Stereometry is an important part general course geometry, which considers the characteristics of spatial figures. One such figure is a quadrangular prism. In this article, we will reveal in more detail the question of how to calculate the volume of a quadrangular prism.

What is a quadrangular prism?

Obviously, before giving the formula for the volume of a quadrangular prism, it is necessary to give a clear definition of this geometric figure. Such a prism is understood as a three-dimensional polyhedron, which is bounded by two arbitrary identical quadrangles lying in parallel planes and four parallelograms.

The marked parallel quadrangles are called the bases of the figure, and the four parallelograms are the sides. It should be clarified here that parallelograms are also quadrangles, but the bases are not always parallelograms. An example of an irregular quadrangle, which may well be the base of a prism, is shown in the figure below.

Any quadrangular prism has 6 sides, 8 vertices, and 12 edges. There are quadrangular prisms different types... For example, a figure can be oblique or straight, irregular and correct. Further in the article we will show how you can calculate the volume of a quadrangular prism, taking into account its type.

Oblique prism with wrong base

This is the most asymmetrical type of quadrangular prism, so calculating its volume will be relatively difficult. The following expression allows you to determine the volume of a figure:

The So symbol here denotes the base area. If this base is a rhombus, parallelogram or rectangle, then it is not difficult to calculate the value of So. So, for a rhombus and a parallelogram, the following formula is valid:

where a is the side of the base, ha is the length of the height dropped to this side from the top of the base.

If the base is not regular polygon(see above), then its area should be divided into more simple shapes(for example, triangles), calculate their areas and find their sum.

In the volume formula, h represents the height of the prism. It is the length of the perpendicular line between two bases. Since the prism is inclined, the calculation of the height h should be carried out using the length of the side edge b and the dihedral angles between the side faces and the base.

Correct figure and its volume

If the base of a quadrangular prism is a square, and the figure itself is straight, then it is called regular. It should be clarified that a straight prism is called when all its lateral sides are rectangles and each of them is perpendicular to the bases. The correct figure is shown below.

The volume of a regular quadrangular prism can be calculated using the same formula as the volume of an irregular figure. Since the base is a square, its area is calculated simply:

The height of the prism h is equal to the length of the lateral rib b (side of the rectangle). Then the volume of a regular quadrangular prism can be calculated using the following formula:

A regular prism with a square base is called rectangular parallelepiped... This parallelepiped, in the case of equality of sides a and b, becomes a cube. The volume of the latter is calculated as follows:

The written formulas for volume V indicate that the higher the symmetry of the figure, the less linear parameters required to calculate this value. So, in the case of a correct prism, the required number of parameters is two, and in the case of a cube, one.

The problem with the correct figure

Having considered the issue of finding the volume of a quadrangular prism from the point of view of theory, we will apply the knowledge gained in practice.

It is known that a regular parallelepiped has a base diagonal length of 12 cm. The diagonal length of its lateral side is 20 cm. It is necessary to calculate the volume of the parallelepiped.

Let us denote the diagonal of the base by the symbol da, and the diagonal of the side face by the symbol db. For the diagonal da, the following expressions are valid:

As for the db value, it is the diagonal of a rectangle with sides a and b. For it, you can write the following equalities:

db2 = a2 + b2 =>

b = √ (db2 - a2)

Substituting the found expression for a into the last equality, we get:

b = √ (db2 - da2 / 2)

Now you can substitute the resulting formulas into the expression for the volume of a regular figure:

V = a2 * b = da2 / 2 * √ (db2 - da2 / 2)

Replacing da and db with the numbers from the problem statement, we arrive at the answer: V ≈ 1304 cm3.

Definition.

It is a hexagon, the bases of which are two equal square and the side faces are equal rectangles

Side rib is the common side of two adjacent side faces

Prism height is a segment perpendicular to the bases of the prism

Diagonal prism- 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) 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 designated by 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
• Side surface - the sum of the areas of all 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 side faces are rectangles
• Side faces are equal to each other
• Side faces are perpendicular to the bases
• The side ribs are parallel and equal
• Perpendicular section perpendicular to all side edges and parallel to the bases
• The corners of the perpendicular section are straight
• 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"it is understood that:

Correct prism- a prism at the base of which a regular polygon lies, and the side edges are perpendicular to the base planes. That is, a regular quadrangular prism contains at its base square... (see above properties of a regular quadrangular prism) Note... This is part of the lesson with geometry problems (section stereometry - prism). Here are the tasks that cause difficulties in solving. If you need to solve a geometry problem that is not here, write about it in the forum. To denote the action of extracting a square root in problem solutions, the symbol√ .

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
√(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

Determine the full surface of a regular quadrangular prism if its diagonal is 5 cm and the diagonal of the side face is 4 cm.

Solution.
Since there is a square at the base of a regular quadrangular prism, we will find the side of the base (denoted as a) 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 = 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.

A prism is a geometric volumetric figure, the characteristics and properties of which are studied in high school. As a rule, when studying it, such quantities as volume and surface area are considered. In this article, we will reveal a slightly different issue: we will give a method for determining the length of the diagonals of a prism using the example of a quadrangular figure.

Which figure is called a prism?

Geometry gives following definition prism: this is a three-dimensional figure, bounded by two polygonal identical sides that are parallel to each other, and a number of parallelograms. The figure below shows an example of a prism corresponding to this definition.

We see that the two red pentagons are equal to each other and are in two parallel planes. Five pink parallelograms connect these pentagons to form a solid object - a prism. The two pentagons are called the bases of the shape, and the parallelograms are the side faces.

Prisms are straight and oblique, which are also called rectangular and oblique. The difference between them lies in the angles between the base and the side edges. For a rectangular prism, these angles are all 90 °.

By the number of sides or vertices of the polygon at the base, they speak of triangular, pentagonal, quadrangular prisms, and so on. Moreover, if this polygon is regular, and the prism itself is straight, then such a figure is called regular.

The prism shown in the previous figure is a pentagonal oblique. Below is a pentagonal straight prism, which is correct.

All calculations, including the method for determining the diagonals of the prism, are convenient to perform precisely for the correct figures.

What elements characterize the prism?

The elements of a figure are the constituent parts that form it. Specifically for a prism, three main types of elements can be distinguished:

• tops;
• edges or sides;
• ribs.

Faces are considered to be the bases and lateral planes that generally represent parallelograms. In a prism, each side always belongs to one of two types: either it is a polygon or a parallelogram.

The edges of the prism are the line segments that define each side of the shape. Like faces, there are also two types of ribs: those belonging to the base and side surface, or belonging only to the side surface. The first are always twice as many as the second, regardless of the type of prism.

Vertices are the intersection points of three prism edges, two of which lie in the plane of the base, and the third belongs to two lateral faces. All vertices of the prism are in the planes of the base of the figure.

The numbers of the described elements are linked into a single equality, which has the following form:

P = B + C - 2.

Here P is the number of edges, B is the vertices, C is the sides. This equality is called Euler's theorem for the polyhedron.

The figure shows a triangular regular prism. Anyone can assume that it has 6 vertices, 5 sides and 9 edges. These numbers are consistent with Euler's theorem.

Prism diagonals

After properties such as volume and surface area, in geometry problems, information about the length of a particular diagonal of the figure under consideration is often found, which is either given or needs to be found by other known parameters. Let's consider what are the diagonals of the prism.

All diagonals can be divided into two types:

1. Lying in the plane of the faces. They connect the non-adjacent vertices of either the polygon at the base of the prism or the parallelogram of the lateral surface. The value of the lengths of such diagonals is determined based on the knowledge of the lengths of the corresponding edges and the angles between them. The properties of triangles are always used to define the diagonals of parallelograms.
2. Prisms lying inside the volume. These diagonals connect the dissimilar tops of the two bases. These diagonals are completely inside the shape. Their lengths are somewhat more difficult to calculate than for the previous type. The calculation method involves taking into account the lengths of the ribs and the base, and parallelograms. For straight and regular prisms, the calculation is relatively simple, since it is carried out using the Pythagorean theorem and the properties of trigonometric functions.

Diagonals of the sides of a rectangular straight prism

The picture above shows four identical straight prisms, and the parameters of their edges are given. On the prisms Diagonal A, Diagonal B and Diagonal C, the dashed red line represents the diagonals of three different faces. Since the prism is straight with a height of 5 cm, and its base is represented by a rectangle with sides of 3 cm and 2 cm, it is not difficult to find the marked diagonals. To do this, you need to use the Pythagorean theorem.

The length of the diagonal of the base of the prism (Diagonal A) is:

D A = √ (3 2 +2 2) = √13 ≈ 3.606 cm.

For the side face of the prism, the diagonal is (see Diagonal B):

D B = √ (3 2 +5 2) = √34 ≈ 5.831 cm.

Finally, the length of one more side diagonal is (see Diagonal C):

D С = √ (2 2 +5 2) = √29 ≈ 5.385 cm.

Inner diagonal length

Now let's calculate the length of the diagonal of the quadrangular prism, which is shown in the previous figure (Diagonal D). This is not so difficult if you notice that it is the hypotenuse of a triangle, in which the legs will be the height of the prism (5 cm) and the diagonal D A shown in the figure above on the left (Diagonal A). Then we get:

D D = √ (D A 2 +5 2) = √ (2 2 +3 2 +5 2) = √38 ≈ 6.164 cm.

Regular prism quadrangular

The diagonal of a regular prism, the base of which is a square, is calculated in the same way as in the example above. The corresponding formula is:

D = √ (2 * a 2 + c 2).

Where a and c are the base side and side rib lengths, respectively.

Note that in the calculations we used only the Pythagorean theorem. To determine the lengths of the diagonals of regular prisms with a large number of vertices (pentagonal, hexagonal, and so on), it is already necessary to use trigonometric functions.

IN school curriculum In the course of stereometry, the study of volumetric figures usually begins with a simple geometric body - a polyhedron of a prism. The role of its foundations is performed by 2 equal polygon lying in parallel planes. A special case is a regular quadrangular prism. Its bases are 2 identical regular quadrangles, to which the lateral sides are perpendicular, in the form of parallelograms (or rectangles if the prism is not inclined).

What a prism looks like

A regular quadrangular prism is called 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.

A drawing showing a quadrilateral prism is shown below.

The picture also shows the most important elements that make up geometric body ... It is customary to refer to them:

Sometimes in problems on geometry one 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 a cutting plane. The section is perpendicular (it intersects 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 diagonals of the base.

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

Various relations 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 ​​its base and height:

V = S main h

Since the base of a regular tetrahedral prism is a square with a side a, you can write the formula in more detail:

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 area of ​​the lateral surface of a prism, you need to imagine its unfolding.

The drawing shows that the side surface is composed of 4 equal rectangles... Its area is calculated as the product of the perimeter of the base and the height of the figure:

Sside = P main h

Taking into account that the perimeter of the square is P = 4a, the formula takes the form:

Sside = 4a h

For a cube:

Sside = 4a²

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

S full = S side + 2S main

With regard to a quadrangular regular prism, the formula is:

S total = 4a · h + 2a²

For the surface area of ​​a cube:

S total = 6a²

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

Finding Prism Elements

Often there are problems in which a 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, the formulas can be derived:

• base side length: a = S side / 4h = √ (V / h);
• length of height or side rib: h = S side / 4a = V / a²;
• base area: Sosn = V / h;
• side face area: S side. gr = S side / 4.

To determine what 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, use the formula:

dprize = √ (2a² + h²)

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

Examples of tasks with solutions

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

Exercise 1.

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

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

V₁ = ha² = 10a²

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

V₂ = h (2a) ² = 4ha²

Because the V₁ = V₂, you can equate expressions:

10a² = 4ha²

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

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

Task 2.

ABCDA₁B₁C₁D₁ is the correct 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 depict a figure.

Since we are talking about the correct prism, we can conclude that at the base there is a square with a diagonal of 6√2. The diagonal of the side face has the same size, 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 a cube:

S total = 6a² = 6 6² = 216

Task 3.

The room is being renovated. It is known that its floor is in the form 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 correct prism... It is necessary to determine the area of ​​its lateral surface.

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

The area 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 own formulas for finding the volume and surface area.

How to find the area of ​​a cube

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

With the help of this lesson, everyone will be able to independently familiarize themselves with the topic “The concept of a polyhedron. Prism. Surface area of ​​a prism ".

Definition... A surface made up of polygons and bounding some 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 made up of six parallelograms (Fig. 2).

Rice. 2

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

Faces are polygons that make up a polyhedron.

The edges are the sides of the faces.

The vertices are the ends of the ribs.

Consider a tetrahedron ABCD(fig. 1). Let's indicate its main elements.

Facets: triangles ABC, ADB, BDC, ADC.

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

Tops: A, B, C, D.

Consider a parallelepiped 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.

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

An important special case of a polyhedron is a prism.

ABCA 1 B 1 C 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, CC 1 are parallel.

I.e ABCA 1 B 1 C 1- a 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, CC 1 are parallel.

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

AA 1, BB 1, CC 1- lateral edges of the prism.

If from an arbitrary point H 1 one plane (for example, β), lower the perpendicular NN 1 on 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, and otherwise - inclined.

Consider a triangular prism ABCA 1 B 1 C 1(fig. 4). This prism is straight. That is, its lateral edges are perpendicular to the bases.

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

Rice. 4

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

Now consider an oblique prism ABCA 1 B 1 C 1(fig. 5). Here, the lateral rib is not perpendicular to the plane of the base. If we omit 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 on the plane ABC.

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

Rice. five

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 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 ║ВВ 1 ║СС 1 ║DD 1.

Definition... A prism diagonal is a line 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 rib AA 1 perpendicular to the plane of the base, then such a prism is called a straight line.

Rice. 6

A particular 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 at 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 ║ВВ 1 ║СС 1 ║DD 1.

Rice. 7

From point A 1 omit the perpendicular AN on the plane ABC... Line segment A 1 H is the height.

Let's consider how the 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 are 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 lateral edge is perpendicular to the plane of the base, then such a hexagonal prism is called a straight line.

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

Consider a regular triangular prism ABCA 1 B 1 C 1.

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Triangular prism ABCA 1 B 1 C 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. This means that all side faces are equal rectangles.

So, if a triangular prism ABCA 1 B 1 C 1- correct, then:

1) The lateral rib is perpendicular to the plane of the base, that is, is the height: AA 1 ⊥ ABC.

2) An equilateral triangle lies at the base: ∆ ABC- correct.

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

Definition... The lateral surface area 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 = 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 using the example of a triangular prism.

Given: ABCA 1 B 1 C 1- straight prism, i.e. AA 1 ⊥ ABC.

AA 1 = h.

Prove: S side = P main ∙ h.

Rice. 10

Proof.

Triangular prism ABCA 1 B 1 C 1- straight, then 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 = AB ∙ h + BC ∙ h + CA ∙ h = (AB + BC + CA) ∙ h = P main ∙ h.

We get S side = P main ∙ h, Q.E.D.

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

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

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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 lateral 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 straight line triangular prism all edges are equal. Its lateral surface area is 27 cm 2. Find the total surface area of ​​the prism.