Lateral edge of this prism. Regular quadrangular prism

Lecture: Prism, its bases, side edges, height, side surface; straight prism; right prism

Prism

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

Prism is a voluminous body that has a large number of faces.

This figure has two polygons at the bases, which are located in parallel planes, and all side faces are in the form 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, the 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 side faces - they consist of parallelograms. For example, BMNC, AGKF, FKJE, etc.

The common surface of all side faces is called side surface.

Each pair of adjacent faces has a common side. Such a common side is called an edge. For example, MB, CE, AB, etc.

If the upper and lower bases 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 a straight line OO 1.

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

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

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

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

If the bases of the 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 volume, area of a prism

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

Similar formulas for finding the surface area of a prism:

Any polygon can lie at the base of the prism - a triangle, a quadrilateral, etc. Both bases are exactly the same, and accordingly, by which the angles of parallel faces are connected to each other, they 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 edges between the side faces are perpendicular to the base. In this case, a polygon with any number of angles can lie at the base of a straight prism. A prism whose base is a parallelogram is called a parallelepiped. Rectangle - special case 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 of this geometric body is rectangular.

How she looks like

There are quite a lot of rectangular prisms in the environment of modern man. This, for example, is the usual cardboard from under shoes, computer components, etc. Look around. Even in a room, you will surely see many rectangular prisms. This is a computer case, and a bookcase, and a refrigerator, and a cabinet, and many other items. The form is extremely popular mainly because it allows you to use the space as efficiently as possible, whether you are decorating the interior or packing things in cardboard before moving.

Properties of a rectangular prism

A rectangular prism has a number of specific properties. Any pair of faces can serve as its, since all adjacent faces are located at the same angle to each other, and this angle is 90 °. Volume and surface area rectangular prism 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, it is enough to multiply these measurements. That is, the formula looks like this: V \u003d 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 by the formula S1=a*b. To get 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. It turns out the formula S2=P*h=2(a+b)*h. To calculate the total surface area of a rectangular prism, add twice the area of the base and the area of the side surface. 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 is a line 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 side edges

Diagonal section- the boundaries of the intersection of the prism and the diagonal plane. The diagonal 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 side edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are marked with the corresponding letters:

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 the side faces of the prism
Full surface- the sum of the areas of all bases and side faces (the 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 sides are rectangles.
Side faces are equal to each other
Side faces are perpendicular to the bases
Lateral ribs are parallel to each other and equal
Perpendicular section perpendicular to all side ribs and parallel to the bases
Perpendicular Section Angles - Right
The diagonal 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" implies 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 above the properties of a regular quadrangular prism) Note. This is part of the lesson with tasks in geometry (section solid geometry - prism). Here are the tasks that cause difficulties in solving. If you need to solve a problem in geometry, which is not here - write about it in the forum. To indicate the action of extracting square root symbol is used in problem solving√ .

A 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 to

√144 = 12 cm.
Whence 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 with the diagonal of the base and the height of the prism right triangle. 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

A task

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

Solution.
Since the base of a regular quadrangular prism is a square, then the side of the base (denoted as a) is found by 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 \u003d 4 2
h 2 + 12.5 = 16
h 2 \u003d 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 \u003d 25 + 10√7 ≈ 51.46 cm 2.

Answer: 25 + 10√7 ≈ 51.46 cm 2.

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 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 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 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, use following parameters:

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 correct 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.

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.

And 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.