Different prisms are different from each other. At the same time, they have a lot in common. To find the area of ​​the base of the prism, you will need to understand what type it has.

General theory

A prism is any polyhedron whose sides have the shape of a parallelogram. Moreover, its base can be any polyhedron - from a triangle to an n-gon. Moreover, the bases of the prism are always equal to each other. What does not apply to the side faces is that they can vary significantly in size.

When solving problems, not only the area of ​​the base of the prism is encountered. It may require knowledge of the lateral surface, that is, all the faces that are not bases. The complete surface will be the union of all the faces that make up the prism.

Sometimes problems involve height. It is perpendicular to the bases. The diagonal of a polyhedron is a segment that connects in pairs any two vertices that do not belong to the same face.

It should be noted that the base area of ​​a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same figures on the top and bottom faces, then their areas will be equal.

Triangular prism

It has at its base a figure with three vertices, that is, a triangle. As you know, it can be different. If so, it is enough to remember that its area is determined by half the product of the legs.

The mathematical notation looks like this: S = ½ av.

To find out the area of ​​the base in general, the formulas are useful: Heron and the one in which half of the side is taken by the height drawn to it.

The first formula should be written as follows: S = √(р (р-а) (р-в) (р-с)). This notation contains a semi-perimeter (p), that is, the sum of three sides divided by two.

Second: S = ½ n a * a.

If you want to find out the area of ​​the base of a triangular prism, which is regular, then the triangle turns out to be equilateral. There is a formula for it: S = ¼ a 2 * √3.

Quadrangular prism

Its base is any of the known quadrangles. It can be a rectangle or square, parallelepiped or rhombus. In each case, in order to calculate the area of ​​the base of the prism, you will need your own formula.

If the base is a rectangle, then its area is determined as follows: S = ab, where a, b are the sides of the rectangle.

When it comes to a quadrangular prism, the area of ​​the base of a regular prism is calculated using the formula for a square. Because it is he who lies at the foundation. S = a 2.

In the case when the base is a parallelepiped, the following equality will be needed: S = a * n a. It happens that the side of a parallelepiped and one of the angles are given. Then, to calculate the height, you will need to use an additional formula: n a = b * sin A. Moreover, angle A is adjacent to side “b”, and height n is opposite to this angle.

If there is a rhombus at the base of the prism, then to determine its area you will need the same formula as for a parallelogram (since it is a special case of it). But you can also use this: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.

Regular pentagonal prism

This case involves dividing the polygon into triangles, the areas of which are easier to find out. Although it happens that figures can have a different number of vertices.

Since the base of the prism is a regular pentagon, it can be divided into five equilateral triangles. Then the area of ​​the base of the prism is equal to the area of ​​one such triangle (the formula can be seen above), multiplied by five.

Regular hexagonal prism

Using the principle described for a pentagonal prism, it is possible to divide the hexagon of the base into 6 equilateral triangles. The formula for the base area of ​​such a prism is similar to the previous one. Only it should be multiplied by six.

The formula will look like this: S = 3/2 a 2 * √3.

Tasks

No. 1. Given a regular straight line, its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of ​​the base of the prism and the entire surface.

Solution. The base of the prism is a square, but its side is unknown. You can find its value from the diagonal of the square (x), which is related to the diagonal of the prism (d) and its height (h). x 2 = d 2 - n 2. On the other hand, this segment “x” is the hypotenuse in a triangle whose legs are equal to the side of the square. That is, x 2 = a 2 + a 2. Thus it turns out that a 2 = (d 2 - n 2)/2.

Substitute the number 22 instead of d, and replace “n” with its value - 14, it turns out that the side of the square is 12 cm. Now just find out the area of ​​the base: 12 * 12 = 144 cm 2.

To find out the area of ​​the entire surface, you need to add twice the base area and quadruple the side area. The latter can be easily found using the formula for a rectangle: multiply the height of the polyhedron and the side of the base. That is, 14 and 12, this number will be equal to 168 cm 2. The total surface area of ​​the prism turns out to be 960 cm 2.

Answer. The area of ​​the base of the prism is 144 cm 2. The entire surface is 960 cm 2.

No. 2. Given At the base there is a triangle with a side of 6 cm. In this case, the diagonal of the side face is 10 cm. Calculate the areas: the base and the side surface.

Solution. Since the prism is regular, its base is an equilateral triangle. Therefore, its area turns out to be equal to 6 squared, multiplied by ¼ and the square root of 3. A simple calculation leads to the result: 9√3 cm 2. This is the area of ​​one base of the prism.

All side faces are the same and are rectangles with sides of 6 and 10 cm. To calculate their areas, just multiply these numbers. Then multiply them by three, because the prism has exactly that many side faces. Then the area of ​​the lateral surface of the wound turns out to be 180 cm 2.

Answer. Areas: base - 9√3 cm 2, lateral surface of the prism - 180 cm 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 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

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.

From Latin as “something sawn off.” This polyhedron always has two bases, which are located in parallel planes and are equal polygons. They can be triangular, quadrangular, or n-gonal.

Remember that the number of remaining (side) faces depends on the type of base. If the base is a triangle, there will be three side faces, a quadrilateral will have four, and so on.

Keep in mind that the ribs the side edge is located at an angle of 90° to the base, the prism is called a straight line. Otherwise - inclined. If the line prisms at the base there will be a regular polygon, it will turn into a regular prism. An example of such a geometric figure is a cube.

To calculate the perimeter of a prism, find the perimeters of the bases and side faces of the prism, and add all the dimensions together. To do this, use a ruler to measure the length of the sides (or edges) of each face. And calculate the perimeter of each polygon.

Simplify your task. Since both bases are the same size, measure the edge lengths of only one of them. Add up the dimensions of all sides and multiply the resulting sum by two.

If the bases have edges of equal size, find the number of identical side faces. Measure the lengths of the sides of one of these faces and calculate its perimeter. Multiply the resulting value by the total number of identical faces.

Separately, calculate the perimeter of each of those side faces that are never repeated.

Add up all the calculated perimeters - two bases, repeating side faces, and those side faces that have no analogue. The total amount will be equal to the perimeter of the prism.

note

The calculation of the perimeter does not depend on the type of prism. It is calculated the same for both straight and inclined prisms.

Sources:

• Prisms

Journalists from the online publication Forbes found that the Department of Internal Policy under the presidential administration began tracking and monitoring the social activity of Russians on the Internet using the Prism terminal. This system has already been installed in the office of the head of the Department, Vyacheslav Voloshin.

The developer of the terminal is the Medialogy company; its website states that the system is designed to track the activity of users of social systems and is capable of processing information flows from 60 million sources in real time. The topics of interest to the user can be any and are configured manually. In particular, the developers claim that the terminal is capable of tracking an increase in the activity of social network users, which is fraught with an increase in social tension. Issues that the system can control include: extremism, participation in riots and unauthorized rallies, protest sentiments, discussion of rising prices, housing and communal services tariffs, salaries and pensions, and the level of medical care.

Prizma terminals operate based on linguistic and semantic analysis of posts on forums and blogs. The system can track both individual blogs and social media accounts. The ones used allow you to analyze and diagnose the positive or negative tone of statements with an error of only 2-3%.

The most current and discussed news on social networks is displayed on the user’s monitor; they are presented in clusters of top stories. If desired, you can find out from which blogs and posts this or that “” news or topic was compiled. For each story, an assessment is given based on the nature of the statements, while both the number of positive and negative assessments are reflected on the monitor. The list of their authors can also be found. The dynamics of statements and assessments can be presented in the form of a graph.

But the system also has weaknesses that are determined by the specifics of network communication. Thus, the use of the notorious “Albanian” language can make it unsuitable for machine perception and subsequent analysis. The same applies to sarcastic, ironic and “quoted” statements, however, sometimes it is not possible to recognize them.

Video on the topic

Sources:

• how terminals work

In mid-August 2012, the online publication Forbes published information on its website that the Kremlin began monitoring social networks using Prism terminals installed in the offices of senior government officials. Despite the assurances of Dmitry Medvedev, who met with United Russia activists, that the government is not interested in the opinions of social network users, the very fact of using such terminals indicates the opposite.

The West already has experience of tracking the political sentiments of the active part of society through social networks. Thus, in the United States, Twitter runs a microblogging service that compares the number of positive and negative reviews about a particular participant in the election campaign with the total number of published entries. Every week, about two million records about Barack Obama or Mitt Romney are analyzed.

The developers of a system similar to the Western one, the Prizma terminal, is the Mediology company. She claims that the development capabilities are quite high - in real time it is possible to process information coming simultaneously from 60 million sources. “Prism” is able to track the dynamics of changes in the number of positive or negative reviews for a particular event, taking into account artificial increases that arise as a result of bot attacks.

The topics selected for statistical samples are configured manually. Information leaked from the Department of Internal Policy of the Presidential Administration claims that the terminal installed there allows you to track the progress of discussions on social networks and blogs on LiveJournal, Twitter, YouTube. A source in the Presidential Administration, whom Forbes calls reliable, claims that monitoring blogs is taken very seriously; the terminal is installed directly in the office of the head of the Office, Vyacheslav Volodin.

The developers’ website claims that using the Prizma terminal it is possible to monitor user activity and determine the degree of social media activity that can lead to increased political and social tension. The system monitors the increase in protest and extremist sentiments, discussions about increasing the price level, housing and communal services problems, discussions of issues related to salaries and pensions, corruption, the level of medical care, etc.

This interest of the authorities in what worries Internet users, who are becoming more and more every year, is, of course, pleasing. The only open question is to what extent they will be able to correctly use the information they receive, and how ready the authorities will be to solve the problems posed to them by the part of the country’s population that uses social networks.

Video on the topic

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