Equal to the area of ​​the prism. Everything you need to know about the prism (2019)

Definition. Prism is a polyhedron, all vertices of which are located in two parallel planes, and in the same two planes there are two prism faces, which are equal polygons with correspondingly parallel sides, and all edges that do not lie in these planes are parallel.

Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).

All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).

All side faces form lateral surface of the prism .

All side faces of the prism are parallelograms .

The ribs that do not lie in the bases are called the lateral ribs of the prism ( AA 1, BB 1, CC 1, DD 1, EE 1).

Diagonal prism is called a segment whose ends are two vertices of a prism that do not lie on one of its faces (AD 1).

The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called height of the prism .

Designation:ABCDE A 1 B 1 C 1 D 1 E 1... (First, the vertices of one base are indicated in the order of traversal, and then, in the same order, the vertices of the other; the ends of each side edge are denoted by the same letters, only the vertices lying in one base are denoted by letters without an index, and in the other - with an index)

The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1, a pentagon lies at the base, therefore the prism is called pentagonal prism... But since such a prism has 7 faces, then it heptahedron(2 faces - prism bases, 5 faces - parallelograms, - its side faces)

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

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

Parallelepiped

Rectangular parallelepiped- a straight parallelepiped, the base of which is a rectangle.

Properties and theorems:

,

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

The idea of ​​a prism is given by:

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

The area of ​​the full and lateral surface of the prism

S full = S side + 2S main,

where S full- total surface area, S side- the area of ​​the lateral surface, S main- base area

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

S side= P main * h,

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

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

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

Prism volume

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

In the school curriculum for the stereometry course, the study of volumetric figures usually begins with a simple geometric body - a polyhedron of a prism. 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 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 can also see the most important elements that make up a geometric body... It is customary to refer to them:

Sometimes in problems in 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 shape 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 its base area 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 + 2 S 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 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, 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 form of a regular quadrangular prism. The height of its level is 10 cm. What will the level of 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

Insofar as V₁ = V₂, you can equate expressions:

10a² = 4ha²

After canceling 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 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. It can be concluded 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 a regular prism. It is necessary to determine the area of ​​its lateral surface.

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

Wallpaper will be pasted over the area Side = 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

In spatial geometry, when solving problems with prisms, there is often a problem with calculating the area of ​​the sides or faces that form these volumetric figures. This article is devoted to the question of determining the area of ​​the base of the prism and its lateral surface.

Figure prism

Before proceeding to the consideration of the formulas for the base area and surface of a prism of one type or another, one should figure out which figure we are talking about.

A prism in geometry is a spatial figure consisting of two parallel polygons, which are equal to each other, and several quadrangles or parallelograms. The number of the latter is always equal to the number of vertices of one polygon. For example, if the figure is formed by two parallel n-gons, then the number of parallelograms will be n.

Parallelograms connecting n-gons are called the lateral sides of the prism, and their total area is the area of ​​the lateral surface of the figure. The n-gons themselves are called bases.

The picture above shows an example of a prism made from paper. The yellow rectangle is its top base. The figure stands on the second similar base. The red and green rectangles are the side faces.

What prisms are there?

There are several types of prisms. They all differ from each other in just two parameters:

• the type of n-gon forming the base;
• the angle between the n-gon and the side faces.

For example, if the bases are triangles, then the prism is called triangular, if quadrangles, as in the previous figure, then the figure is called a quadrangular prism, and so on. In addition, an n-gon can be convex or concave, then this property is also added to the name of the prism.

The angle between the side faces and the base can be either straight or sharp or obtuse. In the first case, they talk about a rectangular prism, in the second - about an inclined or oblique.

Regular prisms are distinguished into a special type of figure. They have the highest symmetry among the other prisms. It will be correct only if it is rectangular and its base is a regular n-gon. The figure below shows a set of regular prisms in which the number of sides of an n-gon varies from three to eight.

Prism surface

The surface of the considered figure of an arbitrary type is understood as the totality of all points that belong to the faces of the prism. It is convenient to study the surface of the prism by looking at its sweep. Below is an example of such a sweep for a triangular prism.

It can be seen that the entire surface is formed by two triangles and three rectangles.

In the case of a general prism, its surface will consist of two n-gonal bases and n quadrangles.

Let us consider in more detail the issue of calculating the surface area of ​​different types of prisms.

The base area of ​​the prism is correct

Perhaps the simplest task when working with prisms is the problem of finding the area of ​​the base of a regular figure. Since it is formed by an n-gon, in which all angles and side lengths are the same, you can always divide it into identical triangles, for which the angles and sides are known. The total area of ​​the triangles will be the area of ​​an n-gon.

Another way to determine the fraction of the surface area of ​​a prism (base) is to use a known formula. It looks like this:

S n = n / 4 * a 2 * ctg (pi / n)

That is, the area S n of an n-gon is uniquely determined based on the knowledge of the length of its side a. Calculating the cotangent can be somewhat difficult when calculating the formula, especially when n> 4 (for n≤4, the cotangent values ​​are tabular data). It is recommended to use a calculator to determine this trigonometric function.

When setting a geometric problem, one should be careful, since it may be necessary to find the area of ​​the bases of the prism. Then the value obtained by the formula should be multiplied by two.

Base area of ​​a triangular prism

Using a triangular prism as an example, consider how you can find the area of ​​the base of this figure.

Let's first consider a simple case - the correct prism. The area of ​​the base is calculated according to the formula given in the paragraph above, you need to substitute n = 3 into it. We get:

S 3 = 3/4 * a 2 * ctg (pi / 3) = 3/4 * a 2 * 1 / √3 = √3 / 4 * a 2

It remains to substitute in the expression the specific values ​​of the length of the side a of an equilateral triangle in order to obtain the area of ​​one base.

Now suppose you have a prism whose base is an arbitrary triangle. Its two sides a and b and the angle α between them are known. This figure is shown below.

How, in this case, to find the area of ​​the base of a triangular prism? It must be remembered that the area of ​​any triangle is equal to half the product of the side and the height lowered to that side. The figure shows the height h to side b. The length h corresponds to the product of the sine of the angle alpha and the length of the side a. Then the area of ​​the entire triangle is:

S = 1/2 * b * h = 1/2 * b * a * sin (α)

This is the area of ​​the base of the depicted triangular prism.

Side surface

We have figured out how to find the area of ​​the base of a prism. The side surface of this figure always consists of parallelograms. For straight prisms, parallelograms become rectangles, so their total area is easy to calculate:

S = ∑ i = 1 n (a i * b)

Here b is the length of the side edge, a i is the length of the side of the i-th rectangle, which coincides with the length of the side of the n-gon. In the case of a regular n-sided prism, we get a simple expression:

If the prism is inclined, then to determine the area of ​​its lateral surface, a perpendicular cut should be made, its perimeter P sr should be calculated and multiplied by the length of the lateral edge.

The picture above shows how to make this slice for an oblique pentagonal prism.

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