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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. A cube is a kind of quadrangular prism in which all faces and edges are equal.
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 needed 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.
Decision: 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.
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:
As , 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 =
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 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 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, "and if it doesn't come 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, bringing him two candles and a manuscript, asked him to begin.
Definition
polyhedron we will call a closed surface composed of polygons and bounding some part of space.
The segments that are the sides of these polygons are called ribs polyhedron, and the polygons themselves - faces. The vertices of the polygons are called the vertices of the polyhedron.
We will consider only convex polyhedra (this is a polyhedron that is on one side of each plane containing its face).
The polygons that make up a polyhedron form its surface. The part of space 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 \(A_1B_1, \A_2B_2, ..., A_nB_n\) are parallel. Polyhedron formed by 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 (\(n\)-coal) prism.
The polygons \(A_1A_2A_3...A_n\) and \(B_1B_2B_3...B_n\) are called the bases of the prism, parallelogram \(A_1B_1B_2A_2, \A_2B_2B_3A_3, ...\)– side faces, segments \(A_1B_1, \A_2B_2, \ ..., A_nB_n\)- side ribs.
Thus, the side edges of the prism are parallel and equal to each other.
Consider an example - a prism \(A_1A_2A_3A_4A_5B_1B_2B_3B_4B_5\), whose base is a convex pentagon.
Height A prism is a perpendicular from any point on one base to the plane of another base.
If the side edges are not perpendicular to the base, then such a prism is called oblique(Fig. 1), otherwise - straight. For 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 right prism, then the prism is called correct.
Definition: concept of volume
The volume unit is a unit cube (cube with dimensions \(1\times1\times1\) units\(^3\) , where unit is some unit of measure).
We can say that the volume of a polyhedron is the amount of space that this polyhedron limits. Otherwise: it is a value whose numerical value indicates 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 value.
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: box
Parallelepiped It is a prism whose base is a parallelogram.
All faces of the parallelepiped (their \(6\) : \(4\) side faces and \(2\) bases) are parallelograms, and the opposite faces (parallel to each other) are equal parallelograms (Fig. 2).
Diagonal of the box is a segment connecting two vertices of a parallelepiped that do not lie in the same face (their \(8\) : \(AC_1, \A_1C, \BD_1, \B_1D\) etc.).
cuboid is a right parallelepiped with a rectangle at its base.
Because is a right parallelepiped, then the side faces are rectangles. So, in general, all the faces of a rectangular parallelepiped are rectangles.
All diagonals of a cuboid are equal (this follows from the equality of triangles \(\triangle ACC_1=\triangle AA_1C=\triangle BDD_1=\triangle BB_1D\) etc.).
Comment
Thus, the parallelepiped has all the properties of a prism.
Theorem
The area of the lateral surface of a rectangular parallelepiped is equal to \
The total surface area of a rectangular parallelepiped is \
Theorem
The volume of a cuboid is equal to the product of three of its edges coming out of one vertex (three dimensions of a cuboid): \
Proof
Because for a rectangular parallelepiped, the lateral edges are perpendicular to the base, then they are also its heights, that is, \(h=AA_1=c\) the base is a rectangle \(S_(\text(main))=AB\cdot AD=ab\). This is where the formula comes from.
Theorem
The diagonal \(d\) of a cuboid is searched for by the formula (where \(a,b,c\) are the dimensions of the cuboid)\
Proof
Consider 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 line in this plane, i.e. \(BB_1\perp BD\) . So \(\triangle BB_1D\) 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 sides of which 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 \(V_(\text(cube))=a^3\) .
2. The cube diagonal is searched by the formula \(d=a\sqrt3\) .
3. Total surface area of a cube \(S_(\text(full cube iterations))=6a^2\).
