Which prism is called correct? Regular quadrangular prism

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

Prism

If you have learned plane figures from previous questions with us, then you are completely ready to study volumetric figures. The first solid we will learn will be a prism.

Prism is a volumetric body that has large number faces.

This figure has two polygons at its bases, which are located in parallel planes, and all 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 direct.

If the bases of the prism lie regular polygons(those whose sides are equal), 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:

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
Full 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 indicate the action of retrieving square root the symbol is used in solving problems√ .

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 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

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.

Definition. Prism is a polyhedron, all of whose vertices are located in two parallel planes, and in these same two planes lie two faces of the prism, which are equal polygons with correspondingly parallel sides, and all edges that do not lie in these planes are parallel.

Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).

All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).

All side faces form lateral surface of the prism .

All lateral faces of the prism are parallelograms .

The edges that do not lie at the bases are called the lateral edges of the prism ( AA 1, BB 1, CC 1, DD 1, EE 1).

Prism diagonal is a segment whose ends are two vertices of a prism that do not lie on the same face (AD 1).

The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called prism height .

Designation:ABCDE A 1 B 1 C 1 D 1 E 1. (First, in traversal order, the vertices of one base are indicated, and then, in the same order, the vertices of another; the ends of each side edge are designated by the same letters, only the vertices lying in one base are designated by letters without an index, and in the other - with an index)

The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1 there is a pentagon at the base, so the prism is called pentagonal prism. But because such a prism has 7 faces, then it heptahedron(2 faces - the bases of the prism, 5 faces - parallelograms, - its side faces)

Among straight prisms, it stands out private view: correct prisms.

A straight prism is called correct, if its bases are regular polygons.

A regular prism has all lateral faces equal rectangles. A special case of a prism is a parallelepiped.

Parallelepiped

Parallelepiped is a quadrangular prism, at the base of which lies a parallelogram (an inclined parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.

Rectangular parallelepiped- a right parallelepiped whose base is a rectangle.

Properties and theorems:

Some properties of a parallelepiped are similar known properties parallelogram. A rectangular parallelepiped having equal measurements, are called cube .All faces of a cube are equal squares. The square of the diagonal is equal to the sum of the squares of its three dimensions

where d is the diagonal of the square;
a is the side of the square.

An idea of a prism is given by:

various architectural structures;
children's toys;
packaging boxes;
designer items, etc.

The area of the total and lateral surface of the prism

Total surface area of the prism is the sum of the areas of all its faces Lateral surface area is called the sum of the areas of its lateral faces. The bases of the prism are equal polygons, then their areas are equal. That's why

S full = S side + 2S main,

Where S full- total surface area, S side-lateral surface area, S base- base area

The lateral surface area of a straight prism is equal to the product of the perimeter of the base and the height of the prism.

S side= P basic * h,

Where S side-area of the lateral surface of a straight prism,

P main - perimeter of the base of a straight prism,

h is the height of the straight prism, equal to the side edge.

Prism volume

Prism volume equal to the product base area to height.

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. 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 for this geometric body is rectangular.

What does she look like

Rectangular prisms surrounded modern man quite a few. 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

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 is 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 cross-sectional perimeter 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's 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 shapes constantly surround us not only in geometry lessons, but also in everyday life There are objects that resemble one or another geometric figure.