface area of a regular truncated pyramid is determined as the product of the sum of the semi-perimeters of the bases and the apothem.

Proof.

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

Since all sides are equal, then

The theorem has been proven.

Sample task

Example 1

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

Solution.

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

Then, by Theorem 3, we get

Here you can find basic information about pyramids and related formulas and concepts. All of them are studied with a mathematics tutor in preparation for the Unified State Exam.

Consider a plane, a polygon , lying in it and a point S, not lying in it. Let's 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 the top of the pyramid. Depending on the number n, the pyramid is called triangular (n=3), quadrangular (n=4), pentagonal (n=5) and so on. An alternative name for a triangular pyramid is tetrahedron. The height of a pyramid is the perpendicular descending from its top to the plane of the base.

A pyramid is called regular if a regular polygon, and the base of the pyramid's altitude (the base of the perpendicular) is its center.

Tutor's comment:
Do not confuse the concepts of “regular pyramid” and “regular tetrahedron”. In a regular pyramid, the side edges are not necessarily equal to the edges of the base, but in a regular tetrahedron, all 6 edges 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 a base height, so a regular tetrahedron is a regular pyramid.

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

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

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

Formula for the volume of a pyramid:
1) , where is the area of the base of the pyramid, and is the height of the pyramid
2) , where is the radius of the inscribed sphere, and is the area of the total surface of the pyramid.
3) , where MN is the distance between any two crossing edges, and is the area of the parallelogram formed by the midpoints of the four remaining edges.

Property of the base of the height of a pyramid:

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 to the base
3) All apothems are equally inclined to the height of the pyramid
4) The height of the pyramid is equally inclined to all side faces

Math tutor's comment: Please note that all points are united by one common property: one way or another, lateral faces are involved everywhere (apothems are their elements). Therefore, the tutor can offer a less accurate, but more convenient for learning, formulation: point P coincides with the center of the inscribed circle, the base of the pyramid, if there is any equal information about its lateral faces. To prove it, it is enough to show that all apothem triangles are equal.

Point P coincides with the center of a circle circumscribed 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 to the base
3) All side ribs are equally inclined to the height

Introduction

When we started studying stereometric figures, we touched on the topic “Pyramid”. We liked this topic because the pyramid is very often used in architecture. And since our future profession of architecture is inspired by this figure, we think that she can push us towards excellent projects.

The strength of architectural structures is their most important quality. Linking strength, firstly, with the materials from which they are created, and, secondly, with the features of design solutions, it turns out that the strength of a structure is directly related to the geometric shape that is basic for it.

In other words, we are talking about a geometric figure that can be considered as a model of the corresponding architectural form. It turns out that geometric shape also determines the strength of an architectural structure.

Since ancient times, the Egyptian pyramids have been considered the most durable architectural structures. As you know, they have the shape of regular quadrangular pyramids.

It is this geometric shape that provides the greatest stability due to the large base area. On the other hand, the pyramid shape ensures that the mass decreases as the height above the ground increases. It is these two properties that make the pyramid stable, and therefore strong under the conditions of gravity.

Project goal: learn something new about pyramids, deepen your knowledge and find practical application.

To achieve this goal, it was necessary to solve the following tasks:

· Learn historical information about the pyramid

· Consider the pyramid as a geometric figure

· Find application in life and architecture

· Find similarities and differences between pyramids located in different parts of the world

Theoretical part

Historical information

Pyramid geometry began in Ancient Egypt and Babylon, but was actively developed in Ancient Greece. The first to establish the volume of the pyramid was Democritus, and Eudoxus of Cnidus proved it. The ancient Greek mathematician Euclid systematized knowledge about the pyramid in the XII volume of his “Elements”, and also derived the first definition of a pyramid: a solid figure bounded by planes that converge from one plane to one point.

Tombs of Egyptian pharaohs. The largest of them - the pyramids of Cheops, Khafre and Mikerin in El Giza - were considered one of the Seven Wonders of the World in ancient times. The construction of the pyramid, in which the Greeks and Romans already saw a monument to the unprecedented pride of kings and cruelty that doomed the entire people of Egypt to meaningless construction, was the most important cult act and was supposed to express, apparently, the mystical identity of the country and its ruler. The population of the country worked on the construction of the tomb during the part of the year free from agricultural work. A number of texts testify to the attention and care that the kings themselves (albeit of a later time) paid to the construction of their tomb and its builders. It is also known about the special cult honors that were given to the pyramid itself.

Basic Concepts

Pyramid is a polyhedron whose base is a polygon, and the remaining faces are triangles that have a common vertex.

Apothem- the height of the side face of a regular pyramid, drawn from its vertex;

Side faces- triangles meeting at a vertex;

Side ribs- common sides of the side faces;

Top of the pyramid- a point connecting the side ribs and not lying in the plane of the base;

Height- a perpendicular segment drawn through the top of the pyramid to the plane of its base (the ends of this segment are the top of the pyramid and the base of the perpendicular);

Diagonal section of a pyramid- section of the pyramid passing through the top and diagonal of the base;

Base- a polygon that does not belong to the vertex of the pyramid.

Basic properties of a regular pyramid

The lateral edges, lateral faces and apothems are respectively equal.

The dihedral angles at the base are equal.

The dihedral angles at the lateral edges are equal.

Each height point is equidistant from all the vertices of the base.

Each height point is equidistant from all side faces.

Basic pyramid formulas

The area of the lateral and total surface of the pyramid.

The area of the lateral surface of a pyramid (full and truncated) is the sum of the areas of all its lateral faces, the total surface area is the sum of the areas of all its faces.

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

p- base perimeter;

h- apothem.

The area of the lateral and full surfaces of a truncated pyramid.

p 1, p 2 - base perimeters;

h- apothem.

R- total surface area of a regular truncated pyramid;

S side- area of the lateral surface of a regular truncated pyramid;

S 1 + S 2- base area

Pyramid volume

Form volume ula is used for pyramids of any kind.

H- height of the pyramid.

Pyramid corners

The angles formed by the side face and the base of the pyramid are called dihedral angles at the base of the pyramid.

A dihedral angle is formed by two perpendiculars.

To determine this angle, you often need to use the three perpendicular theorem.

The angles formed by the lateral edge and its projection onto the base plane are called angles between the side edge and the plane of the base.

The angle formed by two lateral edges is called dihedral angle at the lateral edge of the pyramid.

The angle formed by two lateral edges of one face of the pyramid is called angle at the top of the pyramid.

Pyramid sections

The surface of a pyramid is the surface of a polyhedron. Each of its faces is a plane, therefore the section of a pyramid defined by a cutting plane is a broken line consisting of individual straight lines.

Diagonal section

The section of a pyramid by a plane passing through two lateral edges that do not lie on the same face is called diagonal section pyramids.

Parallel sections

Theorem:

If the pyramid is intersected by a plane parallel to the base, then the lateral edges and heights of the pyramid are divided by this plane into proportional parts;

The section of this plane is a polygon similar to the base;

The areas of the section and the base are related to each other as the squares of their distances from the vertex.

Types of pyramid

Correct pyramid– a pyramid whose base is a regular polygon, and the top of the pyramid is projected into the center of the base.

For a regular pyramid:

1. side ribs are equal

2. side faces are equal

3. apothems are equal

4. dihedral angles at the base are equal

5. dihedral angles at the lateral edges are equal

6. each point of height is equidistant from all vertices of the base

7. each height point is equidistant from all side edges

Truncated pyramid- part of the pyramid enclosed between its base and a cutting plane parallel to the base.

The base and corresponding section of a truncated pyramid are called bases of a truncated pyramid.

A perpendicular drawn from any point of one base to the plane of another is called the height of a truncated pyramid.

Tasks

No. 1. In a regular quadrangular pyramid, point O is the center of the base, SO=8 cm, BD=30 cm. Find the side edge SA.

Problem solving

No. 1. In a regular pyramid, all faces and edges are equal.

Consider OSB: OSB is a rectangular rectangle, because.

SB 2 =SO 2 +OB 2

SB 2 =64+225=289

Pyramid in architecture

A pyramid is a monumental structure in the form of an ordinary regular geometric pyramid, in which the sides converge at one point. According to their functional purpose, pyramids in ancient times were places of burial or cult worship. The base of a pyramid can be triangular, quadrangular, or in the shape of a polygon with an arbitrary number of vertices, but the most common version is the quadrangular base.

There are a considerable number of pyramids built by different cultures of the Ancient World, mainly as temples or monuments. Large pyramids include the Egyptian pyramids.

All over the Earth you can see architectural structures in the form of pyramids. The pyramid buildings are reminiscent of ancient times and look very beautiful.

Egyptian pyramids are the greatest architectural monuments of Ancient Egypt, including one of the “Seven Wonders of the World”, the Pyramid of Cheops. From the foot to the top it reaches 137.3 m, and before it lost the top, its height was 146.7 m

The radio station building in the capital of Slovakia, resembling an inverted pyramid, was built in 1983. In addition to offices and service premises, inside the volume there is a fairly spacious concert hall, which has one of the largest organs in Slovakia.

The Louvre, which is “silent, unchanged and majestic, like a pyramid,” has undergone many changes over the centuries before becoming the greatest museum in the world. It was born as a fortress, erected by Philip Augustus in 1190, which soon became a royal residence. In 1793 the palace became a museum. Collections are enriched through bequests or purchases.

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 the vertices of the polygon, is called a pyramid (Fig. 1).

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 that are equal to each other.

Proof.

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

Figure 4.

Consider the triangle $SOA$. According to the Pythagorean theorem, we get

The theorem has been proven.

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

Definition 3

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

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

Theorem 2

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

Proof.

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

Since, according to Theorem 1, all sides are equal, then

The theorem has been proven.

Another type of pyramid is a truncated pyramid.

Definition 4

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

Figure 5. Truncated pyramid

The lateral faces of the truncated pyramid are trapezoids.

Theorem 3

The lateral surface area of a regular truncated pyramid is determined as the product of the sum of the semi-perimeters of the bases and the apothem.

Proof.

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

Since all sides are equal, then

The theorem has been proven.

Sample task

Example 1

Solution.

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

Then, by Theorem 3, we get