Edge vertex face prism. Prism side surface area

Definition. Prism- this is a polyhedron, all the vertices of which are located in two parallel planes, and in the same two planes there are two faces of the prism, which are equal polygons with respectively 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 side surface of the prism .

All side faces of a prism are parallelograms .

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

Prism Diagonal a segment is called, the ends of which are two vertices of the prism that do not lie on one of its faces (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 the order of the bypass, the vertices of one base are indicated, and then, in the same order, the vertices of the other; the ends of each side edge are indicated by the same letters, only the vertices lying in one base are indicated 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, the base is a pentagon, so the prism is called pentagonal prism. But since such a prism has 7 faces, then it heptahedron(2 faces are the bases of the prism, 5 faces are parallelograms, are its side faces)

Among straight prisms, a particular type stands out: regular prisms.

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

Parallelepiped

cuboid- a right parallelepiped whose base is a rectangle.

Properties and theorems:

,

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

The idea of ​​a prism is given by:

• various architectural structures;
• Kids toys;
• packing boxes;
• designer items, etc.

Total and lateral surface area of ​​the prism

S full \u003d S side + 2S main,

where S full- total surface area, S side- side surface area, S main- base area

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

S side\u003d P main * h,

where S side is the area of ​​the lateral surface of a straight prism,

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

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

Prism Volume

The volume of a prism is equal to the product of the area of ​​the base and the height.

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 right 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

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In the school curriculum for the course of solid geometry, the study of three-dimensional figures usually begins with a simple geometric body - a prism polyhedron. The role of its bases is performed by 2 equal polygons lying in parallel planes. A special case is a regular quadrangular prism. Its bases are 2 identical regular quadrilaterals, to which the sides are perpendicular, having the shape of parallelograms (or rectangles if the prism is not inclined).

What does a prism look like

A regular quadrangular prism is a hexagon, at the bases of which there are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure is a straight parallelepiped.

The figure, which depicts a quadrangular prism, is shown below.

You can also see in the picture the most important elements that make up a geometric body. They are commonly referred to as:

Sometimes in problems in geometry you can find the concept of a section. The definition will sound like this: a section is all points of a volumetric body that belong to the cutting plane. The section is perpendicular (crosses the edges of the figure at an angle of 90 degrees). For a rectangular prism, a diagonal section is also considered (the maximum number of sections that can be built is 2), passing through 2 edges and the diagonals of the base.

If the section is drawn in such a way that the cutting plane is not parallel to either the bases or the side faces, the result is a truncated prism.

Various ratios and formulas are used to find the reduced prismatic elements. Some of them are known from the course of planimetry (for example, to find the area of ​​the base of a prism, it is enough to recall the formula for the area of ​​a square).

Surface area and volume

To determine the volume of a prism using the formula, you need to know the area of ​​\u200b\u200bits base and height:

V = Sprim h

Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in a more detailed form:

V = a² h

If we are talking about a cube - a regular prism with equal length, width and height, the volume is calculated as follows:

To understand how to find the lateral surface area of ​​a prism, you need to imagine its sweep.

It can be seen from the drawing that the side surface is made up of 4 equal rectangles. Its area is calculated as the product of the perimeter of the base and the height of the figure:

Sside = Pos h

Since the perimeter of a square is P = 4a, the formula takes the form:

Sside = 4a h

For cube:

Sside = 4a²

To calculate the total surface area of ​​a prism, add 2 base areas to the side area:

Sfull = Sside + 2Sbase

As applied to a quadrangular regular prism, the formula has the form:

Sfull = 4a h + 2a²

For the surface area of ​​a cube:

Sfull = 6a²

Knowing the volume or surface area, you can calculate the individual elements of a geometric body.

Finding prism elements

Often there are problems in which the volume is given or the value of the lateral surface area is known, where it is necessary to determine the length of the side of the base or the height. In such cases, formulas can be derived:

• base side length: a = Sside / 4h = √(V / h);
• height or side rib length: h = Sside / 4a = V / a²;
• base area: Sprim = V / h;
• side face area: Side gr = Sside / 4.

To determine how much area a diagonal section has, you need to know the length of the diagonal and the height of the figure. For a square d = a√2. Therefore:

Sdiag = ah√2

To calculate the diagonal of the prism, the formula is used:

dprize = √(2a² + h²)

To understand how to apply the above ratios, you can practice and solve a few simple tasks.

Examples of problems with solutions

Here are some of the tasks that appear in the state final exams in mathematics.

Exercise 1.

Sand is poured into a box shaped like a regular quadrangular prism. The height of its level is 10 cm. What will the level of sand be if you move it into a container of the same shape, but with a base length 2 times longer?

It should be argued as follows. The amount of sand in the first and second containers did not change, i.e., its volume in them is the same. You can define the length of the base as a. In this case, for the first box, the volume of the substance will be:

V₁ = ha² = 10a²

For the second box, the length of the base is 2a, but the height of the sand level is unknown:

V₂ = h(2a)² = 4ha²

Because the V₁ = V₂, the expressions can be equated:

10a² = 4ha²

After reducing both sides of the equation by a², we get:

As a result, the new sand level will be h = 10 / 4 = 2.5 cm.

Task 2.

ABCDA₁B₁C₁D₁ is a regular prism. It is known that BD = AB₁ = 6√2. Find the total surface area of ​​the body.

To make it easier to understand which elements are known, you can draw a figure.

Since we are talking about a regular prism, we can conclude that the base is a square with a diagonal of 6√2. The diagonal of the side face has the same value, therefore, the side face also has the shape of a square equal to the base. It turns out that all three dimensions - length, width and height - are equal. We can conclude that ABCDA₁B₁C₁D₁ is a cube.

The length of any edge is determined through the known diagonal:

a = d / √2 = 6√2 / √2 = 6

The total surface area is found by the formula for the cube:

Sfull = 6a² = 6 6² = 216

Task 3.

The room is being renovated. It is known that its floor has the shape of a square with an area of ​​9 m². The height of the room is 2.5 m. What is the lowest cost of wallpapering a room if 1 m² costs 50 rubles?

Since the floor and ceiling are squares, that is, regular quadrangles, and its walls are perpendicular to horizontal surfaces, we can conclude that it is a regular prism. It is necessary to determine the area of ​​its lateral surface.

The length of the room is a = √9 = 3 m.

The square will be covered with wallpaper Sside = 4 3 2.5 = 30 m².

The lowest cost of wallpaper for this room will be 50 30 = 1500 rubles.

Thus, to solve problems for a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and a rectangle, as well as to know the formulas for finding the volume and surface area.

How to find the area of ​​a cube

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