Definition of the correct prism. Prism

1. The smallest number of edges has a tetrahedron - 6.

2. The prism has n faces. What polygon lies at its base?

(n - 2) - a square.

3. Is a prism straight if its two adjacent side faces are perpendicular to the plane of the base?

Yes it is.

4. In which prism are the side edges parallel to its height?

in a straight prism.

5. Is a prism regular if all its edges are equal to each other?

No, it may not be direct.

6. Can the height of one of the side faces of an inclined prism also be the height of the prism?

Yes, if this face is perpendicular to the bases.

7. Is there a prism in which: a) the lateral edge is perpendicular to only one edge of the base; b) only one side face is perpendicular to the base?

a) yes. b) no.

8. A regular triangular prism is divided by a plane passing through the midlines of the bases into two prisms. How are the areas of the lateral surfaces of these prisms?

According to the theorem of item 27, we obtain that the lateral surfaces are related as 5: 3

9. Will the pyramid be regular if its side faces are regular triangles?

10. How many faces perpendicular to the base plane can a pyramid have?

11. Is there a quadrangular pyramid whose opposite side faces are perpendicular to the base?

No, otherwise at least two straight lines, perpendicular to the bases, would pass through the top of the pyramid.

12. Can all faces of a triangular pyramid be right triangles?

Yes (Figure 183).

General information about a straight prism

The lateral surface of the prism (more precisely, the lateral surface area) is called sum side face areas. The total surface of the prism is equal to the sum of the lateral surface and the areas of the bases.

Theorem 19.1. The side surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism, i.e., the length of the side edge.

Proof. The side faces of a straight prism are rectangles. The bases of these rectangles are the sides of the polygon lying at the base of the prism, and the heights are equal to the length of the side edges. Hence it follows that side surface prism is

S = a 1 l + a 2 l + ... + a n l = pl,

where a 1 and n are the lengths of the ribs of the base, p is the perimeter of the base of the prism, and I is the length of the side ribs. The theorem has been proven.

Practical task

Task (22) . In an inclined prism section, perpendicular to the side edges and intersecting all side edges. Find the side surface of the prism if the perimeter of the section is p and the side edges are l.

Solution. The plane of the section drawn divides the prism into two parts (Fig. 411). Let's subject one of them to a parallel translation that combines the bases of the prism. In this case, we obtain a straight prism, in which the section of the original prism serves as the base, and the side edges are equal to l. This prism has the same side surface as the original one. Thus, the side surface of the original prism is equal to pl.

Generalization of the topic

And now let's try with you to summarize the topic of the prism and remember what properties a prism has.

Prism Properties

First, for a prism, all its bases are equal polygons;
Secondly, the prism has all its side faces are parallelograms;
Thirdly, in such a multifaceted figure as a prism, all side edges are equal;

Also, it should be remembered that polyhedra such as prisms can be straight and inclined.

What is a straight prism?

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

It will not be superfluous to recall that the side faces of a straight prism are rectangles.

What is an oblique prism?

But if the side edge of the prism is not located perpendicular to the plane of its base, then we can safely say that this is an inclined prism.

What is the correct prism?

If a regular polygon lies at the base of a straight prism, then such a prism is regular.

Now let's recall the properties that a regular prism has.

Properties of a regular prism

First, regular polygons always serve as the bases of a regular prism;
Secondly, if we consider the side faces of a regular prism, then they are always equal rectangles;
Thirdly, if we compare the sizes of the side ribs, then in the correct prism they are always equal.
Fourth, a regular prism is always straight;
Fifthly, if in a regular prism the side faces are in the form of squares, then such a figure, as a rule, is called a semi-regular polygon.

Prism section

Now let's look at the cross section of a prism:

Homework

And now let's try to consolidate the studied topic by solving problems.

Let's draw an inclined triangular prism, in which the distance between its edges will be: 3 cm, 4 cm and 5 cm, and the side surface of this prism will be equal to 60 cm2. With these parameters, find the lateral edge of the given prism.

And you know that geometric figures constantly surround us not only in geometry lessons, but also in Everyday life there are objects that resemble one or another geometric figure.

Every home, school or work has a computer, system unit which has the shape of a straight prism.

If you pick up a simple pencil, you will see that the main part of the pencil is a prism.

Walking along the main street of the city, we see that under our feet lies a tile that has the shape of a hexagonal prism.

A. V. Pogorelov, Geometry for grades 7-11, Textbook for educational institutions

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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. full surface there will already be a 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 the area of ​​the base in general view, 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-s)). 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 triangular prism, which is correct, 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 we are talking about a quadrangular prism, then the area of ​​\u200b\u200bthe base 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 straight 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.

Solution. 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 (h). 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.

Solution. 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.

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