Prism. Parallelepiped

Prismis called a polyhedron whose two faces are equal n-gons (grounds) lying in parallel planes, and the remaining n faces are parallelograms (side faces) . Side rib a prism is the side of the side face that does not belong to the base.

A prism whose side edges are perpendicular to the planes of the bases is called straight prism (Fig. 1). If the side edges are not perpendicular to the planes of the bases, then the prism is called oblique . Correct a prism is a straight prism, the bases of which are regular polygons.

Heightprism is called the distance between the planes of the bases. Diagonal prism is called a segment that connects two vertices that do not belong to the same face. Diagonal section the section of a prism is called a plane passing through two lateral edges that do not belong to one face. Perpendicular section the section of a prism is called a plane perpendicular to the lateral edge of the prism.

Side surface area prism is called the sum of the areas of all side faces. Full surface area called the sum of the areas of all faces of the prism (i.e. the sum of the areas of the side faces and the areas of the bases).

For an arbitrary prism, the following formulas are valid:

where l - the length of the side rib;

H - height;

P

Q

S side

S full

S main - the area of \u200b\u200bthe bases;

V Is the volume of the prism.

For a straight prism, the formulas are correct:

where p - base perimeter;

l - the length of the side rib;

H - height.

Parallelepiped called a prism, the base of which is a parallelogram. A parallelepiped with side edges perpendicular to the bases is called direct (fig. 2). If the side edges are not perpendicular to the bases, then the parallelepiped is called oblique ... A straight parallelepiped, the base of which is a rectangle, is called rectangular. A rectangular parallelepiped with all edges equal is called cube.

The faces of a parallelepiped that do not have common vertices are called opposing ... The lengths of the edges outgoing from one vertex are called measurements parallelepiped. Since a parallelepiped is a prism, its main elements are defined in the same way as they are defined for prisms.

Theorems.

1. The diagonals of the parallelepiped intersect at one point and are halved by it.

2. In a rectangular parallelepiped, the square of the diagonal length is equal to the sum of the squares of its three dimensions:

3. All four diagonals of a rectangular parallelepiped are equal to each other.

For an arbitrary parallelepiped, the following formulas are true:

where l - the length of the side rib;

H - height;

P - the perimeter of the perpendicular section;

Q - The area of \u200b\u200bthe perpendicular section;

S side - lateral surface area;

S full - total surface area;

S main - the area of \u200b\u200bthe bases;

V Is the volume of the prism.

For a straight parallelepiped, the following formulas are true:

where p - base perimeter;

l - the length of the side rib;

H - the height of the straight parallelepiped.

For a rectangular parallelepiped, the formulas are correct:

(3)

where p - base perimeter;

H - height;

d - diagonal;

a, b, c - measurements of the parallelepiped.

For the cube, the formulas are correct:

where a - rib length;

d Is the diagonal of the cube.

Example 1.The diagonal of a rectangular parallelepiped is 33 dm, and its measurements are related as 2: 6: 9. Find the dimensions of the parallelepiped.

Decision. To find the dimensions of a parallelepiped, we use formula (3), i.e. by the fact that the square of the hypotenuse of a rectangular parallelepiped is equal to the sum of the squares of its dimensions. Let us denote by k proportionality coefficient. Then the dimensions of the parallelepiped will be 2 k, 6k and 9 k... Let us write formula (3) for the problem data:

Solving this equation for k, we get:

This means that the dimensions of the parallelepiped are 6 dm, 18 dm and 27 dm.

Answer: 6 dm, 18 dm, 27 dm.

Example 2. Find the volume of an inclined triangular prism, the base of which is an equilateral triangle with a side of 8 cm, if the lateral edge is equal to the side of the base and is inclined at an angle of 60º to the base.

Decision . Let's make a drawing (fig. 3).

In order to find the volume of an inclined prism, it is necessary to know its base area and height. The base area of \u200b\u200bthis prism is the area of \u200b\u200ban equilateral triangle with a side of 8 cm. We calculate it:

The height of a prism is the distance between its bases. From the top AND 1 of the upper base, we lower the perpendicular to the plane of the lower base AND 1 D... Its length will be the height of the prism. Consider D AND 1 AD: since this is the angle of inclination of the lateral rib AND 1 AND to the plane of the base, AND 1 AND \u003d 8 cm.From this triangle we find AND 1 D:

Now we calculate the volume by the formula (1):

Answer: 192 cm 3.

Example 3. The lateral edge of a regular hexagonal prism is 14 cm. The area of \u200b\u200bthe largest diagonal section is 168 cm 2. Find the total surface area of \u200b\u200bthe prism.

Decision. Let's make a drawing (fig. 4)

Largest Diagonal Section - Rectangle AA 1 DD 1, since the diagonal AD regular hexagon ABCDEF is the greatest. In order to calculate the area of \u200b\u200bthe lateral surface of the prism, it is necessary to know the side of the base and the length of the lateral rib.

Knowing the area of \u200b\u200bthe diagonal section (rectangle), we find the diagonal of the base.

Since, then

Since then AB \u003d 6 cm.

Then the perimeter of the base is:

Find the area of \u200b\u200bthe lateral surface of the prism:

The area of \u200b\u200ba regular hexagon with a side of 6 cm is:

Find the total surface area of \u200b\u200bthe prism:

Answer:

Example 4. A rhombus serves as the base of a straight parallelepiped. The areas of diagonal sections are 300 cm 2 and 875 cm 2. Find the area of \u200b\u200bthe side surface of a parallelepiped.

Decision. Let's make a drawing (fig. 5).

Let us denote the side of the rhombus through and, rhombus diagonals d 1 and d 2, the height of the parallelepiped h... To find the area of \u200b\u200bthe lateral surface of a straight parallelepiped, multiply the perimeter of the base by the height: (formula (2)). Base perimeter p \u003d AB + BC + CD + DA \u003d 4AB \u003d 4a, because ABCD - rhombus. H \u003d AA 1 = h... So Need to find and and h.

Consider diagonal sections. AA 1 SS 1 - rectangle, one side of which is the diagonal of the rhombus AS = d 1, the second is a lateral rib AA 1 = hthen

Similarly for the section BB 1 DD 1 we get:

Using the property of a parallelogram such that the sum of the squares of the diagonals is equal to the sum of the squares of all its sides, we obtain the equality. We obtain the following.

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General information about a straight prism

The lateral surface of the prism (more precisely, the lateral surface area) is called sum areas of the side faces. 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 lateral 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 lateral rib.

Evidence. 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 ribs. Hence it follows that the lateral surface of the prism is

S \u003d a 1 l + a 2 l + ... + a n l \u003d pl,

where a 1 and n are the lengths of the base edges, p is the perimeter of the base of the prism, and I is the length of the side edges. The theorem is proved.

Practical task

Challenge (22) ... In an inclined prism, sectionperpendicular to the side ribs and intersecting all side ribs. Find the side surface of the prism if the cross-sectional perimeter is p and the side edges are l.

Decision. The plane of the section drawn breaks the prism into two parts (Fig. 411). Let's subject one of them to a parallel transfer, which aligns the prism bases. In this case, we get a straight prism, in which the base is the section of the original prism, and the side edges are equal to l. This prism has the same lateral surface as the original. Thus, the lateral surface of the original prism is equal to pl.

Summarizing the covered topic

And now let's try with you to sum up the past topic about a prism and remember what properties a prism has.

Prism properties

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

Also, it should be remembered that polyhedrons such as prisms can be straight and oblique.

Which prism is called a straight line?

If the side edge of the prism is located 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 kind of prism is called oblique?

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.

Which prism is called correct?

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

Now let us recall the properties that the correct prism possesses.

Correct prism properties

First, regular polygons always serve as 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 lateral ribs, then in the correct prism they are always equal.
Fourth, the correct prism is always straight;
Fifth, if in a regular prism the side faces are square, then such a figure is usually called a semi-regular polygon.

Prism section

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

Homework

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

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

Do you know that geometric shapes constantly surround us not only in geometry lessons, but in everyday life there are objects that resemble one or another geometric figure.

Every home, school or work has a computer, the system unit of which is in the form 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