Parallelepiped property definition signs. Rectangular parallelepiped - Knowledge Hypermarket
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.
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 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:
Volume: , where - 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 emanating 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 understand many points kind
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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 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 come out, they will be able to arrange it there." Indeed, while an awkward silence reigned, that insufficiently patriotic face, whom 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.
A parallelepiped is a quadrangular prism whose bases are parallelograms. The height of a parallelepiped is the distance between the planes of its bases. In the figure, the height is shown as a line 
. There are two types of parallelepipeds: straight and oblique. As a rule, a math tutor first gives the appropriate definitions for a prism, and then transfers them to a box. We will do the same.
Let me remind you that a prism is called straight if its side edges are perpendicular to the bases, if there is no perpendicularity, the prism is called oblique. This terminology is also inherited by the parallelepiped. 
A right parallelepiped is nothing more than a kind of straight prism, the lateral edge of which coincides with the height. The definitions of such concepts as a face, an edge, and a vertex, which are common to the entire family of polyhedra, are retained. The concept of opposite faces appears. A parallelepiped has 3 pairs of opposite faces, 8 vertices and 12 edges.
The diagonal of a parallelepiped (the diagonal of a prism) is a segment that connects two vertices of a polyhedron and does not lie in any of its faces.
A diagonal section is a section of a parallelepiped passing through its diagonal and the diagonal of its base.
Oblique box properties:
1) All its faces are parallelograms, and opposite faces are equal parallelograms.
2)
The diagonals of the parallelepiped intersect at one point and bisect at that point.
3)
Each parallelepiped consists of six triangular pyramids of equal volume. To show them to a student, a math tutor must cut off a half of the parallelepeped with its diagonal section and break it separately into 3 pyramids. Their bases must lie on different faces of the original box. A math tutor will find an application for this property in analytic geometry. It is used to derive the volume of the pyramid through the mixed product of vectors.
Formulas for the volume of a parallelepiped:
1) , where is the area of the base, h is the height.
2) The volume of the parallelepiped is equal to the product of the cross-sectional area by the side edge.
math tutor: As you know, the formula is common to all prisms, and if the tutor has already proven it, there is no point in repeating the same for the parallelepiped. However, when working with an average-level student (a weak formula is not useful), it is advisable for the teacher to act exactly the opposite. Leave the prism alone, and carry out an accurate proof for the parallelepiped.
3) , where is the volume of one of the six triangular pyramids that make up the parallelepiped.
4) If , then
The area of the lateral surface of a parallelepiped is the sum of the areas of all its faces:
The total surface of a parallelepiped is the sum of the areas of all its faces, that is, the area + two areas of the base:.
About the work of a tutor with an inclined parallelepiped:
A tutor in mathematics does not often deal with problems on an inclined parallelepiped. The probability of their appearance on the exam is quite 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 might be advised to trim the box to one of its six pyramids (which are discussed in property #3), try to find its volume and multiply it by 6.
If the side edge of the parallelepiped has equal angles with the sides of the base, then H lies on the bisector of angle A of the base ABCD. And if, for example, ABCD is a rhombus, then
Math Tutor Tasks:
1) The faces of a parallelepiped are equal robs with a side of 2 cm and an acute angle. Find the volume of the parallelepiped.
2) In an inclined parallelepiped, the side edge is 5 cm. The section perpendicular to it is a quadrilateral with mutually perpendicular diagonals having lengths of 6 cm and 8 cm. Calculate the volume of the parallelepiped.
3) In an oblique parallelepiped, it is known that , and in the definition of ABCD is a rhombus with a side of 2 cm and an angle of . Determine the volume of the parallelepiped.
Mathematics tutor, Alexander Kolpakov
A parallelepiped is a geometric figure, all 6 faces of which are parallelograms.
Depending on the type of these parallelograms, the following types of parallelepiped are distinguished:
- straight;
- inclined;
- rectangular.
A right parallelepiped is a quadrangular prism whose edges make an angle of 90 ° with the base plane.
A rectangular parallelepiped is a quadrangular prism, all of whose faces are rectangles. The cube is a variety quadrangular prism, in which all faces and edges are equal to each other.
The features of a figure predetermine its properties. These include the following 4 statements:
Remembering all the above properties is simple, they are easy to understand and are derived logically based on the type and features of the geometric body. However, simple statements can be incredibly useful when solving typical USE tasks and will save the time required to pass the test.
Parallelepiped formulas
To find answers to the problem, it is not enough to know only the properties of the figure. You may also need some formulas to find the area and volume of a geometric body.
The area of \u200b\u200bthe bases is also found as the corresponding indicator of a parallelogram or rectangle. You can choose the base of the parallelogram yourself. As a rule, when solving problems, it is easier to work with a prism, which is based on a rectangle.
The formula for finding the side surface of a parallelepiped may also be needed in test tasks.
Examples of solving typical USE tasks
Exercise 1.
Given: a cuboid with measurements of 3, 4 and 12 cm.
Necessary Find the length of one of the main diagonals of the figure.
Solution: Any solution to a geometric problem must begin with the construction of a correct and clear drawing, on which “given” and the desired value will be indicated. The figure below shows an example of the correct formatting of task conditions.
Having considered the drawing made and remembering all the properties of a geometric body, we come to the only correct way to solve it. Applying property 4 of the parallelepiped, we obtain the following expression:
After simple calculations, we obtain the expression b2=169, therefore, b=13. The answer to the task has been found, it should take no more than 5 minutes to search for it and draw it.
Task 2.
Given: an oblique box with a side edge of 10 cm, a KLNM rectangle with dimensions of 5 and 7 cm, which is a section of the figure parallel to the specified edge.
Necessary Find the area of the lateral surface of the quadrangular prism.
Solution: First you need to sketch the data.
To solve this task, you need to use ingenuity. It can be seen from the figure that the sides KL and AD are unequal, as well as the pair ML and DC. However, the perimeters of these parallelograms are obviously equal.
Therefore, the lateral area of the figure will be equal to the cross-sectional area multiplied by the rib AA1, since by the condition the rib is perpendicular to the section. Answer: 240 cm2.
In this lesson, everyone will be able to study the topic "Rectangular box". 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 the parallelepiped. Then we will consider what a cuboid is and discuss its main properties.
Topic: Perpendicularity of lines and planes
Lesson: Cuboid
A surface composed 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).
Rice. 1 Parallelepiped
That is: we have two equal parallelograms ABCD and A 1 B 1 C 1 D 1 (bases), they lie in parallel planes so that the side 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 the parallelograms that make up the parallelepiped.
1. Opposite faces of a parallelepiped are parallel and equal.
(the figures 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 faces 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 bisect that 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 in half by this point (Fig. 2).
Rice. 2 The diagonals of the parallelepiped intersect and bisect the intersection point.
3. There are three quadruples of equal and parallel edges of the 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, SS 1, DD 1.
Definition. A parallelepiped is called straight if its lateral edges are perpendicular to the bases.
Let the side edge AA 1 be perpendicular to the base (Fig. 3). This means that the line AA 1 is perpendicular to the lines AD and AB, which lie in the plane of the base. And, therefore, rectangles lie in the side faces. And the bases are arbitrary parallelograms. Denote, ∠BAD = φ, the angle φ can be any.
Rice. 3 Right box
So, a right box is a box in which the side edges are perpendicular to the bases of the box.
Definition. The parallelepiped is called rectangular, if its lateral edges are perpendicular to the base. The bases are rectangles.
The parallelepiped АВСДА 1 В 1 С 1 D 1 is rectangular (Fig. 4) if:
1. AA 1 ⊥ ABCD (lateral edge is perpendicular to the plane of the base, that is, a straight parallelepiped).
2. ∠BAD = 90°, i.e., the base is a rectangle.
Rice. 4 Cuboid
A rectangular box has all the properties of an arbitrary box. But there are additional properties that are derived from the definition of a cuboid.
So, cuboid is a parallelepiped whose lateral edges are perpendicular to the base. The base of a cuboid is a rectangle.
1. In a cuboid, all six faces are rectangles.
ABCD and A 1 B 1 C 1 D 1 are rectangles by definition.
2. Lateral ribs are perpendicular to the base. This means that all the side faces of a cuboid are rectangles.
3. All dihedral angles of a cuboid are right angles.
Consider, for example, the dihedral angle of a rectangular parallelepiped with an edge AB, i.e., the dihedral angle between the planes ABB 1 and ABC.
AB is an edge, point A 1 lies in one plane - in the plane ABB 1, and point D in the other - in the plane A 1 B 1 C 1 D 1. Then the considered dihedral angle can also be denoted as follows: ∠А 1 АВD.
Take point A on edge AB. AA 1 is perpendicular to the edge AB in the plane ABB-1, AD is perpendicular to the edge AB in the plane ABC. Hence, ∠A 1 AD is the linear angle of the given dihedral angle. ∠A 1 AD \u003d 90 °, which means that the dihedral angle at the edge AB is 90 °.
∠(ABB 1, ABC) = ∠(AB) = ∠A 1 ABD= ∠A 1 AD = 90°.
It is proved similarly that any dihedral angles of a rectangular parallelepiped are right.
The square of the diagonal of a cuboid is equal to the sum of the squares of its three dimensions.
Note. The lengths of the three edges emanating from the same vertex of the cuboid are the measurements of the cuboid. They are sometimes called length, width, height.
Given: ABCDA 1 B 1 C 1 D 1 - a rectangular parallelepiped (Fig. 5).
Prove: .
Rice. 5 Cuboid
Proof:
The line CC 1 is perpendicular to the plane ABC, and hence to the line AC. So triangle CC 1 A is a right triangle. According to the Pythagorean theorem:
Consider a right triangle ABC. According to the Pythagorean theorem:
But BC and AD are opposite sides of the rectangle. So BC = AD. Then:
Because , a , then. Since CC 1 = AA 1, then what was required to be proved.
The diagonals of a rectangular parallelepiped are equal.
Let us designate the dimensions of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =
Lesson Objectives:
1. Educational:
Introduce the concept of a parallelepiped and its types;
- formulate (using the analogy with a parallelogram and a rectangle) and prove the properties of a parallelepiped and a rectangular parallelepiped;
- repeat questions related to parallelism and perpendicularity in space.
2. Developing:
To continue the development of such cognitive processes in students as perception, comprehension, thinking, attention, memory;
- to promote the development of elements of creative activity in students as qualities of thinking (intuition, spatial thinking);
- to form in students the ability to draw conclusions, including by analogy, which helps to understand intrasubject connections in geometry.
3. Educational:
Contribute to the education of organization, the habit of systematic work;
- to promote the formation of aesthetic skills in the preparation of records, the execution of drawings.
Type of lesson: lesson-learning new material (2 hours).
Lesson structure:
1. Organizational moment.
2. Actualization of knowledge.
3. Learning new material.
4. Summing up and setting homework.
Equipment: posters (slides) with evidence, models of various geometric bodies, including all types of parallelepipeds, a graph projector.
During the classes.
1. Organizational moment.
2. Actualization of knowledge.
Reporting the topic of the lesson, formulating goals and objectives together with students, showing the practical significance of studying the topic, repeating previously studied issues related to this topic.
3. Learning new material.
3.1. Parallelepiped and its types.
Models of parallelepipeds are demonstrated with the identification of their features that help formulate the definition of a parallelepiped using the concept of a prism.
Definition:
Parallelepiped A prism whose base is a parallelogram is called.
A parallelepiped is drawn (Figure 1), the elements of the parallelepiped are listed as a special case of a prism. Slide 1 is shown.
Schematic notation of the definition:
Conclusions are drawn from the definition:
1) If ABCDA 1 B 1 C 1 D 1 is a prism and ABCD is a parallelogram, then ABCDA 1 B 1 C 1 D 1 is parallelepiped.
2) If ABCDA 1 B 1 C 1 D 1 – parallelepiped, then ABCDA 1 B 1 C 1 D 1 is a prism and ABCD is a parallelogram.
3) If ABCDA 1 B 1 C 1 D 1 is not a prism or ABCD is not a parallelogram, then
ABCDA 1 B 1 C 1 D 1 - not parallelepiped.
four) . If ABCDA 1 B 1 C 1 D 1 is not parallelepiped, then ABCDA 1 B 1 C 1 D 1 is not a prism or ABCD is not a parallelogram.
Next, special cases of a parallelepiped are considered with the construction of a classification scheme (see Fig. 3), models are demonstrated and the characteristic properties of a straight and rectangular parallelepipeds are distinguished, their definitions are formulated.
Definition:
A parallelepiped is called straight if its side edges are perpendicular to the base.
Definition:
The parallelepiped is called rectangular, if its side edges are perpendicular to the base, and the base is a rectangle (see Figure 2).
After writing the definitions in a schematic form, the conclusions from them are formulated.
3.2. Properties of parallelepipeds.
Search for planimetric figures, the spatial analogues of which are a parallelepiped and a rectangular parallelepiped (parallelogram and rectangle). In this case, we are dealing with the visual similarity of the figures. Using the inference rule by analogy, the tables are filled.
Inference rule by analogy:
1. Choose among previously studied figures figure similar to this one.
2. Formulate a property of the selected figure.
3. Formulate a similar property of the original figure.
4. Prove or refute the formulated statement.
After the formulation of the properties, the proof of each of them is carried out according to the following scheme:
- discussion of the proof plan;
- proof slide demonstration (slides 2-6);
- registration of evidence in notebooks by students.
3.3 Cube and its properties.
Definition: A cube is a cuboid with all three dimensions equal.
By analogy with a parallelepiped, students independently make a schematic record of the definition, derive consequences from it, and formulate the properties of the cube.
4. Summing up and setting homework.
Homework:
- Using the lesson outline, according to the geometry textbook for grades 10-11, L.S. Atanasyan and others, study ch.1, §4, p.13, ch.2, §3, p.24.
- Prove or disprove the property of a parallelepiped, item 2 of the table.
- Answer security questions.
Test questions.
1. It is known that only two side faces of a parallelepiped are perpendicular to the base. What type of parallelepiped?
2. How many side faces of a rectangular shape can a parallelepiped have?
3. Is it possible to have a parallelepiped with only one side face:
1) perpendicular to the base;
2) has the shape of a rectangle.
4. In a right parallelepiped, all diagonals are equal. Is it rectangular?
5. Is it true that in a right parallelepiped the diagonal sections are perpendicular to the planes of the base?
6. Formulate a theorem converse to the theorem on the square of the diagonal of a rectangular parallelepiped.
7. What additional features distinguish a cube from a cuboid?
8. Will a cube be a parallelepiped in which all edges are equal at one of the vertices?
9. Formulate a theorem on the square of the diagonal of a rectangular parallelepiped for the case of a cube.