Or (equivalently) a polyhedron with six faces and each of them - parallelogram.

Types of box

There are several types of parallelepipeds:

• A cuboid is a cuboid whose faces are all rectangles.
• A right parallelepiped is a parallelepiped with 4 side faces that are rectangles.
• An oblique box is a box 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 line segment connecting opposite vertices is called the diagonal of the parallelepiped. The lengths of three edges of a cuboid that have a common vertex are called its dimensions.

Properties

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

Basic Formulas

Right parallelepiped

Lateral surface area S b \u003d R o * h, where R o is the perimeter of the base, h is the height

Total surface area S p \u003d S b + 2S o, where S o is the area of ​​\u200b\u200bthe base

Volume V=S o *h

cuboid

Lateral surface area S b \u003d 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 \u003d 2 (ab + bc + ac)

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

Cube

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

Arbitrary box

The volume and ratios in a skew box are often defined using vector algebra. The volume of a parallelepiped is equal to the absolute value of the mixed product of three vectors defined by the three sides of the parallelepiped coming from one vertex. The ratio between the lengths of the sides of the parallelepiped and the angles between them gives the statement that the Gram determinant of these three vectors is equal to the square of their mixed product: 215 .

In mathematical analysis

In mathematical analysis, under an n-dimensional rectangular parallelepiped $B$ 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 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,” said Anna Pavlovna, coming up. - 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, - said Anna Pavlovna with a smile over 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. I was told that she was a little better. Oh, without a doubt, this is the most beautiful woman in the world. We belong to different camps, but this does not prevent me from respecting her according to her merits. She is so unhappy.] Anna Pavlovna added.
Believing that with these words Anna Pavlovna slightly lifted the veil of secrecy over the countess's illness, one careless young man allowed himself to express surprise that famous doctors were not called, but a charlatan who could give dangerous means was treating the countess.
“Vos informations peuvent etre meilleures que les miennes,” Anna Pavlovna suddenly lashed out venomously at the inexperienced young man. 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, picking up the skin and, apparently, about to dissolve it, to say un mot, spoke about the Austrians.
- Je trouve que c "est charmant! [I find it charming!] - he said about a diplomatic paper, under which the Austrian banners taken by Wittgenstein were sent to Vienna, le heros de Petropol [the hero of Petropolis] (as he was called in Petersburg).
- How, how is it? Anna Pavlovna turned to him, rousing silence to hear mot, which she already knew.
And Bilibin repeated the following authentic words of the diplomatic dispatch he had compiled:
- L "Empereur renvoie les drapeaux Autrichiens," Bilibin said, "drapeaux amis et egares qu" il a trouve hors de la route, [The Emperor sends the Austrian banners, friendly and misguided banners that he found off the real road.] - finished Bilibin loosening the skin.
- Charmant, charmant, [Charming, charming,] - said Prince Vasily.
- C "est la route de Varsovie peut etre, [This is the Warsaw road, maybe.] - Prince Hippolyte said loudly and unexpectedly. Everyone looked at him, not understanding what he wanted to say with this. Prince Hippolyte also looked around with cheerful surprise around him. He, like others, did not understand what the words he said meant. During his diplomatic career, he noticed more than once that words suddenly spoken in this way turned out to be very witty, and just in case, he said these words, "Maybe it will turn out very well," he thought, "but if it doesn't, they'll be able to arrange it there." Anna Pavlovna, and she, smiling and shaking her finger at Ippolit, invited Prince Vasily to the table, and, bringing him two candles and a manuscript, asked him to begin.

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A parallelepiped is a prism whose bases are parallelograms. In this case, all edges will 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 BC "C" and ADA "D").
The body under consideration has twelve edges, four equal and parallel to each other.
Theorem 3 . The diagonals of the parallelepiped intersect at one point, coinciding with the midpoint 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", which has 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 base plane.
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 coming out of the same vertex with it, and will therefore be perpendicular to the planes of the faces defined by these edges. In contrast, a straight, but not rectangular, box can be viewed as a right prism in only one way.
Definition 9 . The lengths of three edges of a cuboid, of which no two are parallel to each other (for example, three edges coming out of 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 and the angles of all faces are equal or complementary is called a rhombohedron.
All faces of a rhombohedron are equal rhombuses. (The shape of a rhombohedron has some crystals of great importance, for example, crystals of Iceland spar.) In a rhombohedron, one can find such a vertex (and even two opposite vertices) that all 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 three dimensions.
In a rectangular parallelepiped ABCDA "B" C "D" (Fig. 6), the diagonals AC "and BD" are equal, since the quadrilateral ABC "D" is a rectangle (line AB is perpendicular to the plane BC "C", in which lies BC ") .
In addition, AC" 2 =BD" 2 = AB2+AD" 2 based on the hypotenuse square theorem. But based on the same theorem AD" 2 = AA" 2 + +A"D" 2; hence we have:
AC "2 \u003d AB 2 + AA" 2 + A "D" 2 \u003d AB 2 + AA "2 + AD 2.

Often students indignantly ask: “How will this be useful to me in life?”. On any topic of each subject. The topic about the volume of a parallelepiped is no exception. And here it is just possible to say: "It will come in handy."

How, for example, to find out if a parcel will fit in a mailbox? Of course, you can choose the right one by trial and error. What if there is no such possibility? Then calculations will come to the rescue. Knowing the capacity of the box, you can calculate the volume of the parcel (at least approximately) and answer the question.

Parallelepiped and its types

If we literally translate its name from ancient Greek, it turns out that this is a figure consisting of parallel planes. There are such equivalent definitions of a parallelepiped:

• a prism with a base in the form of a parallelogram;
• polyhedron, each face of which is a parallelogram.

Its types are distinguished depending on which figure lies at its base and how the side ribs are directed. In general, one speaks of oblique parallelepiped whose base and all faces are parallelograms. If the side faces of the previous view become rectangles, then it will need to be called already direct. And at rectangular and the base also has 90º angles.

Moreover, in geometry they try to depict the latter in such a way that it is noticeable that all the edges are parallel. Here, by the way, the main difference between mathematicians and artists is observed. It is important for the latter to convey the body in compliance with the law of perspective. And in this case, the parallelism of the edges is completely invisible.

About the introduced notation

In the formulas below, the designations indicated in the table are valid.

Formulas for an oblique box

The first and second for areas:

The third one is for calculating the volume of the box:

Since the base is a parallelogram, to calculate its area, you will need to use the appropriate expressions.

Formulas for a cuboid

Similarly to the first paragraph - two formulas for areas:

And one more for volume:

First task

Condition. Given a rectangular parallelepiped whose volume is to be found. The diagonal is known - 18 cm - and the fact that it forms angles of 30 and 45 degrees with the plane of the side face and the side edge, respectively.

Solution. To answer the question of the problem, you need to find out all the sides in three right triangles. They will give the necessary edge values ​​for which you need to calculate the volume.

First you need to figure out where the 30º angle is. To do this, you need to draw a diagonal of the side face from the same vertex from which the main diagonal of the parallelogram was drawn. The angle between them will be what you need.

The first triangle, which will give one of the sides of the base, will be the following. It contains the desired side and two diagonals drawn. It is rectangular. Now you need to use the ratio of the opposite leg (base side) and the hypotenuse (diagonal). It is equal to the sine of 30º. That is, the unknown side of the base will be determined as the diagonal multiplied by the sine of 30º or ½. Let it be marked with the letter "a".

The second will be a triangle containing a known diagonal and an edge with which it forms 45º. It is also rectangular, and you can again use the ratio of the leg to the hypotenuse. In other words, the side edge to the diagonal. It is equal to the cosine of 45º. That is, "c" is calculated as the product of the diagonal and the cosine of 45º.

c = 18 * 1/√2 = 9 √2 (cm).

In the same triangle, you need to find another leg. This is necessary in order to then calculate the third unknown - "in". Let it be marked with the letter "x". It is easy to calculate using the Pythagorean theorem:

x \u003d √ (18 2 - (9 √ 2) 2) \u003d 9 √ 2 (cm).

Now we need to consider another right triangle. It contains the already known sides "c", "x" and the one that needs to be counted, "c":

c \u003d √ ((9 √ 2) 2 - 9 2 \u003d 9 (cm).

All three quantities are known. You can use the formula for volume and calculate it:

V \u003d 9 * 9 * 9√2 \u003d 729√2 (cm 3).

Answer: the volume of the parallelepiped is 729√2 cm 3 .

Second task

Condition. Find the volume of the parallelepiped. It knows the sides of the parallelogram that lies at the base, 3 and 6 cm, as well as its acute angle - 45º. The lateral rib has an inclination to the base of 30º and is equal to 4 cm.

Solution. To answer the question of the problem, you need to take the formula that was written for the volume of an inclined parallelepiped. But both quantities are unknown in it.

The area of ​​\u200b\u200bthe base, that is, the parallelogram, will be determined by the formula in which you need to multiply the known sides and the sine of the acute angle between them.

S o \u003d 3 * 6 sin 45º \u003d 18 * (√2) / 2 \u003d 9 √2 (cm 2).

The second unknown is the height. It can be drawn from any of the four vertices above the base. It can be found from a right triangle, in which the height is the leg, and the side edge is the hypotenuse. In this case, an angle of 30º lies opposite the unknown height. So, you can use the ratio of the leg to the hypotenuse.

n \u003d 4 * sin 30º \u003d 4 * 1/2 \u003d 2.

Now all values ​​​​are known and you can calculate the volume:

V \u003d 9 √2 * 2 \u003d 18 √2 (cm 3).

Answer: the volume is 18 √2 cm 3 .

Third task

Condition. Find the volume of the parallelepiped if it is known to be a straight line. The sides of its base form a parallelogram and are equal to 2 and 3 cm. The acute angle between them is 60º. The smaller diagonal of the parallelepiped is equal to the larger diagonal of the base.

Solution. In order to find out the volume of a parallelepiped, we use the formula with the base area and height. Both quantities are unknown, but they are easy to calculate. The first one is height.

Since the smaller diagonal of the parallelepiped is the same size as the larger base, they can be denoted by the same letter d. The largest angle of a parallelogram is 120º, since it forms 180º with an acute one. Let the second diagonal of the base be denoted by the letter "x". Now, for the two diagonals of the base, cosine theorems can be written:

d 2 \u003d a 2 + in 2 - 2av cos 120º,

x 2 \u003d a 2 + in 2 - 2ab cos 60º.

Finding values ​​without squares does not make sense, since then they will be raised to the second power again. After substituting the data, it turns out:

d 2 \u003d 2 2 + 3 2 - 2 * 2 * 3 cos 120º \u003d 4 + 9 + 12 * ½ \u003d 19,

x 2 \u003d a 2 + in 2 - 2ab cos 60º \u003d 4 + 9 - 12 * ½ \u003d 7.

Now the height, which is also the side edge of the parallelepiped, will be the leg in the triangle. The hypotenuse will be the known diagonal of the body, and the second leg will be "x". You can write the Pythagorean Theorem:

n 2 \u003d d 2 - x 2 \u003d 19 - 7 \u003d 12.

Hence: n = √12 = 2√3 (cm).

Now the second unknown quantity is the area of ​​the base. It can be calculated using the formula mentioned in the second problem.

S o \u003d 2 * 3 sin 60º \u003d 6 * √3/2 \u003d 3 √3 (cm 2).

Combining everything into a volume formula, we get:

V = 3√3 * 2√3 = 18 (cm 3).

Answer: V \u003d 18 cm 3.

The fourth task

Condition. It is required to find out the volume of a parallelepiped that meets the following conditions: the base is a square with a side of 5 cm; side faces are rhombuses; one of the vertices above the base is equidistant from all the vertices lying at the base.

Solution. First you need to deal with the condition. There are no questions with the first paragraph about the square. The second, about rhombuses, makes it clear that the parallelepiped is inclined. Moreover, all its edges are equal to 5 cm, since the sides of the rhombus are the same. And from the third it becomes clear that the three diagonals drawn from it are equal. These are two that lie on the side faces, and the last one is inside the parallelepiped. And these diagonals are equal to the edge, that is, they also have a length of 5 cm.

To determine the volume, you will need a formula written for an inclined parallelepiped. Again, there are no known quantities in it. However, the area of ​​the base is easy to calculate because it is a square.

S o \u003d 5 2 \u003d 25 (cm 2).

A little more difficult is the case with height. It will be such in three figures: a parallelepiped, a quadrangular pyramid and an isosceles triangle. The last circumstance should be used.

Since it is a height, it is a leg in a right triangle. The hypotenuse in it will be a known edge, and the second leg is equal to half the diagonal of the square (the height is also the median). And the diagonal of the base is easy to find:

d = √(2 * 5 2) = 5√2 (cm).

The height will need to be calculated as the difference of the second degree of the edge and the square of half the diagonal and do not forget to extract the square root:

n = √ (5 2 - (5/2 * √2) 2) = √(25 - 25/2) = √(25/2) = 2.5 √2 (cm).

V \u003d 25 * 2.5 √2 \u003d 62.5 √2 (cm 3).

Answer: 62.5 √2 (cm 3).

or (equivalently) a polyhedron with six faces that are parallelograms. Hexagon.

The parallelograms that make up the parallelepiped are faces this parallelepiped, the sides of these parallelograms are parallelepiped edges, and the vertices of the parallelograms are peaks parallelepiped. Each face of a parallelepiped is parallelogram.

As a rule, any 2nd opposite faces are distinguished and called them the bases of the parallelepiped, and the remaining faces side faces of the parallelepiped. The edges of the parallelepiped that do not belong to the bases are side ribs.

The 2 faces of a cuboid that share an edge are related, and those that do not have common edges - opposite.

A segment that connects 2 vertices that do not belong to the 1st face is the diagonal of the parallelepiped.

The lengths of the edges of a cuboid that are not parallel are linear dimensions (measurements) a parallelepiped. A rectangular parallelepiped has 3 linear dimensions.

Types of parallelepiped.

There are several types of parallelepipeds:

Direct is a parallelepiped with an edge perpendicular to the plane of the base.

A cuboid with all 3 dimensions equal in magnitude is cube. Each of the faces of the cube is equal squares .

Arbitrary parallelepiped. The volume and ratios in a skew box are mostly defined using vector algebra. The volume of the box is equal to the absolute value of the mixed product of 3 vectors, which are determined by the 3 sides of the box (which come from the same vertex). The ratio between the lengths of the sides of the parallelepiped and the angles between them shows the statement that the Gram determinant of the given 3 vectors is equal to the square of their mixed product.

Properties of a parallelepiped.

• The parallelepiped is symmetrical about the midpoint of its diagonal.
• Any segment with ends that belong to the surface of the parallelepiped and which passes through the midpoint of its diagonal is divided by it into two equal parts. All diagonals of the parallelepiped intersect at the 1st point and are divided by it into two equal parts.
• Opposite faces of a parallelepiped are parallel and have equal dimensions.
• The square of the length of the diagonal of a cuboid is