Stereometry is a branch of geometry that studies figures that do not lie in the same plane. One of the objects of study of stereometry are prisms. In the article we will give a definition of a prism from a geometric point of view, and also briefly list the properties that are characteristic of it.

Geometric figure

The definition of a prism in geometry is as follows: it is a spatial figure consisting of two identical n-gons located in parallel planes, connected to each other by their vertices.

Getting a prism is not difficult. Imagine that there are two identical n-gons, where n is the number of sides or vertices. Let's place them so that they are parallel to each other. After that, the vertices of one polygon should be connected to the corresponding vertices of another. The formed figure will consist of two n-gonal sides, which are called bases, and n quadrangular sides, which in the general case are parallelograms. The set of parallelograms forms the side surface of the figure.

There is another way to geometrically obtain the figure in question. So, if we take an n-gon and transfer it to another plane using parallel segments of equal length, then in the new plane we will get the original polygon. Both polygons and all parallel segments drawn from their vertices form a prism.

The picture above shows it so called because its bases are triangles.

The elements that make up a figure

The definition of a prism was given above, from which it is clear that the main elements of a figure are its faces or sides, limiting all the internal points of the prism from external space. Any face of the figure under consideration belongs to one of two types:

• lateral;
• grounds.

There are n side pieces, and they are parallelograms or their particular types (rectangles, squares). In general, the side faces differ from each other. There are only two faces of the base, they are n-gons and are equal to each other. Thus, every prism has n+2 sides.

In addition to the sides, the figure is characterized by its vertices. They are points where three faces touch at the same time. Moreover, two of the three faces always belong to the side surface, and one - to the base. Thus, in a prism there is no specially selected one vertex, as, for example, in a pyramid, all of them are equal. The number of vertices of the figure is 2*n (n pieces for each base).

Finally, the third important element of the prism is its edges. These are segments of a certain length, which are formed as a result of the intersection of the sides of the figure. Like faces, edges also have two different types:

• or formed only by the sides;
• or arise at the junction of the parallelogram and the side of the n-gonal base.

The number of edges is thus 3*n, and 2*n of them belong to the second of the named types.

Prism types

There are several ways to classify prisms. However, they are all based on two features of the figure:

• on the type of n-coal base;
• on side type.

To begin with, let us turn to the second singularity and give a definition of a straight line. If at least one side is a parallelogram of a general type, then the figure is called oblique or oblique. If all parallelograms are rectangles or squares, then the prism will be straight.

You can also give a definition a little differently: a straight figure is a prism in which the side edges and faces are perpendicular to its bases. The figure shows two quadrangular figures. The left one is straight, the right one is oblique.

Now let's move on to classification according to the type of n-gon lying in the bases. It can have the same sides and angles or different. In the first case, the polygon is called regular. If the figure under consideration contains a polygon with equal sides and angles at the base and is a straight line, then it is called regular. According to this definition, a regular prism at its base can have an equilateral triangle, a square, a regular pentagon, or a hexagon, and so on. The listed correct figures are shown in the figure.

Linear parameters of prisms

To describe the dimensions of the figures under consideration, the following parameters are used:

• height;
• sides of the base;
• side rib lengths;
• volumetric diagonals;
• diagonal sides and bases.

For regular prisms, all the named quantities are related to each other. For example, the lengths of the side ribs are the same and equal to the height. For a specific n-gonal regular figure, there are formulas that allow us to determine all the rest from any two linear parameters.

Figure surface

If we turn to the definition of a prism given above, then it will not be difficult to understand what the surface of the figure represents. The surface is the area of ​​all the faces. For a straight prism, it is calculated by the formula:

S = 2*S o + P o *h

where S o is the area of ​​the base, P o is the perimeter of the n-gon at the base, h is the height (distance between the bases).

figure volume

Along with the surface for practice, it is important to know the volume of the prism. It can be determined using the following formula:

This expression is true for absolutely any kind of prism, including those that are oblique and formed by irregular polygons.

For correct, it is a function of the length of the side of the base and the height of the figure. For the corresponding n-gonal prism, the formula for V has a specific form.

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

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 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

Polyhedra

The main object of study of stereometry are three-dimensional bodies. Body is a part of space bounded by some surface.

polyhedron A body whose surface consists of a finite number of plane polygons is called. A polyhedron is called convex if it lies on one side of the plane of every flat polygon on its surface. The common part of such a plane and the surface of a polyhedron is called edge. The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called edges of the polyhedron, and the vertices vertices of the polyhedron.

For example, a cube consists of six squares that are its faces. It contains 12 edges (sides of squares) and 8 vertices (vertices of squares).

The simplest polyhedra are prisms and pyramids, which we will study further.

Prism

Definition and properties of a prism

prism is called a polyhedron consisting of two flat polygons lying in parallel planes combined by parallel translation, and all segments connecting the corresponding points of these polygons. The polygons are called prism bases, and the segments connecting the corresponding vertices of the polygons are side edges of the prism.

Prism height called the distance between the planes of its bases (). A segment connecting two vertices of a prism that do not belong to the same face is called prism diagonal(). The prism is called n-coal if its base is an n-gon.

Any prism has the following properties, which follow from the fact that the bases of the prism are combined by parallel translation:

1. The bases of the prism are equal.

2. The side edges of the prism are parallel and equal.

The surface of a prism is made up of bases and lateral surface. The lateral surface of the prism consists of parallelograms (this follows from the properties of the prism). The area of ​​the lateral surface of a prism is the sum of the areas of the lateral faces.

straight prism

The prism is called straight if its side edges are perpendicular to the bases. Otherwise, the prism is called oblique.

The faces of a straight prism are rectangles. The height of a straight prism is equal to its side faces.

full prism surface is the sum of the lateral surface area and the areas of the bases.

Correct prism is called a right prism with a regular polygon at the base.

Theorem 13.1. The area of ​​the lateral surface of a straight prism is equal to the product of the perimeter and the height of the prism (or, equivalently, to the lateral edge).

Proof. The side faces of a straight prism are rectangles whose bases are the sides of the polygons at the bases of the prism, and the heights are the side edges of the prism. Then, by definition, the lateral surface area is:

,

where is the perimeter of the base of a straight prism.

Parallelepiped

If parallelograms lie at the bases of a prism, then it is called parallelepiped. All the faces of a parallelepiped are parallelograms. In this case, the opposite faces of the parallelepiped are parallel and equal.

Theorem 13.2. The diagonals of the parallelepiped intersect at one point and the intersection point is divided in half.

Proof. Consider two arbitrary diagonals, for example, and . Because the faces of the parallelepiped are parallelograms, then and , which means that according to T about two straight lines parallel to the third . In addition, this means that the lines and lie in the same plane (the plane). This plane intersects parallel planes and along parallel lines and . Thus, a quadrilateral is a parallelogram, and by the property of a parallelogram, its diagonals and intersect and the intersection point is divided in half, which was required to be proved.

A right parallelepiped whose base is a rectangle is called cuboid. All faces of a cuboid are rectangles. The lengths of non-parallel edges of a rectangular parallelepiped are called its linear dimensions (measurements). There are three sizes (width, height, length).

Theorem 13.3. In a cuboid, the square of any diagonal is equal to the sum of the squares of its three dimensions (proved by applying Pythagorean T twice).

A rectangular parallelepiped in which all edges are equal is called cube.

Tasks

13.1 How many diagonals does n- carbon prism

13.2 In an inclined triangular prism, the distances between the side edges are 37, 13, and 40. Find the distance between the larger side face and the opposite side edge.

13.3 Through the side of the lower base of a regular triangular prism, a plane is drawn that intersects the side faces along segments, the angle between which is . Find the angle of inclination of this plane to the base of the prism.

Definition.

This is a hexagon, the bases of which are two equal squares, and the side faces are equal rectangles.

Side rib is the common side of two adjacent side faces

Prism Height 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.