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 quadrangles, 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 hexahedron, 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

Insofar as 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 quadrilaterals, 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 on 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

With the help of this video tutorial, everyone will be able to independently get acquainted with the topic “The concept of a polyhedron. Prism. Prism surface area. During the lesson, the teacher will talk about what geometric shapes such as a polyhedron and prisms are, give the appropriate definitions and explain their essence with specific examples.

With the help of this lesson, everyone will be able to independently get acquainted with the topic “The concept of a polyhedron. Prism. Prism surface area.

Definition. A surface composed of polygons and bounding a certain geometric body will be called a polyhedral surface or a polyhedron.

Consider the following examples of polyhedra:

1. Tetrahedron ABCD is a surface made up of four triangles: ABC, adb, bdc and ADC(Fig. 1).

Rice. one

2. Parallelepiped ABCDA 1 B 1 C 1 D 1 is a surface composed of six parallelograms (Fig. 2).

Rice. 2

The main elements of a polyhedron are faces, edges, vertices.

The faces are the polygons that make up the polyhedron.

Edges are sides of faces.

The vertices are the ends of the edges.

Consider a tetrahedron ABCD(Fig. 1). Let us indicate its main elements.

Facets: triangles ABC, ADB, BDC, ADC.

Ribs: AB, AC, BC, DC, AD, BD.

Peaks: A, B, C, D.

Consider a box ABCDA 1 B 1 C 1 D 1(Fig. 2).

Facets: parallelograms AA 1 D 1 D, D 1 DCC 1, BB 1 C 1 C, AA 1 B 1 B, ABCD, A 1 B 1 C 1 D 1 .

Ribs: AA 1 , BB 1 , SS 1 , DD 1 , AD, A 1 D 1 , B 1 C 1 , BC, AB, A 1 B 1 , D 1 C 1 , DC.

Peaks: A, B, C, D, A 1 ,B 1 ,C 1 ,D 1 .

An important special case of a polyhedron is a prism.

ABSA 1 IN 1 WITH 1(Fig. 3).

Rice. 3

Equal Triangles ABC and A 1 B 1 C 1 are located in parallel planes α and β so that the edges AA 1 , BB 1 , SS 1 are parallel.

I.e ABSA 1 IN 1 WITH 1- triangular prism, if:

1) Triangles ABC and A 1 B 1 C 1 are equal.

2) Triangles ABC and A 1 B 1 C 1 located in parallel planes α and β: ABC║A 1 B 1 C (α ║ β).

3) Ribs AA 1 , BB 1 , SS 1 are parallel.

ABC and A 1 B 1 C 1- the base of the prism.

AA 1 , BB 1 , SS 1- side ribs of the prism.

If from an arbitrary point H 1 one plane (for example, β) drop the perpendicular HH 1 onto the plane α, then this perpendicular is called the height of the prism.

Definition. If the lateral edges are perpendicular to the bases, then the prism is called straight, otherwise it is called oblique.

Consider a triangular prism ABSA 1 IN 1 WITH 1(Fig. 4). This prism is straight. That is, its side edges are perpendicular to the bases.

For example, rib AA 1 perpendicular to the plane ABC. Edge AA 1 is the height of this prism.

Rice. 4

Note that the side face AA 1 V 1 V perpendicular to the bases ABC and A 1 B 1 C 1, since it passes through the perpendicular AA 1 to the foundations.

Now consider an inclined prism ABSA 1 IN 1 WITH 1(Fig. 5). Here the lateral edge is not perpendicular to the plane of the base. If we drop from the point A 1 perpendicular A 1 H on the ABC, then this perpendicular will be the height of the prism. Note that the segment AN is the projection of the segment AA 1 to the plane ABC.

Then the angle between the line AA 1 and plane ABC is the angle between the line AA 1 and her AN projection onto a plane, that is, the angle A 1 AN.

Rice. 5

Consider a quadrangular prism ABCDA 1 B 1 C 1 D 1(Fig. 6). Let's see how it turns out.

1) Quadrilateral ABCD equal to a quadrilateral A 1 B 1 C 1 D 1: ABCD = A 1 B 1 C 1 D 1.

2) Quadrangles ABCD and A 1 B 1 C 1 D 1 ABC║A 1 B 1 C (α ║ β).

3) Quadrangles ABCD and A 1 B 1 C 1 D 1 arranged so that the lateral ribs are parallel, that is: AA 1 ║BB 1 ║SS 1 ║DD 1.

Definition. The diagonal of a prism is a segment that connects two vertices of a prism that do not belong to the same face.

For example, AC 1- diagonal of a quadrangular prism ABCDA 1 B 1 C 1 D 1.

Definition. If the side edge AA 1 perpendicular to the plane of the base, then such a prism is called a straight line.

Rice. 6

A special case of a quadrangular prism is the known parallelepiped. Parallelepiped ABCDA 1 B 1 C 1 D 1 shown in fig. 7.

Let's see how it works:

1) Equal figures lie in the bases. In this case - equal parallelograms ABCD and A 1 B 1 C 1 D 1: ABCD = A 1 B 1 C 1 D 1.

2) Parallelograms ABCD and A 1 B 1 C 1 D 1 lie in parallel planes α and β: ABC║A 1 B 1 C 1 (α ║ β).

3) Parallelograms ABCD and A 1 B 1 C 1 D 1 arranged in such a way that the side ribs are parallel to each other: AA 1 ║BB 1 ║SS 1 ║DD 1.

Rice. 7

From a point A 1 drop the perpendicular AN to the plane ABC. Line segment A 1 H is the height.

Consider how a hexagonal prism is arranged (Fig. 8).

1) Equal hexagons lie at the base ABCDEF and A 1 B 1 C 1 D 1 E 1 F 1: ABCDEF= A 1 B 1 C 1 D 1 E 1 F 1.

2) Planes of hexagons ABCDEF and A 1 B 1 C 1 D 1 E 1 F 1 parallel, that is, the bases lie in parallel planes: ABC║A 1 B 1 C (α ║ β).

3) Hexagons ABCDEF and A 1 B 1 C 1 D 1 E 1 F 1 arranged so that all side edges are parallel to each other: AA 1 ║BB 1 …║FF 1.

Rice. eight

Definition. If any side edge is perpendicular to the plane of the base, then such a hexagonal prism is called a straight line.

Definition. A right prism is called regular if its bases are regular polygons.

Consider a regular triangular prism ABSA 1 IN 1 WITH 1.

Rice. nine

triangular prism ABSA 1 IN 1 WITH 1- correct, this means that regular triangles lie at the bases, that is, all sides of these triangles are equal. Also, this prism is straight. This means that the side edge is perpendicular to the plane of the base. And this means that all side faces are equal rectangles.

So if a triangular prism ABSA 1 IN 1 WITH 1 is correct, then:

1) The side edge is perpendicular to the plane of the base, that is, it is the height: AA 1 ⊥ ABC.

2) The base is a regular triangle: ∆ ABC- right.

Definition. The total surface area of ​​a prism is the sum of the areas of all its faces. Denoted S full.

Definition. The area of ​​the lateral surface is the sum of the areas of all lateral faces. Denoted S side.

The prism has two bases. Then the total surface area of ​​the prism is:

S full \u003d S side + 2S main.

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.

The proof will be carried out on the example of a triangular prism.

Given: ABSA 1 IN 1 WITH 1- direct prism, i.e. AA 1 ⊥ ABC.

AA 1 = h.

Prove: S side \u003d R main ∙ h.

Rice. ten

Proof.

triangular prism ABSA 1 IN 1 WITH 1- straight, so AA 1 B 1 B, AA 1 C 1 C, BB 1 C 1 C - rectangles.

Find the area of ​​the lateral surface as the sum of the areas of the rectangles AA 1 B 1 B, AA 1 C 1 C, BB 1 C 1 C:

S side \u003d AB ∙ h + BC ∙ h + CA ∙ h \u003d (AB + BC + CA) ∙ h \u003d P main ∙ h.

We get S side \u003d R main ∙ h, Q.E.D.

We got acquainted with polyhedrons, a prism, its varieties. We proved the theorem on the lateral surface of a prism. In the next lesson, we will solve problems on a prism.

1. Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, corrected and supplemented - M .: Mnemosyne, 2008. - 288 p. : ill.
2. Geometry. Grade 10-11: A textbook for general educational institutions / Sharygin I. F. - M .: Bustard, 1999. - 208 p.: ill.
3. Geometry. Grade 10: Textbook for general educational institutions with in-depth and profile study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th edition, stereotype. - M. : Bustard, 008. - 233 p. :ill.

1. Iclass().
2. Shkolo.ru ().
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4. wikihow().

1. What is the minimum number of faces a prism can have? How many vertices, edges does such a prism have?
2. Is there a prism that has exactly 100 edges?
3. The side rib is inclined to the base plane at an angle of 60°. Find the height of the prism if the side edge is 6 cm.
4. In a right triangular prism, all edges are equal. Its lateral surface area is 27 cm 2 . Find the total surface area of ​​the prism.

Different prisms are different from each other. At the same time, they have a lot in common. To find the area of ​​\u200b\u200bthe base of a prism, you need to figure out what kind it looks like.

General theory

A prism is any polyhedron whose sides have the form of a parallelogram. Moreover, any polyhedron can be at its base - 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 - they can vary significantly in size.

When solving problems, it is not only the area of ​​\u200b\u200bthe base of the prism that is encountered. It may be necessary to know the lateral surface, that is, all faces that are not bases. The full surface will already be the union of all the faces that make up the prism.

Sometimes heights appear in tasks. 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 area of ​​the base of a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same figures in the upper and lower faces, then their areas will be equal.

triangular prism

It has at the base a figure with three vertices, that is, a triangle. It is known to be different. If then it is enough to recall that its area is determined by half the product of the legs.

Mathematical notation looks like this: S = ½ av.

To find out the area of ​​\u200b\u200bthe base in a general form, the formulas are useful: Heron and the one in which half of the side is taken to the height drawn to it.

The first formula should be written like this: S \u003d √ (p (p-a) (p-in) (p-c)). This entry 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 know the area of ​​​​the base of a triangular prism, which is regular, then the triangle is equilateral. It has its own formula: S = ¼ a 2 * √3.

quadrangular prism

Its base is any of the known quadrilaterals. It can be a rectangle or a square, a parallelepiped or a rhombus. In each case, in order to calculate the area of ​​\u200b\u200bthe base of the prism, you will need your own formula.

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

When it comes to a quadrangular prism, the base area of ​​a regular prism is calculated using the formula for a square. Because it is he who lies at the base. S \u003d a 2.

In the case when the base is a parallelepiped, the following equality will be needed: S \u003d a * n a. It happens that a side of a parallelepiped and one of the angles are given. Then, to calculate the height, you will need to use an additional formula: na \u003d b * sin A. Moreover, the angle A is adjacent to the side "b", and the height is na opposite to this angle.

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

Regular pentagonal prism

This case involves splitting the polygon into triangles, the areas of which are easier to find out. Although it happens that the figures can be with 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 ​​\u200b\u200bthe 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

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

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

Tasks

No. 1. A regular line is given. Its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of ​​\u200b\u200bthe base of the prism and the entire surface.

Decision. The base of a prism is a square, but its side is not known. 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 (n). x 2 \u003d 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 \u003d a 2 + a 2. Thus, it turns out that a 2 \u003d (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 it’s easy to find out the base area: 12 * 12 \u003d 144 cm 2.

To find out the area of ​​\u200b\u200bthe entire surface, you need to add twice the value of the base area and quadruple the side. The latter is easy to find by 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 is found to be 960 cm 2 .

Answer. The base area of ​​the prism is 144 cm2. The entire surface - 960 cm 2 .

No. 2. Dana At the base lies 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.

Decision. Since the prism is regular, its base is an equilateral triangle. Therefore, its area turns out to be equal to 6 squared times 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, it is enough to multiply these numbers. Then multiply them by three, because the prism has exactly so many side faces. Then the area of ​​the side surface is wound 180 cm 2 .

Answer. Areas: base - 9√3 cm 2, side surface of the prism - 180 cm 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
• Total 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 denote the action of extracting a 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.

Decision.
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 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

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.

Decision.
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.

In the school course of stereometry, one of the simplest figures that has non-zero dimensions along three spatial axes is a quadrangular prism. Consider in the article what kind of figure it is, what elements it consists of, and also how you can calculate its surface area and volume.

The concept of a prism

In geometry, a prism is a spatial figure, which is formed by two identical bases and side surfaces that connect the sides of these bases. Note that both bases are transformed into each other using the operation of parallel translation by some vector. This task of the prism leads to the fact that all its sides are always parallelograms.

The number of sides of the base can be arbitrary, starting from three. When this number tends to infinity, the prism smoothly turns into a cylinder, since its base becomes a circle, and the side parallelograms, connecting, form a cylindrical surface.

Like any polyhedron, a prism is characterized by sides (planes that bound the figure), edges (segments along which any two sides intersect) and vertices (meeting points of three sides, for a prism two of them are lateral, and the third is the base). The numbers of these three elements of the figure are interconnected by the following expression:

Here P, C and B are the number of edges, sides and vertices, respectively. This expression is a mathematical notation of Euler's theorem.

Above is a picture showing two prisms. At the base of one of them (A) lies a regular hexagon, and the side sides are perpendicular to the bases. Figure B shows another prism. Its sides are no longer perpendicular to the bases, and the base is a regular pentagon.

quadrangular?

As is clear from the description above, the type of prism is primarily determined by the type of polygon that forms the base (both bases are the same, so we can talk about one of them). If this polygon is a parallelogram, then we get a quadrangular prism. So all sides of this are parallelograms. A quadrangular prism has its own name - a parallelepiped.

The number of sides of the parallelepiped is six, with each side having an analogous parallel to it. Since the bases of the parallelepiped are two sides, the remaining four are lateral.

The number of vertices of the parallelepiped is eight, which is easy to see if we remember that the vertices of the prism are formed only at the vertices of the base polygons (4x2=8). Applying Euler's theorem, we obtain the number of edges:

P \u003d C + B - 2 \u003d 6 + 8 - 2 \u003d 12

Of the 12 ribs, only 4 are formed independently by the sides. The remaining 8 lie in the planes of the bases of the figure.

Types of parallelepipeds

The first type of classification lies in the features of the parallelogram that lies at the base. It may look like this:

• ordinary, in which the angles are not equal to 90 o;
• rectangle;
• a square is a regular quadrilateral.

The second type of classification is the angle at which the side crosses the base. Two different cases are possible here:

• this angle is not straight, then the prism is called oblique or oblique;
• the angle is 90 o, then such a prism is rectangular or just straight.

The third type of classification is related to the height of the prism. If the prism is rectangular, and the base is either a square or a rectangle, then it is called a cuboid. If there is a square at the base, the prism is rectangular, and its height is equal to the length of the side of the square, then we get the well-known cube figure.

The surface of the prism and its area

The set of all points that lie on the two bases of the prism (parallelograms) and on its sides (four parallelograms) form the surface of the figure. The area of ​​this surface can be calculated by calculating the area of ​​the base and this value for the side surface. Then their sum will give the desired value. Mathematically, this is written like this:

Here S o and S b are the area of ​​the base and side surface, respectively. The number 2 before S o appears because there are two bases.

Note that the written formula is valid for any prism, and not just for the area of ​​a quadrangular prism.

It is useful to recall that the area of ​​the parallelogram S p is calculated by the formula:

Where the symbols a and h denote the length of one of its sides and the height drawn to this side, respectively.

Area of ​​a rectangular prism with a square base

The base is a square. For definiteness, we denote its side by the letter a. To calculate the area of ​​a regular quadrangular prism, you should know its height. According to the definition for this quantity, it is equal to the length of the perpendicular dropped from one base to another, that is, equal to the distance between them. Let's denote it by the letter h. Since all side faces are perpendicular to the bases for the type of prism under consideration, the height of a regular quadrangular prism will be equal to the length of its side edge.

There are two terms in the general formula for the surface area of ​​a prism. The area of ​​​​the base in this case is easy to calculate, it is equal to:

To calculate the area of ​​the side surface, we argue as follows: this surface is formed by 4 identical rectangles. Moreover, the sides of each of them are equal to a and h. This means that the area S b will be equal to:

Note that the product 4*a is the perimeter of the square base. If we generalize this expression to the case of an arbitrary base, then for a rectangular prism the side surface can be calculated as follows:

Where P o is the perimeter of the base.

Returning to the problem of calculating the area of ​​a regular quadrangular prism, we can write the final formula:

S = 2*S o + S b = 2*a 2 + 4*a*h = 2*a*(a+2*h)

Area of ​​an oblique parallelepiped

Calculating it is somewhat more difficult than for a rectangular one. In this case, the base area of ​​a quadrangular prism is calculated using the same formula as for a parallelogram. The changes relate to the method of determining the area of ​​the lateral surface.

For this, the same formula is used through the perimeter, which is given in the paragraph above. Only now it will have slightly different multipliers. The general formula for S b in the case of an oblique prism is:

Here, c is the length of the side edge of the figure. The value P sr is the perimeter of the rectangular slice. This environment is built as follows: it is necessary to intersect all the side faces with a plane so that it is perpendicular to all of them. The resulting rectangle will be the desired slice.

The figure above shows an example of an oblique box. Its cross-hatched section forms right angles with the sides. The perimeter of the section is P sr . It is formed by four heights of lateral parallelograms. For this quadrangular prism, the lateral surface area is calculated using the above formula.

The length of the diagonal of a cuboid

The diagonal of a parallelepiped is a segment that connects two vertices that do not have common sides that form them. There are only four diagonals in any quadrangular prism. For a cuboid with a rectangle at its base, the lengths of all diagonals are equal to each other.

The figure below shows the corresponding figure. The red segment is its diagonal.

D = √(A 2 + B 2 + C 2)

Here D is the length of the diagonal. The remaining symbols are the lengths of the sides of the parallelepiped.

Many people confuse the diagonal of a parallelepiped with the diagonals of its sides. Below is a figure where the diagonals of the sides of the figure are shown with colored segments.

The length of each of them is also determined by the Pythagorean theorem and is equal to the square root of the sum of the squares of the corresponding side lengths.

Prism Volume

In addition to the area of ​​\u200b\u200ba regular quadrangular prism or other types of prisms, in order to solve some geometric problems, their volume should also be known. This value for absolutely any prism is calculated by the following formula:

If the prism is rectangular, then it is enough to calculate the area of ​​its base and multiply it by the length of the edge of the side to get the volume of the figure.

If the prism is a regular quadrilateral, then its volume will be equal to:

It is easy to see that this formula is converted into an expression for the volume of a cube if the length of the side edge h is equal to the side of the base a.

Problem with a cuboid

To consolidate the studied material, we will solve the following problem: there is a rectangular parallelepiped whose sides are 3 cm, 4 cm and 5 cm. It is necessary to calculate its surface area, diagonal length and volume.

S \u003d 2 * S o + S b \u003d 2 * 12 + 5 * 14 \u003d 24 + 70 \u003d 94 cm 2

To determine the length of the diagonal and the volume of the figure, you can directly use the above expressions:

D \u003d √ (3 2 +4 2 +5 2) \u003d 7.071 cm;

V \u003d 3 * 4 * 5 \u003d 60 cm 3.

Problem with an oblique parallelepiped

The figure below shows an oblique prism. Its sides are equal: a=10 cm, b=8 cm, c=12 cm. It is necessary to find the surface area of ​​this figure.

First, let's determine the area of ​​the base. It can be seen from the figure that the acute angle is 50 o. Then its area is:

S o \u003d h * a \u003d sin (50 o) * b * a

To determine the lateral surface area, find the perimeter of the shaded rectangle. The sides of this rectangle are a*sin(45o) and b*sin(60o). Then the perimeter of this rectangle is:

P sr = 2*(a*sin(45o)+b*sin(60o))

The total surface area of ​​this parallelepiped is:

S = 2*S o + S b = 2*(sin(50 o)*b*a + a*c*sin(45 o) + b*c*sin(60 o))

We substitute the data from the condition of the problem for the lengths of the sides of the figure, we get the answer:

From the solution of this problem, it can be seen that trigonometric functions are used to determine the areas of oblique figures.