A parallelepiped is a prism whose bases are parallelograms. In this case, all edges will be parallelograms.
Each parallelepiped can be considered as a prism in three different ways, since every two opposite faces can be taken as bases (in Fig. 5, faces ABCD and A"B"C"D", or ABA"B" and CDC"D", or BCB "C" and ADA"D").
The body in question has twelve edges, four equal and parallel to each other.
Theorem 3 . The diagonals of a parallelepiped intersect at one point, coinciding with the middle of each of them.
The parallelepiped ABCDA"B"C"D" (Fig. 5) has four diagonals AC", BD", CA", DB". We must prove that the midpoints of any two of them, for example AC and BD", coincide. This follows from the fact that the figure ABC"D", having equal and parallel sides AB and C"D", is a parallelogram.
Definition 7 . A right parallelepiped is a parallelepiped that is also a straight prism, that is, a parallelepiped whose side edges are perpendicular to the plane of the base.
Definition 8 . A rectangular parallelepiped is a right parallelepiped whose base is a rectangle. In this case, all its faces will be rectangles.
A rectangular parallelepiped is a right prism, no matter which of its faces we take as the base, since each of its edges is perpendicular to the edges emerging from the same vertex, and will, therefore, be perpendicular to the planes of the faces defined by these edges. In contrast, a straight, but not rectangular, parallelepiped can be viewed as a right prism in only one way.
Definition 9 . The lengths of three edges of a rectangular parallelepiped, of which no two are parallel to each other (for example, three edges emerging from the same vertex), are called its dimensions. Two rectangular parallelepipeds having correspondingly equal dimensions are obviously equal to each other.
Definition 10 .A cube is a rectangular parallelepiped, all three dimensions of which are equal to each other, so that all its faces are squares. Two cubes whose edges are equal are equal.
Definition 11 . An inclined parallelepiped in which all edges are equal to each other and the angles of all faces are equal or complementary is called a rhombohedron.
All faces of a rhombohedron are equal rhombuses. (Some crystals of great importance have a rhombohedron shape, for example, Iceland spar crystals.) In a rhombohedron you can find a vertex (and even two opposite vertices) such that all the angles adjacent to it are equal to each other.
Theorem 4 . The diagonals of a rectangular parallelepiped are equal to each other. The square of the diagonal is equal to the sum of the squares of the three dimensions.
In the rectangular parallelepiped ABCDA"B"C"D" (Fig. 6), the diagonals AC" and BD" are equal, since the quadrilateral ABC"D" is a rectangle (the straight line AB is perpendicular to the plane ECB"C", in which BC lies") .
In addition, AC" 2 =BD" 2 = AB2+AD" 2 based on the theorem about the square of the hypotenuse. But based on the same theorem AD" 2 = AA" 2 + +A"D" 2; hence we have:
AC" 2 = AB 2 + AA" 2 + A" D" 2 = AB 2 + AA" 2 + AD 2.

Or (equivalently) a polyhedron, which has six faces and each of them - parallelogram.

Types of parallelepiped

There are several types of parallelepipeds:

• A cuboid is a parallelepiped whose faces are all rectangles.
• A right parallelepiped is a parallelepiped with 4 lateral faces that are rectangles.
• An inclined parallelepiped is a parallelepiped whose side faces are not perpendicular to the bases.

Essential elements

Two faces of a parallelepiped that do not have a common edge are called opposite, and those that have a common edge are called adjacent. Two vertices of a parallelepiped that do not belong to the same face are called opposite. The segment connecting opposite vertices is called the diagonal of the parallelepiped. The lengths of three edges of a rectangular parallelepiped that have a common vertex are called its dimensions.

Properties

• The parallelepiped is symmetrical about the middle of its diagonal.
• Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided in half by it; in particular, all diagonals of a parallelepiped intersect at one point and are bisected by it.
• The opposite faces of a parallelepiped are parallel and equal.
• The square of the diagonal length of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Basic formulas

Right parallelepiped

Lateral surface area S b =P o *h, where P o is the perimeter of the base, h is the height

Total surface area S p =S b +2S o, where S o is the base area

Volume V=S o *h

Rectangular parallelepiped

Lateral surface area S b =2c(a+b), where a, b are the sides of the base, c is the side edge of the rectangular parallelepiped

Total surface area S p =2(ab+bc+ac)

Volume V=abc, where a, b, c are the dimensions of a rectangular parallelepiped.

Cube

Surface area: $S=6a^2$
Volume: $V=a^3$, Where $a$- edge of a cube.

Any parallelepiped

The volume and ratios in an inclined parallelepiped are often determined using vector algebra. The volume of a parallelepiped is equal to the absolute value of the mixed product of three vectors determined by the three sides of the parallelepiped emanating from one vertex. The relationship between the lengths of the sides of the parallelepiped and the angles between them gives the statement that the Gram determinant of the indicated three vectors is equal to the square of their mixed product: 215.

In mathematical analysis

In mathematical analysis under an n-dimensional cuboid $B$ understand many points $x\; =\; (x\_1,\backslash ldots,x\_n)$ kind $B\; =\; \backslash (x|a\_1\backslash leqslant\; x\_1\backslash leqslant\; b\_1,\backslash ldots,a\_n\backslash leqslant\; x\_n\backslash leqslant\; b\_n\backslash )$

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An excerpt characterizing the Parallelepiped

- On dit que les rivaux se sont reconcilies grace a l "angine... [They say that the rivals were reconciled thanks to this illness.]
The word angine was repeated with great pleasure.
– Le vieux comte est touchant a ce qu"on dit. Il a pleure comme un enfant quand le medecin lui a dit que le cas etait dangereux. [The old count is very touching, they say. He cried like a child when the doctor said that dangerous case.]
- Oh, ce serait une perte terrible. C"est une femme ravissante. [Oh, that would be a great loss. Such a lovely woman.]
“Vous parlez de la pauvre comtesse,” Anna Pavlovna said, approaching. “J"ai envoye savoir de ses nouvelles. On m"a dit qu"elle allait un peu mieux. Oh, sans doute, c"est la plus charmante femme du monde," Anna Pavlovna said with a smile at her enthusiasm. – Nous appartenons a des camps differents, mais cela ne m"empeche pas de l"estimer, comme elle le merite. Elle est bien malheureuse, [You are talking about the poor countess... I sent to find out about her health. They told me she was feeling a little better. Oh, without a doubt, this is the loveliest woman in the world. We belong to different camps, but that doesn't stop me from respecting her on her merits. She is so unhappy.] – added Anna Pavlovna.
Believing that with these words Anna Pavlovna was slightly lifting the veil of secrecy over the countess’s illness, one careless young man allowed himself to express surprise that famous doctors were not called in, but that the countess was being treated by a charlatan who could give dangerous remedies.
“Vos informations peuvent etre meilleures que les miennes,” Anna Pavlovna suddenly attacked the inexperienced young man venomously. – Mais je sais de bonne source que ce medecin est un homme tres savant et tres habile. C"est le medecin intime de la Reine d"Espagne. [Your news may be more accurate than mine... but I know from good sources that this doctor is a very learned and skillful person. This is the life physician of the Queen of Spain.] - And thus destroying the young man, Anna Pavlovna turned to Bilibin, who, in another circle, picked up the skin and, apparently, about to loosen it to say un mot, spoke about the Austrians.
“Je trouve que c"est charmant! [I find it charming!],” he said about the diplomatic paper with which the Austrian banners taken by Wittgenstein were sent to Vienna, le heros de Petropol [the hero of Petropol] (as he was called in Petersburg).
- How, how is this? - Anna Pavlovna turned to him, awakening silence to hear the mot, which she already knew.
And Bilibin repeated the following original words of the diplomatic dispatch he composed:
“L"Empereur renvoie les drapeaux Autrichiens,” said Bilibin, “drapeaux amis et egares qu"il a trouve hors de la route, [The Emperor sends the Austrian banners, friendly and lost banners that he found outside the real road.],” Bilibin finished , loosening the skin.
“Charmant, charmant, [Lovely, charming," said Prince Vasily.
“C"est la route de Varsovie peut être, [This is the Warsaw road, maybe.] - Prince Hippolyte said loudly and unexpectedly. Everyone looked back at him, not understanding what he wanted to say by this. Prince Hippolyte also looked back with cheerful surprise around him. He, like others, did not understand what the words he said meant. During his diplomatic career, he more than once noticed that the words spoken in this way suddenly turned out to be very witty, and he said these words just in case. the first to come to his mind. “Maybe it will work out very well,” he thought, “and if it doesn’t work out, they will be able to arrange it there.” Indeed, while an awkward silence reigned, that insufficiently patriotic face entered. Anna Pavlovna, and she, smiling and shaking her finger at Ippolit, invited Prince Vasily to the table, and, presenting him with two candles and a manuscript, asked him to begin. Everything fell silent.

Rectangular parallelepiped

A rectangular parallelepiped is a right parallelepiped whose all faces are rectangles.

It is enough to look around us, and we will see that the objects around us have a shape similar to a parallelepiped. They can be distinguished by color, have a lot of additional details, but if these subtleties are discarded, then we can say that, for example, a cabinet, box, etc., have approximately the same shape.

We come across the concept of a rectangular parallelepiped almost every day! Look around and tell me where you see rectangular parallelepipeds? Look at the book, it's exactly the same shape! A brick, a matchbox, a block of wood have the same shape, and even right now you are inside a rectangular parallelepiped, because the classroom is the brightest interpretation of this geometric figure.

Exercise: What examples of parallelepiped can you name?

Let's take a closer look at the cuboid. And what do we see?

First, we see that this figure is formed from six rectangles, which are the faces of a cuboid;

Secondly, a cuboid has eight vertices and twelve edges. The edges of a cuboid are the sides of its faces, and the vertices of the cuboid are the vertices of the faces.

Exercise:

1. What is the name of each of the faces of a rectangular parallelepiped? 2. Thanks to what parameters can a parallelogram be measured? 3. Define opposite faces.

Types of parallelepipeds

But parallelepipeds are not only rectangular, but they can also be straight and inclined, and straight lines are divided into rectangular, non-rectangular and cubes.

Assignment: Look at the picture and say what parallelepipeds are shown on it. How does a rectangular parallelepiped differ from a cube?

Properties of a rectangular parallelepiped

A rectangular parallelepiped has a number of important properties:

Firstly, the square of the diagonal of this geometric figure is equal to the sum of the squares of its three main parameters: height, width and length.

Secondly, all four of its diagonals are absolutely identical.

Thirdly, if all three parameters of a parallelepiped are the same, that is, the length, width and height are equal, then such a parallelepiped is called a cube, and all its faces will be equal to the same square.

Exercise

1. Does a rectangular parallelepiped have equal sides? If there are any, then show them in the figure. 2. What geometric shapes do the faces of a rectangular parallelepiped consist of? 3. What is the arrangement of equal edges in relation to each other? 4. Name the number of pairs of equal faces of this figure. 5. Find the edges in a rectangular parallelepiped that indicate its length, width, height. How many did you count?

Task

To beautifully decorate a birthday present for her mother, Tanya took a box in the shape of a rectangular parallelepiped. The size of this box is 25cm*35cm*45cm. To make this packaging beautiful, Tanya decided to cover it with beautiful paper, the cost of which is 3 hryvnia per 1 dm2. How much money should you spend on wrapping paper?

Do you know that the famous illusionist David Blaine spent 44 days in a glass parallelepiped suspended over the Thames as part of an experiment. For these 44 days he did not eat, but only drank water. In his voluntary prison, David took only writing materials, a pillow and mattress, and handkerchiefs.

It will be useful for high school students to learn how to solve Unified State Examination problems to find the volume and other unknown parameters of a rectangular parallelepiped. The experience of previous years confirms the fact that such tasks are quite difficult for many graduates.

At the same time, high school students with any level of training should understand how to find the volume or area of ​​a rectangular parallelepiped. Only in this case will they be able to count on receiving competitive scores based on the results of passing the unified state exam in mathematics.

Key points to remember

• The parallelograms that make up a parallelepiped are its faces, their sides are its edges. The vertices of these figures are considered the vertices of the polyhedron itself.
• All diagonals of a rectangular parallelepiped are equal. Since this is a straight polyhedron, the side faces are rectangles.
• Since a parallelepiped is a prism with a parallelogram at its base, this figure has all the properties of a prism.
• The lateral edges of a rectangular parallelepiped are perpendicular to the base. Therefore, they are its heights.

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Definition

Polyhedron we will call a closed surface composed of polygons and bounding a certain part of space.

The segments that are the sides of these polygons are called ribs polyhedron, and the polygons themselves are edges. The vertices of polygons are called polyhedron vertices.

We will consider only convex polyhedra (this is a polyhedron that is located on one side of each plane containing its face).

The polygons that make up a polyhedron form its surface. The part of space that is bounded by a given polyhedron is called its interior.

Definition: prism

Consider two equal polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) located in parallel planes so that the segments \(A_1B_1, \A_2B_2, ..., A_nB_n\) parallel. A polyhedron formed by the polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) , as well as parallelograms \(A_1B_1B_2A_2, \A_2B_2B_3A_3, ...\), is called (\(n\)-gonal) prism.

Polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) are called prism bases, parallelograms \(A_1B_1B_2A_2, \A_2B_2B_3A_3, ...\)– side faces, segments \(A_1B_1, \ A_2B_2, \ ..., A_nB_n\)- lateral ribs.
Thus, the lateral edges of the prism are parallel and equal to each other.

Let's look at an example - a prism \(A_1A_2A_3A_4A_5B_1B_2B_3B_4B_5\), at the base of which lies a convex pentagon.

Height prisms are a perpendicular dropped from any point of one base to the plane of another base.

If the side edges are not perpendicular to the base, then such a prism is called inclined(Fig. 1), otherwise – straight. In a straight prism, the side edges are heights, and the side faces are equal rectangles.

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

Definition: concept of volume

The unit of volume measurement is a unit cube (a cube measuring \(1\times1\times1\) units\(^3\), where unit is a certain unit of measurement).

We can say that the volume of a polyhedron is the amount of space that this polyhedron limits. Otherwise: this is a quantity whose numerical value shows how many times a unit cube and its parts fit into a given polyhedron.

Volume has the same properties as area:

1. The volumes of equal figures are equal.

2. If a polyhedron is composed of several non-intersecting polyhedra, then its volume is equal to the sum of the volumes of these polyhedra.

3. Volume is a non-negative quantity.

4. Volume is measured in cm\(^3\) (cubic centimeters), m\(^3\) (cubic meters), etc.

Theorem

1. The area of ​​the lateral surface of the prism is equal to the product of the perimeter of the base and the height of the prism.
The lateral surface area is the sum of the areas of the lateral faces of the prism.

2. The volume of the prism is equal to the product of the base area and the height of the prism: \

Definition: parallelepiped

Parallelepiped is a prism with a parallelogram at its base.

All faces of the parallelepiped (there are \(6\) : \(4\) side faces and \(2\) bases) are parallelograms, and the opposite faces (parallel to each other) are equal parallelograms (Fig. 2).

Diagonal of a parallelepiped is a segment connecting two vertices of a parallelepiped that do not lie on the same face (there are \(8\) of them: \(AC_1,\A_1C,\BD_1,\B_1D\) etc.).

Rectangular parallelepiped is a right parallelepiped with a rectangle at its base.
Because Since this is a right parallelepiped, the side faces are rectangles. This means that in general all the faces of a rectangular parallelepiped are rectangles.

All diagonals of a rectangular parallelepiped are equal (this follows from the equality of triangles \(\triangle ACC_1=\triangle AA_1C=\triangle BDD_1=\triangle BB_1D\) etc.).

Comment

Thus, a parallelepiped has all the properties of a prism.

Theorem

The lateral surface area of ​​a rectangular parallelepiped is \

The total surface area of ​​a rectangular parallelepiped is \

Theorem

The volume of a cuboid is equal to the product of its three edges emerging from one vertex (three dimensions of the cuboid): \

Proof

Because In a rectangular parallelepiped, the lateral edges are perpendicular to the base, then they are also its heights, that is, \(h=AA_1=c\) Because the base is a rectangle, then \(S_(\text(main))=AB\cdot AD=ab\). This is where this formula comes from.

Theorem

The diagonal \(d\) of a rectangular parallelepiped is found using the formula (where \(a,b,c\) are the dimensions of the parallelepiped) \

Proof

Let's look at Fig. 3. Because the base is a rectangle, then \(\triangle ABD\) is rectangular, therefore, by the Pythagorean theorem \(BD^2=AB^2+AD^2=a^2+b^2\) .

Because all lateral edges are perpendicular to the bases, then \(BB_1\perp (ABC) \Rightarrow BB_1\) perpendicular to any straight line in this plane, i.e. \(BB_1\perp BD\) . This means that \(\triangle BB_1D\) is rectangular. Then, by the Pythagorean theorem \(B_1D=BB_1^2+BD^2=a^2+b^2+c^2\), thd.

Definition: cube

Cube is a rectangular parallelepiped, all of whose faces are equal squares.

Thus, the three dimensions are equal to each other: \(a=b=c\) . So the following are true

Theorems

1. The volume of a cube with edge \(a\) is equal to \(V_(\text(cube))=a^3\) .

2. The diagonal of the cube is found using the formula \(d=a\sqrt3\) .

3. Total surface area of ​​a cube \(S_(\text(full cube))=6a^2\).