The base of the direct parallelepiped is a diagonal rhombus. Geometric figures
either (equivalent) a polyhedron with six faces, which are parallelograms. Hexagon.
Parallelograms from which the parallelepiped is citizensthis parallelepipeda, the parties of these parallelograms are perbrallepiped ribs, and the tops of parallelograms - verters parallelepipeda. Par Allepipeda has every face parallelogram.
As a rule, any 2nd opposite faces are distinguished and called them bases of parallelepipeda, and the remaining faces - side edges of parallelepipeda. The ribs of parallelepiped, which do not belong to the grounds are side ribs.
2 faces of parallelepiped who have a common edge are adjacent, and those that do not have common ribs - opposite.
A segment that connects 2 vertices that do not belong to the 1st face is diagonal of parallelepipeda.
The length of the ribs of rectangular parallelepiped, which are not parallel, are linear dimensions (measurements) Pollolepipeda. Rectangular parallelepiped 3 linear sizes.
Types of parallelepiped.
There are several types of parallelepipeds:
Direct It is a parallelepiped with an edge perpendicular to the foundation plane.
Rectangular parallelepipedwhich has all 3 measurements have an equal value, is cuba . Each of the faces of the cube is equal squares .
Arbitrary parallelepiped.Volume and relationships in inclined parallelepiped Basically are determined by vector algebra. The amount of parallelepiped is equally the magnitude of the mixed product of 3 vectors, which are determined by the 3 sides of the parallelepiped (which come from one vertex). The ratio between the lengths of the parallelepiped side and the corners between them shows the assertion that the determinant of the gram of data of 3 vectors is equal to the square of their mixed product.
Properties of parallelepiped.
- The parallelepipide is symmetrical about the middle of it is diagonal.
- Every segment with the ends that belong to the surface of the parallelepiped and which passes through the middle of it is diagonally, it is divided into two equal parts. All diagonals of the parallelepiped intersect in the 1st point and share it into two equal parts.
- Opposite faces of the parallelepiped parallel and have equal dimensions.
- The square of the diagonal length of the rectangular parallelepiped is equal
In this lesson, everyone will be able to explore the theme "rectangular parallelepiped". At the beginning of the lesson we will repeat what arbitrary and direct parallelepipeda are, remember the properties of their opposite faces and diagonals of the parallelepiped. Then consider what is rectangular parallelepiped, and discuss its basic properties.
Topic: Perpendicularity of straight and planes
Lesson: rectangular parallelepiped
The surface composed of two equal parallelograms of the ABSD and A 1 in 1 C 1 D 1 and four parallelograms of ABV 1 A 1, ASC 1 in 1, CDD 1 C 1, DAA 1 D 1, called parallelepiped (Fig. 1).
Fig. 1 parallelepiped
That is: We have two equal parallelogram of the ABSD and A 1 in 1 C 1 D 1 (base), they lie in parallel planes so that the lateral ribs AA 1, BB 1, DD 1, SS 1 are parallel. Thus, composed of the parallelogram surface is called parallelepiped.
Thus, the surface of the parallelepiped is the sum of all parallelograms from which the parallelepiped is compiled.
1. The opposite faces of the parallelepiped are parallel and equal.
(Figures are equal, that is, they can be combined with imposition)
For example:
AVD \u003d A 1 in 1 C 1 D 1 (equal parallelograms by definition),
AA 1 in 1 V \u003d DD 1 C 1 C (as AA 1 in 1 V and DD 1 with 1 C - the 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 s is the opposite faces of the parallelepiped).
2. The diagonals of the parallelepiped intersect at one point and are divided by this point in half.
The diagonal of the parallelepiped AC 1, in 1 D, and 1 C, D 1 in intersect at one point O, and each diagonal is divided by this point in half (Fig. 2).
Fig. 2 diagonals of the parallelepiped intersect and divide the intersection point in half.
3. There are three fours of equal and parallel edges of the parallelepiped: 1 - AB, A 1 in 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. The parallelepiped is called direct if its side ribs are perpendicular to the grounds.
Let the lateral edge of AA 1 perpendicular to the base (Fig. 3). This means that the straight AA 1 is perpendicular to the direct AD and AB, which lie in the base plane. And, it means that the rectangles lie in the side of the sidelines. And in the bases are arbitrary parallelograms. Denote by ∠bad \u003d φ, the angle φ can be any.
Fig. 3 straight parallelepiped
So, direct parallelepiped is a parallelepiped, in which the side ribs are perpendicular to the bases of the parallelepiped.
Definition. Parallelepiped is called rectangular, If its side ribs are perpendicular to the base. Basins are rectangles.
Parallelepiped AVDA 1 in 1 C 1 D 1 - rectangular (Fig. 4), if:
1. AA 1 ⊥ AVD (side edge perpendicular to the foundation plane, that is, parallelepiped direct).
2. ∠vd \u003d 90 °, i.e., at the base is a rectangle.
Fig. 4 rectangular parallelepiped
The rectangular parallelepiped has all the properties of arbitrary parallelepiped. But there is additional propertieswhich are derived from the definition of a rectangular parallelepiped.
So, rectangular parallelepiped - This is a parallelepipide, whose side ribs are perpendicular to the base. The base of the rectangular parallelepipeda is a rectangle.
1. In a rectangular parallelepiped, all six faces of rectangles.
ABSD and A 1 in 1 C 1 D 1 - rectangles by definition.
2. Side edges perpendicular to the base. So, all the side faces of the rectangular parallelepiped are rectangles.
3. All dumarted corners of the rectangular parallelepiped direct.
Consider, for example, a dihedral corner of a rectangular parallelepiped with an edge of AV, that is, the dihedral angle between the AVB 1 and ABS planes.
Av - edge, point A 1 lies in the same plane - in the plane of ABV 1, and point D in the other - in the plane A 1 in 1 s 1 D 1. Then the dihedral angle conspiraced can still be indicated as follows: ∠A 1 AVD.
Take the point A on the edge of AB. AA 1 - perpendicular to the edge of AV in the plane of ABV-1, AD perpendicular to the edge of AB in the ABC plane. So, ∠a 1 AD is the linear angle of this dihedral angle. ∠A 1 AD \u003d 90 °, which means that the dwarbon angle at the edge of the AV is 90 °.
∠ (AVB 1, ABC) \u003d ∠ (AV) \u003d ∠A 1 AVD \u003d ∠A 1 AD \u003d 90 °.
Similarly, it is proved that any dug in the corners of the rectangular parallelepiped direct.
Square diagonal of rectangular parallelepiped equal to sum Squares of its three dimensions.
Note. The length of the three ribs emanating from one vertex of the rectangular parallelepiped are measurements of a rectangular parallelepiped. They are sometimes called the length, width, height.
It is given: AVDA 1 in 1 C 1 D 1 - rectangular parallelepiped (Fig. 5).
Prove:
Fig. 5 rectangular parallelepiped
Evidence:
Direct SS 1 perpendicular to the ABC plane, and hence the straight speaker. So, the SS triangle 1 A is rectangular. According to Pythagore's theorem:
Consider right triangle ABC. According to Pythagore's theorem:
But Sun and AD are opposite directions of the rectangle. So, Sun \u003d AD. Then:
As , but then. Since the SS 1 \u003d AA 1, then what was required to prove.
The diagonals of the rectangular parallelepiped are equal.
Denote by the measurements of the parallelepiped ABC as a, b, c (see Fig. 6), then the AU 1 \u003d Ca 1 \u003d in 1 d \u003d DB 1 \u003d
Parlialelepiped is called quadrangular prism, in the bases of which are parallelograms. The height of the parallelepiped is called the distance between the planes of its bases. In the figure, the height is shown by a segment 
. There are two types of parallelepipeds: straight and inclined. As a rule, the mathematics tutor first gives the corresponding definitions for the prism, and then transfers them to the parallelepiped. We will also do.
Let me remind you that prism is called straight, if its side ribs are perpendicular to the grounds, if there is no perpendicularity - the prism is called inclined. This terminology inherits parallelepiped. 
Straight parallelepiped - nothing but a variety of direct prism, the lateral edge of which coincides with the height. The definitions of such concepts as the edge, edge and therapy are preserved, which are common to the entire family of polyhedra. The concept of opposite faces appear. Par Allepipeda has 3 pairs of opposite faces, 8 vertices of TI 12 ribs.
The diagonal of the parallelepiped (diagonal of the prism) is a segment connecting the two vertices of the polyhedron and not lying in any of its faces.
Diagonal section - a cross section of parallelepiped, passing through its diagonal and a diagonal of its base.
Properties of inclined parallelepipeda:
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 into this point in half.
3)
Each parallelepiped consists of six equal in the volume of triangular pyramids. To show their student a tutor in mathematics should cut off from the parallel support of half a diagonal cross section and break it separately on 3 pyramids. Their foundations must lie in different facilities of the initial parallepiped. Mathematics tutor will find the use of this property in analytical geometry. It is used to output the pyramid volume through a mixed product of vectors.
Parallelepiped volume formulas:
1), where - the base area, H is height.
2) The volume of parallelepiped is equal to the product of the cross-sectional area on the side edge.
Tutor in mathematics: As you know, the formula is common to all prisms and if the tutor has already proven it, it makes no sense to repeat the same for the parallelepiped. However, in working with a mid-level student (weak formula is not useful) to the teacher, it is advisable to act with accuracy to the opposite. The prism should be left alone, and for parallelepiped to conduct a neat proof.
3), where -bigination of one of the six triangular pyramids of which consists of parallelepiped.
4) if, then
The area of \u200b\u200bthe side surface of the parallelepiped is the sum of the areas of all its faces:
The complete surface of the 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 inclined parallelepiped:
Tasks on the inclined parallelepiped tutor in mathematics do not often do. The probability of their appearance on the exam is quite small, and didactics indecently poor. A more or less decent task on the volume of inclined parallelepiped causes serious problems associated with the location of the point H - the base of its height. In this case, the tutorial in mathematics can be advised to trim the parallelepiped to one of the six pyramids (which are discussed in the property number 3), try to find its volume and multiply it to 6.
If the side edge of the parallelepiped has equal angles With the sides of the base, Nu lies on the bisector of the angle A ABCD base. And if, for example, ABCD - rhombus, then
Task tutor in mathematics:
1) The faces of the parallelepiped equal surfaces with a side 2cm and a sharp angle. Find the volume of parallelepiped.
2) In the inclined parallelepiped, the side edge is 5 cm. The cross section perpendicular to it is a quadrangle with mutually perpendicular diagonals having lengths 6cm and 8 cm. Calculate the volume of parallepipeda.
3) In inclined parallelepiped, it is known that, and in the annoyance ABCD is a rhombus with a side of 2 cm and angle. Determine the volume of parallelepiped.
Tutor in mathematics, Alexander Kolpakov
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