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 define 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. Let's 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 this, the vertices of one polygon should be connected to the corresponding vertices of the other. The resulting figure will consist of two n-gonal sides, which are called bases, and n quadrangular sides, which in general are parallelograms. The set of parallelograms forms the lateral 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 will get the original polygon. Both polygons and all parallel segments drawn from their vertices form a prism.

The picture above demonstrates this. It is called so because its bases are triangles.

Elements that make up a figure

Above, the definition of a prism was given, from which it is clear that the main elements of the figure are its edges or sides, which limit all the internal points of the prism from the external space. Any face of the figure in question belongs to one of two types:

• lateral;
• grounds.

There are n lateral 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 represent points where three faces touch simultaneously. 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 allocated one vertex, as, for example, in a pyramid, they are all equal. The number of vertices of the figure is 2*n (n pieces for each base).

Finally, the third important element of a prism is its ribs. These are segments of a certain length that are formed as a result of the intersection of the sides of a 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 equal to 3*n, and 2*n of them belong to the second of the named types.

Types of prisms

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

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

First, let's turn to the second feature and give a definition of a straight line. If at least one side is a general parallelogram, then the figure is called oblique or oblique. If all parallelograms are rectangles or squares, then the prism will be straight.

The definition can also be given a little differently: a straight figure is a prism whose side edges and faces are perpendicular to its bases. The figure shows two quadrangular figures. The left one is straight, the right one is inclined.

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

Linear parameters of prisms

To describe the sizes of the figures in question, the following parameters are used:

• height;
• sides of the base;
• length of lateral ribs;
• volumetric diagonals;
• diagonals of the sides and bases.

For regular prisms, all these 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 you to determine all the others using any two linear parameters.

Surface of a figure

If we refer to the definition of a prism given above, then it will not be difficult to understand what the surface of the figure represents. Surface is the area of ​​all 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 (the 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 valid for absolutely any type of prism, including those that are inclined and formed by irregular polygons.

For correct ones, 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.

Lecture: Prism, its bases, side ribs, height, lateral surface; straight prism; correct prism

Prism

If you learned flat figures with us from previous questions, then you are completely ready to study three-dimensional figures. The first solid we will learn will be a prism.

Prism is a three-dimensional body that has a large number of faces.

This figure has two polygons at the bases, which are located in parallel planes, and all the side faces have the shape of a parallelogram.

Fig. 1. Fig. 2

So, let's figure out what a prism consists of. To do this, pay attention to Fig. 1

As mentioned earlier, a prism has two bases that are parallel to each other - these are the pentagons ABCEF and GMNJK. Moreover, these polygons are equal to each other.

All other faces of the prism are called lateral faces - they consist of parallelograms. For example BMNC, AGKF, FKJE, etc.

The total surface of all lateral faces is called lateral surface.

Each pair of adjacent faces has a common side. This common side is called an edge. For example MV, SE, AB, etc.

If the upper and lower base of the prism are connected by a perpendicular, then it will be called the height of the prism. In the figure, the height is marked as straight line OO 1.

There are two main types of prism: oblique and straight.

If the lateral edges of the prism are not perpendicular to the bases, then such a prism is called inclined.

If all the edges of a prism are perpendicular to the bases, then such a prism is called straight.

If the bases of a prism contain regular polygons (those with equal sides), then such a prism is called correct.

If the bases of a prism are not parallel to each other, then such a prism will be called truncated.

You can see it in Fig. 2

Formulas for finding the volume and area of ​​a prism

There are three basic formulas for finding volume. They differ from each other in application:

Similar formulas for finding the surface area of ​​a prism:

General information about straight prism

The lateral surface of a prism (more precisely, the lateral surface area) is called sum areas of the side faces. 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 lateral 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 lateral 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 base edges, p is the perimeter of the base of the prism, and I is the length of the side edges. The theorem has been proven.

Practical task

Problem (22) . In an inclined prism it is carried out section, perpendicular to the side ribs and intersecting all the side ribs. Find the lateral surface of the prism if the perimeter of the section is equal to p and the side edges are equal to l.

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

Summary of the covered topic

Now let’s try to summarize the topic we covered about prisms and remember what properties a prism has.

Prism properties

Firstly, a prism has all its bases as equal polygons;
Secondly, in a prism all its lateral faces are parallelograms;
Thirdly, in such a multifaceted figure as a prism, all lateral edges are equal;

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

Which prism is called a straight prism?

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

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

What type of prism is called oblique?

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

Which prism is called correct?

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

Now let us remember the properties that a regular prism has.

Properties of a regular prism

Firstly, regular polygons always serve as the bases of a regular prism;
Secondly, if we consider the side faces of a regular prism, they are always equal rectangles;
Thirdly, if you compare the sizes of the side ribs, then in a regular prism they are always equal.
Fourthly, a correct prism is always straight;
Fifthly, if in a regular prism the lateral faces have the shape of squares, then such a figure is usually called a semi-regular polygon.

Prism cross section

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

Homework

Now let's try to consolidate the topic we've learned by solving problems.

Let's draw an inclined triangular prism, the distance between its edges will be equal to: 3 cm, 4 cm and 5 cm, and the lateral surface of this prism will be equal to 60 cm2. Having these parameters, find the side edge of this 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 whose system unit is shaped like 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 central 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

The base of the prism can be any polygon - triangle, quadrangle, etc. Both bases are absolutely identical, and accordingly, with which the corners of parallel edges are connected to each other, are always parallel. At the base of a regular prism lies a regular polygon, that is, one in which all sides are equal. In a straight prism, the ribs between the side faces are perpendicular to the base. In this case, the base of a straight prism can contain a polygon with any number of angles. A prism whose base is a parallelogram is called a parallelepiped. A rectangle is a special case of a parallelogram. If this figure lies at the base, and the side faces are located at right angles to the base, the parallelepiped is called rectangular. The second name for this geometric body is rectangular.

How does she look

There are quite a lot of rectangular prisms in the environment of modern man. This is, for example, ordinary cardboard for shoes, computer components, etc. Look around. Even in a room you will probably see many rectangular prisms. This includes a computer case, a bookcase, a refrigerator, a wardrobe, and many other items. The shape is extremely popular mainly because it allows you to make the most of your space, whether you're decorating your interior or packing things into cardboard before moving.

Properties of a rectangular prism

A rectangular prism has a number of specific properties. Any pair of faces can serve as it, since all adjacent faces are located at the same angle to each other, and this angle is 90°. The volume and surface area of ​​a rectangular prism are easier to calculate than any other. Take any object that has the shape of a rectangular prism. Measure its length, width and height. To find the volume, just multiply these measurements. That is, the formula looks like this: V=a*b*h, where V is the volume, a and b are the sides of the base, h is the height that coincides with the side edge of this geometric body. The base area is calculated using the formula S1=a*b. For the side surface, you must first calculate the perimeter of the base using the formula P=2(a+b), and then multiply it by the height. The resulting formula is S2=P*h=2(a+b)*h. To calculate the total surface area of ​​a rectangular prism, add twice the base area and the side surface area. The formula is S=2S1+S2=2*a*b+2*(a+b)*h=2

Definition.

This is a hexagon, the bases of which are two equal squares, and the side faces are equal rectangles

Side rib- is the common side of two adjacent side faces

Prism height- this is a segment perpendicular to the bases of the prism

Prism diagonal- a segment connecting two vertices of the bases that do not belong to the same face

Diagonal plane- a plane that passes through the diagonal of the prism and its lateral edges

Diagonal section- the boundaries of the intersection of the prism and the diagonal plane. The diagonal cross section of a regular quadrangular prism is a rectangle

Perpendicular section (orthogonal section)- this is the intersection of a prism and a plane drawn perpendicular to its lateral edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are indicated by the corresponding letters:

• The bases ABCD and A 1 B 1 C 1 D 1 are equal and parallel to each other
• Side faces AA 1 D 1 D, AA 1 B 1 B, BB 1 C 1 C and CC 1 D 1 D, each of which is a rectangle
• Lateral surface - the sum of the areas of all lateral faces of the prism
• Total surface - the sum of the areas of all bases and side faces (sum of the area of ​​the side surface and bases)
• Side ribs AA 1, BB 1, CC 1 and DD 1.
• Diagonal B 1 D
• Base diagonal BD
• Diagonal section BB 1 D 1 D
• Perpendicular section A 2 B 2 C 2 D 2.

Properties of a regular quadrangular prism

• The bases are two equal squares
• The bases are parallel to each other
• The side faces are rectangles
• The side edges are equal to each other
• Side faces are perpendicular to the bases
• The lateral ribs are parallel to each other and equal
• Perpendicular section perpendicular to all side ribs and parallel to the bases
• Angles of perpendicular section - straight
• The diagonal cross section of a regular quadrangular prism is a rectangle
• Perpendicular (orthogonal section) parallel to the bases

Formulas for a regular quadrangular prism

Instructions for solving problems

When solving problems on the topic " regular quadrangular prism" means that:

Correct prism- a prism at the base of which lies a regular polygon, and the side edges are perpendicular to the planes of the base. That is, a regular quadrangular prism contains at its base square. (see properties of a regular quadrangular prism above) Note. This is part of a lesson with geometry problems (section stereometry - prism). Here are problems that are difficult to solve. If you need to solve a geometry problem that is not here, write about it in the forum. To denote the action of extracting the square root in solving problems, the symbol is used√ .

Task.

In a regular quadrangular prism, the base area is 144 cm 2 and the height is 14 cm. Find the diagonal of the prism and the total surface area.

Solution.
A regular quadrilateral is a square.
Accordingly, the side of the base will be equal

√144 = 12 cm.
From where the diagonal of the base of a regular rectangular prism will be equal to
√(12 2 + 12 2 ) = √288 = 12√2

The diagonal of a regular prism forms a right triangle with the diagonal of the base and the height of the prism. Accordingly, according to the Pythagorean theorem, the diagonal of a given regular quadrangular prism will be equal to:
√((12√2) 2 + 14 2 ) = 22 cm

Answer: 22 cm

Task

Determine the total surface of a regular quadrangular prism if its diagonal is 5 cm and the diagonal of its side face is 4 cm.

Solution.
Since the base of a regular quadrangular prism is a square, we find the side of the base (denoted as a) using the Pythagorean theorem:

A 2 + a 2 = 5 2
2a 2 = 25
a = √12.5

The height of the side face (denoted as h) will then be equal to:

H 2 + 12.5 = 4 2
h 2 + 12.5 = 16
h 2 = 3.5
h = √3.5

The total surface area will be equal to the sum of the lateral surface area and twice the base area

S = 2a 2 + 4ah
S = 25 + 4√12.5 * √3.5
S = 25 + 4√43.75
S = 25 + 4√(175/4)
S = 25 + 4√(7*25/4)
S = 25 + 10√7 ≈ 51.46 cm 2.

Answer: 25 + 10√7 ≈ 51.46 cm 2.