What lies at the base of the parallelepiped. Rectangular parallelepiped - Knowledge Hypermarket

In this lesson, we will give a definition of a parallelepiped, discuss its structure and its elements (parallelepiped diagonals, parallelepiped sides and their properties). And also consider the properties of the faces and diagonals of the parallelogram. Next, we will solve a typical problem of constructing a section in a parallelepiped.

Topic: Parallelism of lines and planes

Lesson: Parallelepiped. Properties of faces and diagonals of a parallelepiped

In this lesson we will give a definition of a parallelepiped, discuss its structure, properties and its elements (sides, diagonals).

The parallelepiped is formed by two equal parallelograms ABCD and A 1 B 1 C 1 D 1, which are in parallel planes. Designation: ABCDA 1 B 1 C 1 D 1 or AD 1 (Fig. 1.).

2. Festival of pedagogical ideas "Open Lesson" ()

1. Geometry. Grades 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th edition, revised and supplemented - M .: Mnemozina, 2008. - 288 p .: ill.

Tasks 10, 11, 12 p. 50

2. Construct a section of a rectangular parallelepiped ABCDA1B1C1D1plane passing through the points:

a) A, C, B1

b) B1, D1and the middle of the rib AA1.

3. The edge of the cube is equal to a. Construct a section of the cube with a plane passing through the midpoints of three edges extending from one vertex, and calculate its perimeter and area.

4. What shapes can be obtained as a result of intersection of a parallelepiped plane?

A parallelepiped is a quadrangular prism, at the bases of which are parallelograms. The height of a parallelepiped is the distance between the planes of its bases. In the figure, the height is shown by the line ... There are two types of parallelepipeds: straight and oblique. Typically, a math tutor first gives the appropriate definitions for the prism, and then transfers them to the box. We will do the same.

Let me remind you that a prism is called straight if its lateral edges are perpendicular to the bases, if there is no perpendicularity, the prism is called inclined. This terminology is inherited by the parallelepiped. A straight parallelepiped is nothing more than a kind of straight prism, the side edge of which coincides with the height. The definitions of concepts such as face, edge, and vertex, which are common to the entire family of polyhedra, are retained. The concept of opposite sides appears. The parallelepiped has 3 pairs of opposite faces, 8 vertices and 12 edges.

Diagonal of a parallelepiped (diagonal of a prism) is a segment that connects two vertices of a polyhedron and does not lie in any of its faces.

Diagonal section - a section of a parallelepiped passing through its diagonal and the diagonal of its base.

Sloped Box Properties:
1) All its faces are parallelograms, and the opposite faces are equal parallelograms.
2) The diagonals of the parallelepiped intersect at one point and are divided in half at this point.
3) Each parallelepiped consists of six triangular pyramids of equal volume. To show them to the student, the math tutor must cut off a half of its diagonal section from the parallelepedo and break it separately into 3 pyramids. Their foundations must lie in different facets the original parallelepiped. A math tutor will find an application of this property in analytic geometry... It is used to output the volume of a pyramid through a mixed product of vectors.

Volume formulas for a parallelepiped:
1), where is the area of \u200b\u200bthe base, h is the height.
2) The volume of a parallelepiped is equal to the product of the cross-sectional area by the side edge.
Tutor in mathematics: As you know, the formula is common for all prisms, and if the tutor has already proved it, there is no point in repeating the same thing for a parallelepiped. However, when working with a student of an intermediate level (a weak formula is not useful), it is advisable for the teacher to act exactly the opposite. Leave the prism alone and carry out a neat proof for the parallelepiped.
3), where is the volume of one of the six triangular pyramids of which the parallelepiped consists.
4) If, then

The area of \u200b\u200bthe side surface of a parallelepiped is the sum of the areas of all its faces:
The full surface of a parallelepiped is the sum of the areas of all its faces, that is, area + two areas of the base:.

About the work of a tutor with an inclined parallelepiped:
A math tutor does not often deal with problems on an oblique parallelepiped. The likelihood of their appearance on the exam is rather small, and the didactics is indecently poor. A more or less decent problem on the volume of an inclined parallelepiped causes serious problems associated with determining the location of the point H - the base of its height. In this case, the math tutor may be advised to cut the parallelepiped to one of its six pyramids (which are referred to in property # 3), try to find its volume and multiply it by 6.

If the side edge of the parallelepiped has equal angles with base sides, then H lies on the bisector of the angle A of the base ABCD. And if, for example, ABCD is a rhombus, then

Math Tutor Tasks:
1) The edges of the parallelepiped are equal ribs with a side of 2 cm and an acute angle. Find the volume of a parallelepiped.
2) In an inclined parallelepiped, the side edge is 5 cm. The section perpendicular to it is a quadrangle with mutually perpendicular diagonals having lengths of 6 cm and 8 cm. Calculate the volume of the parallelepiped.
3) In an inclined parallelepiped it is known that, and in ABCD is a rhombus with a side of 2 cm and an angle. Determine the volume of the parallelepiped.

Tutor in mathematics, Alexander Kolpakov

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

Types of parallelepiped

There are several types of parallelepipeds:

• A rectangular parallelepiped is a parallelepiped with all faces as rectangles.
• A rectangular parallelepiped is a parallelepiped with 4 side faces rectangles.
• An oblique parallelepiped is a parallelepiped whose side faces are not perpendicular to the bases.

Main 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 box that do not belong to the same face are called opposite. The segment connecting opposite vertices is called the parallelepiped's diagonal. The lengths of three edges of a rectangular parallelepiped that have a common vertex are called measurements.

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 by it in half; in particular, all the diagonals of the parallelepiped meet at one point and are halved by it.
• Opposite faces of the box are parallel and equal.
• The square of the length of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Basic formulas

Straight parallelepiped

Side surface area S b \u003d P o * h, where P 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 base area

Volume V \u003d S o * h

Rectangular parallelepiped

Side surface area S b \u003d 2c (a + b), where a, b are the sides of the base, c is the lateral edge of a rectangular parallelepiped

Total surface area S p \u003d 2 (ab + bc + ac)

Volume V \u003d abc, where a, b, c - measurements of a rectangular parallelepiped.

Cube

Surface area: $S\; \backslash u003d\; 6a\; ^\; 2$
Volume: $V\; \backslash u003d\; a\; ^\; 3$where $a$ - the edge of the cube.

Arbitrary parallelepiped

The volume and ratios in an oblique parallelepiped are often defined using vector algebra. The volume of the 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 ratio between the lengths of the sides of the parallelepiped and the angles between them gives the assertion 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 $x\; \backslash u003d\; (x\_1,\; \backslash \backslash \; ldots,\; x\_n)$ kind $B\; \backslash u003d\; \backslash \backslash \; (x\; |\; a\_1\; \backslash \backslash \; leqslant\; x\_1\; \backslash \backslash \; leqslant\; b\_1,\; \backslash \backslash \; ldots,\; a\_n\; \backslash \backslash \; leqslant\; x\_n\; \backslash \backslash \; leqslant\; b\_n\; \backslash \backslash )$

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An excerpt characterizing the Box

- On dit que les rivaux se sont reconcilies grace a l "angine ... [It is said that the rivals were reconciled due to this disease.]
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 wept 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, 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, "Anna Pavlovna said 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 inquire about her health. I was told she was a little better. Oh, without a doubt, this is the prettiest woman in the world. We belong to different camps, but this does not prevent me from respecting her according to her merits. She's so unhappy.] - added Anna Pavlovna.
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 up, but he was treating the countess with a charlatan who could give dangerous remedies.
“Vos informations peuvent etre meilleures que les miennes,” Anna Pavlovna suddenly snapped 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 truer than mine ... but I know from good sources that this doctor is a very learned and skillful person. This is the health care provider 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 in order to say un mot, was talking about the Austrians.
- Je trouve que c "est charmant! [I find it charming!] - he said about the diplomatic paper in which the Austrian banners were sent to Vienna, taken by Wittgenstein, le heros de Petropol [hero of Petropolis] (as he was called in Petersburg).
- How, how is it? Anna Pavlovna turned to him, arousing silence to hear the mot, which she already knew.
And Bilibin repeated the following authentic words of the diplomatic dispatch he had composed:
- L "Empereur renvoie les drapeaux Autrichiens," said Bilibin, "drapeaux amis et egares qu" il a trouve hors de la route, [The emperor sends Austrian banners, friendly and lost 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 a Warsaw road, maybe.]" Said Prince Ippolit loudly and unexpectedly. Everyone looked at him, not understanding what he wanted to say by this. Prince Ippolit also looked around with cheerful surprise He, like the others, did not understand what the words he said meant. During his diplomatic career, he noticed more than once that the words spoken in this way suddenly turned out to be very witty, and he said these words just in case, the first ones who came to his tongue. “Maybe it will work out very well,” he thought, “but 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, which was waiting for the appeal Anna Pavlovna, and she, smiling and shaking her finger at Ippolita, invited Prince Vasily to the table, and, bringing him two candles and a manuscript, asked him to begin.

In this lesson, everyone will be able to study the topic "Rectangular parallelepiped". At the beginning of the lesson, we will repeat what an arbitrary and straight parallelepipeds are, recall the properties of their opposite faces and diagonals of a parallelepiped. Then we will consider what a rectangular parallelepiped is and discuss its basic properties.

Topic: Perpendicularity of lines and planes

Lesson: Rectangular Parallelepiped

A surface made up of two equal parallelograms ABCD and A 1 B 1 C 1 D 1 and four parallelograms ABB 1 A 1, BCC 1 B 1, CDD 1 C 1, DAA 1 D 1 is called parallelepiped (fig. 1).

Figure: 1 Parallelepiped

That is: we have two equal parallelograms ABCD and A 1 B 1 C 1 D 1 (base), they lie in parallel planes so that the lateral edges AA 1, BB 1, DD 1, CC 1 are parallel. Thus, a surface composed of parallelograms is called parallelepiped.

Thus, the surface of a parallelepiped is the sum of all parallelograms that make up the parallelepiped.

1. Opposite faces of the box are parallel and equal.

(the shapes are equal, that is, they can be combined by overlay)

For example:

ABCD \u003d A 1 B 1 C 1 D 1 (equal parallelograms by definition),

AA 1 B 1 B \u003d DD 1 C 1 C (since AA 1 B 1 B and DD 1 C 1 C are opposite sides of the parallelepiped),

AA 1 D 1 D \u003d BB 1 C 1 C (since AA 1 D 1 D and BB 1 C 1 C are opposite faces of the parallelepiped).

2. The diagonals of the parallelepiped intersect at one point and are halved by this point.

The diagonals of the parallelepiped AC 1, B 1 D, A 1 C, D 1 B intersect at one point O, and each diagonal is divided by this point in half (Fig. 2).

Figure: 2 The diagonals of the parallelepiped intersect and are halved by the intersection point.

3. There are three fours equal and parallel ribs parallelepiped: 1 - AB, A 1 B 1, D 1 C 1, DC, 2 - AD, A 1 D 1, B 1 C 1, BC, 3 - AA 1, BB 1, CC 1, DD 1.

Definition. A parallelepiped is called straight if its lateral edges are perpendicular to the bases.

Let the lateral edge AA 1 be perpendicular to the base (Fig. 3). This means that straight line AA 1 is perpendicular to straight lines AD and AB, which lie in the plane of the base. This means that rectangles lie in the side faces. And the bases are arbitrary parallelograms. We denote ∠BAD \u003d φ, the angle φ can be any.

Figure: 3 Straight parallelepiped

So, a straight parallelepiped is a parallelepiped in which the side edges are perpendicular to the bases of the parallelepiped.

Definition. The parallelepiped is called rectangular, if its lateral ribs are perpendicular to the base. The bases are rectangles.

Parallelepiped ABCDA 1 B 1 C 1 D 1 - rectangular (Fig. 4), if:

1. AA 1 ⊥ ABCD (lateral edge perpendicular to the plane of the base, that is, a straight parallelepiped).

2. ∠BAD \u003d 90 °, that is, there is a rectangle at the base.

Figure: 4 Rectangular parallelepiped

A rectangular parallelepiped has all the properties of an arbitrary parallelepiped. But there are additional properties that are derived from the definition of a rectangular parallelepiped.

So, rectangular parallelepiped is a parallelepiped with lateral edges perpendicular to the base. The base of the rectangular parallelepiped is a rectangle.

1. In a rectangular parallelepiped, all six faces are rectangles.

ABCD and A 1 B 1 C 1 D 1 - rectangles by definition.

2. Side ribs perpendicular to the base... This means that all the side faces of a rectangular parallelepiped are rectangles.

3. All dihedral corners of a rectangular parallelepiped are straight.

Consider, for example, the dihedral angle of a rectangular parallelepiped with an edge AB, that is, the dihedral angle between the planes ABB 1 and ABC.

AB is an edge, point A 1 lies in one plane - in plane ABB 1, and point D in another - in plane A 1 B 1 C 1 D 1. Then the considered dihedral angle can also be denoted as follows: ∠A 1 ABD.

Take point A on edge AB. AA 1 - perpendicular to the edge AB in the plane ABB-1, AD perpendicular to the edge AB in the plane ABC. Hence, ∠А 1 АD is the linear angle of the given dihedral angle. ∠А 1 АD \u003d 90 °, which means that the dihedral angle at the edge AB is 90 °.

∠ (ABB 1, ABC) \u003d ∠ (AB) \u003d ∠A 1 ABD \u003d ∠A 1 AD \u003d 90 °.

It is proved similarly that any dihedral angles of a rectangular parallelepiped are straight.

The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Note. The lengths of the three edges emanating from one vertex of the rectangle are the dimensions of the rectangular parallelepiped. They are sometimes called length, width, height.

Given: ABCDA 1 B 1 C 1 D 1 - rectangular parallelepiped (Fig. 5).

Prove:.

Figure: 5 Rectangular parallelepiped

Evidence:

Straight line SS 1 is perpendicular to the plane ABC, and hence to straight line AC. This means that triangle CC 1 A is rectangular. By the Pythagorean theorem:

Consider right triangle ABC. By the Pythagorean theorem:

But BC and AD are opposite sides of the rectangle. Hence, BC \u003d AD. Then:

As , and then. Since CC 1 \u003d AA 1, then what was required to prove.

The diagonals of a rectangular parallelepiped are equal.

Let's designate the measurements of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 \u003d CA 1 \u003d B 1 D \u003d DB 1 \u003d

In geometry, the key concepts are plane, point, line and angle. Using these terms, you can describe any geometric shape. Polyhedra are usually described in terms of simpler shapes that lie in the same plane, such as a circle, triangle, square, rectangle, etc. In this article, we will consider what a parallelepiped is, describe the types of parallelepipeds, its properties, what elements it consists of, and also give the basic formulas for calculating the area and volume for each type of parallelepiped.

Definition

A parallelepiped in three-dimensional space is a prism, all sides of which are parallelograms. Accordingly, it can only have three pairs of parallel parallelograms or six faces.

To render a box, imagine a regular standard brick. Brick is a good example of a rectangular parallelepiped that even a child can imagine. Other examples include multi-story panel houses, cabinets, suitably shaped food storage containers, etc.

Varieties of figure

There are only two types of parallelepipeds:

1. Rectangular, all side faces of which are at an angle of 90 ° to the base and are rectangles.
2. Inclined, the side edges of which are located at a certain angle to the base.

What elements can this figure be divided into?

• As in any other geometric figure, in a parallelepiped, any 2 faces with a common edge are called adjacent, and those that do not have it are parallel (based on the property of a parallelogram that has pairwise parallel opposite sides).
• The vertices of a parallelepiped that do not lie on the same face are called opposite.
• The line segment connecting such vertices is a diagonal.
• The lengths of the three edges of a rectangular parallelepiped connecting at one vertex are its measurements (namely, its length, width, and height).

Shape properties

1. It is always built symmetrically in relation to the middle of the diagonal.
2. The intersection point of all diagonals divides each diagonal into two equal segments.
3. Opposing faces are equal in length and lie on parallel straight lines.
4. If you add the squares of all the dimensions of the parallelepiped, the resulting value will be equal to the square of the length of the diagonal.

Calculation formulas

Formulas for each particular case of a parallelepiped will be different.

For an arbitrary parallelepiped, it is true that its volume is equal to the absolute value of the triple dot product vectors of three sides outgoing from one vertex. However, there is no formula for calculating the volume of an arbitrary parallelepiped.

For a rectangular parallelepiped, the following formulas apply:

• V \u003d a * b * c;
• Sb \u003d 2 * c * (a + b);
• Sп \u003d 2 * (a * b + b * c + a * c).

• V is the volume of the figure;
• Sb - lateral surface area;
• Sп - total surface area;
• a - length;
• b - width;
• c - height.

Another special case of a parallelepiped, in which all sides are squares, is a cube. If any of the sides of the square is designated by the letter a, then the following formulas can be used for the surface area and volume of this figure:

• S \u003d 6 * a * 2;
• V \u003d 3 * a.

• S - figure area,
• V is the volume of the figure,
• a - the length of the face of the figure.

The last type of parallelepiped we are considering is a straight parallelepiped. What is the difference between a rectangular parallelepiped and a rectangular parallelepiped, you ask. The fact is that the base of a rectangular parallelepiped can be any parallelogram, and only a rectangle can be the base of a straight line. If we designate the perimeter of the base, equal to the sum of the lengths of all sides, as Po, and designate the height with the letter h, we have the right to use the following formulas to calculate the volume and areas of the full and lateral surfaces.