A rectangular parallelepiped with a rhombus base. Rectangular parallelepiped
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. As a rule, a math tutor first gives the appropriate definitions for the prism, and then transfers them to a parallelepiped. 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 also 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 such concepts 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.
Oblique 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 halved at this point.
3)
Each parallelepiped consists of six triangular pyramids of equal volume. To show them to the student, the mathematics tutor must cut off a half of its diagonal section from the parallelepedo and break it separately into 3 pyramids. Their bases must lie on different faces of the original parallelelepiped. A math tutor will find an application of this property in analytical 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 the base, h is the height.
2) The volume of the parallelepiped is equal to the product of the cross-sectional area and the side edge.
Math tutor: 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 (the formula is not useful to a weak one), it is desirable 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 the lateral 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 lateral edge of the parallelepiped has equal angles with the sides of the base, 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 2cm 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 box.
Tutor in mathematics, Alexander Kolpakov
or (equivalently) a polyhedron with six parallelogram faces. Hexagon.
The parallelograms that make up the parallelepiped are facets of this parallelepiped, the sides of these parallelograms are edges of a parallelepiped, and the vertices of the parallelograms are peaks parallelepiped... For a parallelepiped, each face is parallelogram.
As a rule, any 2 opposite faces are distinguished and called the bases of the parallelepiped, and the remaining faces are side faces of the parallelepiped... The edges of the box that do not belong to the bases are lateral ribs.
2 faces of a box that have a common edge are related, and those that do not have common edges - opposite.
The segment that connects 2 vertices that do not belong to the 1st face is by the diagonal of the parallelepiped.
The lengths of the edges of a rectangular parallelepiped that are not parallel are linear dimensions (measurements) of 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 rectangular parallelepiped, in which all 3 dimensions are of equal magnitude, is cube... Each of the faces of the cube is equal squares .
Arbitrary parallelepiped. The volume and ratios in an oblique parallelepiped are mainly determined using vector algebra. The volume of the parallelepiped is equal to the absolute value of the mixed product of 3 vectors, which are determined by the 3 sides of the parallelepiped (which originate from one 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 these 3 vectors is equal to the square of their mixed product.
Box properties.
- The parallelepiped is symmetrical about the middle of its diagonal.
- Any segment with ends that belong to the surface of the parallelepiped and that passes through the middle of its diagonal is divided by it into two equal parts. All the diagonals of the parallelepiped intersect at the 1st point and are divided by it into two equal parts.
- Opposite faces of the box are parallel and equal in size.
- The square of the length of the diagonal of a rectangular parallelepiped is
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 main 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).
Fig. 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 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 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 = A 1 B 1 C 1 D 1 (equal parallelograms by definition),
AA 1 B 1 B = 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 = 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).
Fig. 2 The diagonals of the parallelepiped intersect and are halved by the intersection point.
3. There are three quadruples of equal and parallel parallelepiped edges: 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 in the bases are arbitrary parallelograms. Denote, ∠BAD = φ, the angle φ can be any.
Fig. 3 Straight parallelepiped
So, a straight parallelepiped is a parallelepiped in which the side edges are perpendicular to the parallelepiped's bases.
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 = 90 °, that is, there is a rectangle at the base.
Fig. 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 side 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 are 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 = 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 in a similar way 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 outgoing 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:.
Fig. 5 Rectangular parallelepiped
Evidence:
Straight CC 1 is perpendicular to the plane ABC, and hence to the straight line AC. This means that triangle CC 1 A is rectangular. By the Pythagorean theorem:
Consider a right-angled triangle ABC. By the Pythagorean theorem:
But BC and AD are opposite sides of the rectangle. Hence, BC = AD. Then:
As , but then. Since CC 1 = 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 = CA 1 = B 1 D = DB 1 =
