Schoolchildren who are preparing for passing the Unified State Exam in mathematics, you should definitely learn how to solve problems to find the area of ​​a straight line and correct prism. Many years of practice confirm the fact that similar tasks Geometry is considered quite difficult by many students.

At the same time, high school students with any level of training should be able to find the area and volume of a regular and straight prism. Only in this case will they be able to count on receiving competitive scores based on the results of passing the Unified State Exam.

Key Points to Remember

• If lateral ribs prisms are perpendicular to the base, it is called a straight line. All side faces of this figure are rectangles. The height of a straight prism coincides with its edge.
• A correct prism is one whose side edges are perpendicular to the base in which it is located. regular polygon. The lateral edges of this figure are equal rectangles. A correct prism is always straight.

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Specialists educational project“Shkolkovo” proposes to go from simple to complex: first we give theory, basic formulas, theorems and elementary problems with solutions, and then gradually move on to expert-level tasks.

Basic information is systematized and clearly presented in the “Theoretical Information” section. If you have already managed to repeat the necessary material, we recommend that you practice solving problems on finding the area and volume of a right prism. The “Catalogue” section presents a large selection of exercises varying degrees difficulties.

Try to calculate the area of ​​a straight and regular prism or right now. Analyze any task. If it does not cause any difficulties, you can safely move on to expert-level exercises. And if certain difficulties do arise, we recommend that you regularly prepare for the Unified State Exam online together with the Shkolkovo mathematical portal, and tasks on the topic “Straight and Regular Prism” will be easy for you.

Suppose we need to find the volume of a right triangular prism, the base area of ​​which is equal to S, and the height is equal to h= AA’ = BB’ = CC’ (Fig. 306).

Let us separately draw the base of the prism, i.e. triangle ABC (Fig. 307, a), and build it up to a rectangle, for which we draw a straight line KM through vertex B || AC and from points A and C we lower perpendiculars AF and CE onto this line. We get rectangle ACEF. Drawing the height ВD of triangle ABC, we see that rectangle ACEF is divided into 4 right triangles. Moreover, \(\Delta\)ALL = \(\Delta\)BCD and \(\Delta\)BAF = \(\Delta\)BAD. This means that the area of ​​the rectangle ACEF is doubled more area triangle ABC, i.e. equal to 2S.

To this prism with base ABC we will attach prisms with bases ALL and BAF and height h(Fig. 307, b). We obtain a rectangular parallelepiped with an ACEF base.

If we dissect this parallelepiped with a plane passing through straight lines BD and BB’, we will see that the rectangular parallelepiped consists of 4 prisms with bases BCD, ALL, BAD and BAF.

Prisms with bases BCD and BC can be combined, since their bases are equal (\(\Delta\)BCD = \(\Delta\)BCE) and their side edges, which are perpendicular to the same plane, are also equal. This means that the volumes of these prisms are equal. The volumes of prisms with bases BAD and BAF are also equal.

Thus, it turns out that the volume of a given triangular prism with base ABC is half the volume rectangular parallelepiped with ACEF base.

We know that the volume of a rectangular parallelepiped equal to the product area of ​​its base by height, i.e. in in this case equal to 2S h. Hence the volume of this right triangular prism is equal to S h.

The volume of a right triangular prism is equal to the product of the area of ​​its base and its height.

2. Volume of a right polygonal prism.

To find the volume of a line polygonal prism, for example pentagonal, with base area S and height h, let's divide it into triangular prisms (Fig. 308).

Denoting the base areas of triangular prisms by S 1, S 2 and S 3, and the volume of a given polygonal prism by V, we obtain:

V = S 1 h+ S 2 h+ S 3 h, or

V = (S 1 + S 2 + S 3) h.

And finally: V = S h.

In the same way, the formula for the volume of a right prism with any polygon at its base is derived.

Means, The volume of any right prism is equal to the product of the area of ​​its base and its height.

Prism volume

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

First we prove this theorem for a triangular prism, and then for a polygonal one.

1) Let us draw (Fig. 95) through edge AA 1 of the triangular prism ABCA 1 B 1 C 1 a plane parallel to face BB 1 C 1 C, and through edge CC 1 a plane parallel to face AA 1 B 1 B; then we will continue the planes of both bases of the prism until they intersect with the drawn planes.

Then we get a parallelepiped BD 1, which is divided by the diagonal plane AA 1 C 1 C into two triangular prisms (one of which is this one). Let us prove that these prisms are equal in size. To do this, we draw a perpendicular section abcd. The cross-section will produce a parallelogram whose diagonal ac divisible by two equal triangle. This prism is equal in size to a straight prism whose base is \(\Delta\) abc, and the height is edge AA 1. Other triangular prism equal in area to a straight line whose base is \(\Delta\) adc, and the height is edge AA 1. But two straight prisms with equally And equal heights are equal (because when nested they are combined), which means that the prisms ABCA 1 B 1 C 1 and ADCA 1 D 1 C 1 are equal in size. It follows from this that the volume of this prism is half the volume of the parallelepiped BD 1; therefore, denoting the height of the prism by H, we get:

$$ V_(\Delta ex.) = \frac(S_(ABCD)\cdot H)(2) = \frac(S_(ABCD))(2)\cdot H = S_(ABC)\cdot H $$

2) Let us draw diagonal planes AA 1 C 1 C and AA 1 D 1 D through the edge AA 1 of the polygonal prism (Fig. 96).

Then this prism will be cut into several triangular prisms. The sum of the volumes of these prisms constitutes the required volume. If we denote the areas of their bases by b 1 , b 2 , b 3, and the total height through H, we get:

volume of polygonal prism = b 1H+ b 2H+ b 3 H =( b 1 + b 2 + b 3) H =

= (area ABCDE) H.

Consequence. If V, B and H are numbers expressing in the corresponding units the volume, base area and height of the prism, then, according to what has been proven, we can write:

Other materials

In physics, a triangular prism made of glass is often used to study the spectrum of white light because it can resolve it into its individual components. In this article we will consider the volume formula

What is a triangular prism?

Before giving the volume formula, let's consider the properties of this figure.

To get this, you need to take a triangle of any shape and move it parallel to itself to some distance. The vertices of the triangle in the initial and final positions should be connected by straight segments. Received volumetric figure called a triangular prism. It consists of five sides. Two of them are called bases: they are parallel and equal to each other. The bases of the prism in question are triangles. The three remaining sides are parallelograms.

In addition to the sides, the prism in question is characterized by six vertices (three for each base) and nine edges (6 edges lie in the planes of the bases and 3 edges are formed by the intersection of the sides). If the side edges are perpendicular to the bases, then such a prism is called rectangular.

The difference between a triangular prism and all other figures of this class is that it is always convex (four-, five-, ..., n-gonal prisms may also be concave).

This rectangular figure, which is based on equilateral triangle.

Volume of a general triangular prism

How to find the volume of a triangular prism? Formula in general view similar to that for any type of prism. It has the following mathematical notation:

Here h is the height of the figure, that is, the distance between its bases, S o is the area of ​​the triangle.

The value of S o can be found if some parameters for the triangle are known, for example, one side and two angles or two sides and one angle. The area of ​​a triangle is equal to half the product of its height and the length of the side by which this height is lowered.

As for the height h of the figure, it is easiest to find for rectangular prism. IN the latter case h coincides with the length of the side edge.

Volume of a regular triangular prism

General formula volume of a triangular prism, which is given in the previous section of the article, can be used to calculate the corresponding value for a regular triangular prism. Since its base is an equilateral triangle, its area is equal to:

Anyone can get this formula if they remember that in an equilateral triangle all angles are equal to each other and amount to 60 o. Here the symbol a is the length of the side of the triangle.

The height h is the length of the edge. It is in no way connected with the base of a regular prism and can take arbitrary values. As a result, the formula for the volume of a triangular prism is the right kind looks like that:

Having calculated the root, you can rewrite this formula as follows:

Thus, to find the volume of a regular prism with triangular base, it is necessary to square the side of the base, multiply this value by the height and multiply the resulting value by 0.433.

Definition.

This is a hexagon whose bases are two equal square, and the side faces are equal rectangles

Side rib- This common side two adjacent side faces

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

Prism diagonal- a segment connecting two vertices of the bases that do not belong to the same face

Diagonal plane- a plane that passes through the diagonal of the prism and its lateral edges

Diagonal section - the boundaries of the intersection of the prism and the diagonal plane. The diagonal cross section of a regular quadrangular prism is a rectangle

Perpendicular section (orthogonal section)- this is the intersection of a prism and a plane drawn perpendicular to its lateral edges

Elements of a regular quadrangular prism

The figure shows two regular quadrangular prisms, which are indicated by the corresponding letters:

• The bases ABCD and A 1 B 1 C 1 D 1 are equal and parallel to each other
• Side faces AA 1 D 1 D, AA 1 B 1 B, BB 1 C 1 C and CC 1 D 1 D, each of which is a rectangle
• Side surface- the sum of the areas of all lateral faces of the prism
• Total surface - the sum of the areas of all bases and side faces (sum of the area of ​​the side surface and bases)
• Side ribs AA 1, BB 1, CC 1 and DD 1.
• Diagonal B 1 D
• Base diagonal BD
• Diagonal section BB 1 D 1 D
• Perpendicular section A 2 B 2 C 2 D 2.

Properties of a regular quadrangular prism

• The bases are two equal squares
• The bases are parallel to each other
• The side faces are rectangles
• The side edges are equal to each other
• Side faces are perpendicular to the bases
• The lateral ribs are parallel to each other and equal
• Perpendicular section perpendicular to all side ribs and parallel to the bases
• Angles perpendicular section- straight
• The diagonal cross section of a regular quadrangular prism is a rectangle
• Perpendicular (orthogonal section) parallel to the bases

Formulas for a regular quadrangular prism

Instructions for solving problems

When solving problems on the topic " correct quadrangular prism " means that:

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

Task.

In a regular quadrangular prism, the base area is 144 cm 2 and the height is 14 cm. Find the diagonal of the prism and the total surface area.

Solution.
A regular quadrilateral is a square.
Accordingly, the side of the base will be equal

√144 = 12 cm.
From where the diagonal of the base of a regular rectangular prism will be equal to
√(12 2 + 12 2 ) = √288 = 12√2

The diagonal of a regular prism forms 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

Task

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

Solution.
Since the base of a regular quadrangular prism is a square, we find the side of the base (denoted as a) using the Pythagorean theorem:

A 2 + a 2 = 5 2
2a 2 = 25
a = √12.5

The height of the side face (denoted as h) will then be equal to:

H 2 + 12.5 = 4 2
h 2 + 12.5 = 16
h 2 = 3.5
h = √3.5

The total surface area will be equal to the sum of the lateral surface area and twice the base area

S = 2a 2 + 4ah
S = 25 + 4√12.5 * √3.5
S = 25 + 4√43.75
S = 25 + 4√(175/4)
S = 25 + 4√(7*25/4)
S = 25 + 10√7 ≈ 51.46 cm 2.

Answer: 25 + 10√7 ≈ 51.46 cm 2.

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

General theory

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

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

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

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

Triangular prism

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

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

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

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

Second: S = ½ n a * a.

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

Quadrangular prism

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

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

When we're talking about about a quadrangular prism, then the area of ​​the base of a regular prism is calculated using the formula for a square. Because it is he who lies at the foundation. S = a 2.

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

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

Regular pentagonal prism

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

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

Regular hexagonal prism

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

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

Tasks

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

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

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

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

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

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

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

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

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