direct prism. Definition of a prism, its elements and types. The main characteristics of the figure. Properties of a regular prism

Stereometry is a branch of geometry that studies figures that do not lie in the same plane. One of the objects of study of stereometry are prisms. In the article we will give a definition of a prism from a geometric point of view, and also briefly list the properties that are characteristic of it.

Geometric figure

The definition of a prism in geometry is as follows: it is a spatial figure consisting of two identical n-gons located in parallel planes, connected to each other by their vertices.

Getting a prism is not difficult. Imagine that there are two identical n-gons, where n is the number of sides or vertices. Let's place them so that they are parallel to each other. After that, the vertices of one polygon should be connected to the corresponding vertices of another. The formed figure will consist of two n-gonal sides, which are called bases, and n quadrangular sides, which in the general case are parallelograms. The set of parallelograms forms the side surface of the figure.

There is another way to geometrically obtain the figure in question. So, if we take an n-gon and transfer it to another plane using parallel segments of equal length, then in the new plane we get the original polygon. Both polygons and all parallel segments drawn from their vertices form a prism.

The picture above shows it so called because its bases are triangles.

The elements that make up a figure

The definition of a prism was given above, from which it is clear that the main elements of a figure are its faces or sides, limiting all the internal points of the prism from external space. Any face of the figure under consideration belongs to one of two types:

• lateral;
• grounds.

There are n side pieces, and they are parallelograms or their particular types (rectangles, squares). In general, the side faces differ from each other. There are only two faces of the base, they are n-gons and are equal to each other. Thus, every prism has n+2 sides.

In addition to the sides, the figure is characterized by its vertices. They are points where three faces touch at the same time. Moreover, two of the three faces always belong to the side surface, and one - to the base. Thus, in a prism there is no specially selected one vertex, as, for example, in a pyramid, all of them are equal. The number of vertices of the figure is 2*n (n pieces for each base).

Finally, the third important element of the prism is its edges. These are segments of a certain length, which are formed as a result of the intersection of the sides of the figure. Like faces, edges also have two different types:

• or formed only by the sides;
• or arise at the junction of the parallelogram and the side of the n-gonal base.

The number of edges is thus 3*n, and 2*n of them belong to the second of the named types.

Prism types

There are several ways to classify prisms. However, they are all based on two features of the figure:

• on the type of n-coal base;
• on side type.

To begin with, let us turn to the second singularity and give a definition of a straight line. If at least one side is a parallelogram of a general type, then the figure is called oblique or oblique. If all parallelograms are rectangles or squares, then the prism will be straight.

You can also give a definition a little differently: a straight figure is a prism in which the side edges and faces are perpendicular to its bases. The figure shows two quadrangular figures. The left one is straight, the right one is oblique.

Now let's move on to classification according to the type of n-gon lying in the bases. It can have the same sides and angles or different. In the first case, the polygon is called regular. If the figure under consideration contains a polygon with equal sides and angles at the base and is a straight line, then it is called regular. According to this definition, a regular prism at its base can have an equilateral triangle, a square, a regular pentagon, or a hexagon, and so on. The listed correct figures are shown in the figure.

Linear parameters of prisms

To describe the dimensions of the figures under consideration, the following parameters are used:

• height;
• sides of the base;
• side rib lengths;
• volumetric diagonals;
• diagonal sides and bases.

For regular prisms, all the named quantities are related to each other. For example, the lengths of the side ribs are the same and equal to the height. For a specific n-gonal regular figure, there are formulas that allow us to determine all the rest from any two linear parameters.

Figure surface

If we turn to the definition of a prism given above, then it will not be difficult to understand what the surface of the figure represents. The surface is the area of ​​all the faces. For a straight prism, it is calculated by the formula:

S = 2*S o + P o *h

where S o is the area of ​​the base, P o is the perimeter of the n-gon at the base, h is the height (distance between the bases).

figure volume

Along with the surface for practice, it is important to know the volume of the prism. It can be determined using the following formula:

This expression is true for absolutely any kind of prism, including those that are oblique and formed by irregular polygons.

For correct, it is a function of the length of the side of the base and the height of the figure. For the corresponding n-gonal prism, the formula for V has a specific form.

Definition 1. Prismatic surface
Theorem 1. On parallel sections of a prismatic surface
Definition 2. Perpendicular section of a prismatic surface
Definition 3. Prism
Definition 4. Prism height
Definition 5. Direct prism
Theorem 2. The area of ​​the lateral surface of the prism

Parallelepiped :
Definition 6. Parallelepiped
Theorem 3. On the intersection of the diagonals of a parallelepiped
Definition 7. Right parallelepiped
Definition 8. Rectangular parallelepiped
Definition 9. Dimensions of a parallelepiped
Definition 10. Cube
Definition 11. Rhombohedron
Theorem 4. On the diagonals of a rectangular parallelepiped
Theorem 5. Volume of a prism
Theorem 6. Volume of a straight prism
Theorem 7. Volume of a rectangular parallelepiped

prism a polyhedron is called, in which two faces (bases) lie in parallel planes, and the edges that do not lie in these faces are parallel to each other.
Faces other than bases are called lateral.
The sides of the side faces and bases are called prism edges, the ends of the edges are called the tops of the prism. Lateral ribs called edges that do not belong to the bases. The union of side faces is called side surface of the prism, and the union of all faces is called full surface of the prism. Prism height called the perpendicular dropped from the point of the upper base to the plane of the lower base or the length of this perpendicular. straight prism called a prism, in which the side edges are perpendicular to the planes of the bases. correct called a straight prism (Fig. 3), at the base of which lies a regular polygon.

Designations:
l - side rib;
P - base perimeter;
S o - base area;
H - height;
P ^ - perimeter of the perpendicular section;
S b - side surface area;
V - volume;
S p - area of ​​the total surface of the prism.

V=SH S p \u003d S b + 2S o S b = P^l

Definition 1 . A prismatic surface is a figure formed by parts of several planes parallel to one straight line limited by those straight lines along which these planes successively intersect one another *; these lines are parallel to each other and are called edges of the prismatic surface.
*It is assumed that every two consecutive planes intersect and that the last plane intersects the first.

Theorem 1 . Sections of a prismatic surface by planes parallel to each other (but not parallel to its edges) are equal polygons.
Let ABCDE and A"B"C"D"E" be sections of a prismatic surface by two parallel planes. To verify that these two polygons are equal, it is enough to show that triangles ABC and A"B"C" are equal and have the same direction of rotation and that the same holds for the triangles ABD and A"B"D", ABE and A"B"E". But the corresponding sides of these triangles are parallel (for example, AC is parallel to A "C") as the lines of intersection of a certain plane with two parallel planes; it follows that these sides are equal (for example, AC equals A"C") as opposite sides of a parallelogram, and that the angles formed by these sides are equal and have the same direction.

Definition 2 . A perpendicular section of a prismatic surface is a section of this surface by a plane perpendicular to its edges. Based on the previous theorem, all perpendicular sections of the same prismatic surface will be equal polygons.

Definition 3 . A prism is a polyhedron bounded by a prismatic surface and two planes parallel to each other (but not parallel to the edges of the prismatic surface)
The faces lying in these last planes are called prism bases; faces belonging to a prismatic surface - side faces; edges of the prismatic surface - side edges of the prism. By virtue of the previous theorem, the bases of the prism are equal polygons. All side faces of the prism parallelograms; all side edges are equal to each other.
It is obvious that if the base of the prism ABCDE and one of the edges AA" are given in magnitude and direction, then it is possible to construct a prism by drawing the edges BB", CC", .., equal and parallel to the edge AA".

Definition 4 . The height of a prism is the distance between the planes of its bases (HH").

Definition 5 . A prism is called a straight line if its bases are perpendicular sections of a prismatic surface. In this case, the height of the prism is, of course, its side rib; side edges will rectangles.
Prisms can be classified by the number of side faces, equal to the number of sides of the polygon that serves as its base. Thus, prisms can be triangular, quadrangular, pentagonal, etc.

Theorem 2 . The area of ​​the lateral surface of the prism is equal to the product of the lateral edge and the perimeter of the perpendicular section.
Let ABCDEA"B"C"D"E" be the given prism and abcde be its perpendicular section, so that the segments ab, bc, .. are perpendicular to its side edges. Face ABA"B" is a parallelogram; its area is equal to the product of the base AA " to a height that matches ab; the area of ​​\u200b\u200bthe face BCV "C" is equal to the product of the base BB" by the height bc, etc. Therefore, the side surface (i.e., the sum of the areas of the side faces) is equal to the product of the side edge, in other words, the total length of the segments AA", BB", .., by the sum ab+bc+cd+de+ea.

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General information about a straight prism

The lateral surface of the prism (more precisely, the lateral surface area) is called sum side face areas. The total surface of the prism is equal to the sum of the lateral surface and the areas of the bases.

Theorem 19.1. The side surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism, i.e., the length of the side edge.

Proof. The side faces of a straight prism are rectangles. The bases of these rectangles are the sides of the polygon lying at the base of the prism, and the heights are equal to the length of the side edges. It follows that the lateral surface of the prism is equal to

S = a 1 l + a 2 l + ... + a n l = pl,

where a 1 and n are the lengths of the ribs of the base, p is the perimeter of the base of the prism, and I is the length of the side ribs. The theorem has been proven.

Practical task

Task (22) . In an inclined prism section, perpendicular to the side edges and intersecting all side edges. Find the side surface of the prism if the perimeter of the section is p and the side edges are l.

Solution. The plane of the section drawn divides the prism into two parts (Fig. 411). Let's subject one of them to a parallel translation that combines the bases of the prism. In this case, we obtain a straight prism, in which the section of the original prism serves as the base, and the side edges are equal to l. This prism has the same side surface as the original one. Thus, the side surface of the original prism is equal to pl.

Generalization of the topic

And now let's try with you to summarize the topic of the prism and remember what properties a prism has.

Prism Properties

First, for a prism, all its bases are equal polygons;
Secondly, for a prism, all its side faces are parallelograms;
Thirdly, in such a multifaceted figure as a prism, all side edges are equal;

Also, it should be remembered that polyhedra such as prisms can be straight and inclined.

What is a straight prism?

If the side edge of a prism is perpendicular to the plane of its base, then such a prism is called a straight line.

It will not be superfluous to recall that the side faces of a straight prism are rectangles.

What is an oblique prism?

But if the side edge of the prism is not located perpendicular to the plane of its base, then we can safely say that this is an inclined prism.

What is the correct prism?

If a regular polygon lies at the base of a straight prism, then such a prism is regular.

Now let's recall the properties that a regular prism has.

Properties of a regular prism

First, regular polygons always serve as the bases of a regular prism;
Secondly, if we consider the side faces of a regular prism, then they are always equal rectangles;
Thirdly, if we compare the sizes of the side ribs, then in the correct prism they are always equal.
Fourth, a regular prism is always straight;
Fifthly, if in a regular prism the side faces are in the form of squares, then such a figure, as a rule, is called a semi-regular polygon.

Prism section

Now let's look at the cross section of a prism:

Homework

And now let's try to consolidate the studied topic by solving problems.

Let's draw an inclined triangular prism, in which the distance between its edges will be: 3 cm, 4 cm and 5 cm, and the side surface of this prism will be equal to 60 cm2. With these parameters, find the lateral edge of the given prism.

Do you know that geometric figures constantly surround us not only in geometry lessons, but also in everyday life there are objects that resemble one or another geometric figure.

Every home, school or work has a computer, the system unit of which is in the form of a straight prism.

If you pick up a simple pencil, you will see that the main part of the pencil is a prism.

Walking along the main street of the city, we see that under our feet lies a tile that has the shape of a hexagonal prism.

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions

Description of the presentation on individual slides:

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Description of the slide:

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Description of the slide:

Definition 1. A polyhedron, two faces of which are polygons of the same name lying in parallel planes, and any two edges not lying in these planes are parallel, is called a prism. The term “prism” is of Greek origin and literally means “sawn off” (body). Polygons lying in parallel planes are called the bases of the prism, and the remaining faces are called lateral faces. The surface of a prism, therefore, consists of two equal polygons (bases) and parallelograms (side faces). There are triangular, quadrangular, pentagonal prisms, etc. depending on the number of base vertices.

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All prisms are divided into straight and inclined. (Fig. 2) If the lateral edge of the prism is perpendicular to the plane of its base, then such a prism is called a straight line; if the lateral edge of the prism is perpendicular to the plane of its base, then such a prism is called inclined. In a straight prism, the side faces are rectangles. The perpendicular to the planes of the bases, the ends of which belong to these planes, is called the height of the prism.

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prism properties. 1. The bases of the prism are equal polygons. 2. The side faces of the prism are parallelograms. 3. The side edges of the prism are equal.

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The surface area of ​​the prism and the lateral surface area of ​​the prism. The surface of a polyhedron consists of a finite number of polygons (faces). The surface area of ​​a polyhedron is the sum of the areas of all its faces. The surface area of ​​the prisms (Spr) is equal to the sum of the areas of its side faces (the area of ​​the lateral surface Sside) and the areas of the two bases (2Sbase) - equal polygons: Ssur=Sside+2Sbase. Theorem. The area of ​​the lateral surface of the prism is equal to the product of the perimeter of its perpendicular section and the length of the lateral edge.

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Proof. The side faces of a straight prism are rectangles, the bases of which are the sides of the base of the prism, and the heights are equal to the height h of the prism. Sside of the prism surface is equal to the sum S of the indicated triangles, i.e. is equal to the sum of the products of the sides of the base and the height h. Taking the factor h out of brackets, we get in brackets the sum of the sides of the base of the prism, i.e. perimeter P. So, Sside = Ph. The theorem has been proven. Consequence. The area of ​​the lateral surface of a straight prism is equal to the product of the perimeter of its base and the height. Indeed, for a straight prism, the base can be considered as a perpendicular section, and the side edge is the height.

7 slide

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Section of a prism 1. Section of a prism by a plane parallel to the base. A polygon is formed in the section, equal to the polygon lying at the base. 2. Section of a prism by a plane passing through two non-neighboring side ribs. A parallelogram is formed in the section. Such a section is called the diagonal section of the prism. In some cases, a rhombus, rectangle or square can be obtained.

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Definition 2. A straight prism whose base is a regular polygon is called a regular prism. Properties of a Regular Prism 1. The bases of a regular prism are regular polygons. 2. The side faces of a regular prism are equal rectangles. 3. The side edges of a regular prism are equal.

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Cross section of a regular prism. 1. Section of a regular prism by a plane parallel to the base. A regular polygon is formed in the section, equal to the polygon lying at the base. 2. Section of a regular prism by a plane passing through two non-adjacent side edges. The section forms a rectangle. In some cases, a square may form.

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Symmetry of a regular prism 1. The center of symmetry with an even number of sides of the base is the point of intersection of the diagonals of a regular prism (Fig. 6)