A parallelepiped is a quadrangular prism with parallelograms at its base. The height of a parallelepiped is the distance between the planes of its bases. In the figure, the height is shown by the segment 
. There are two types of parallelepipeds: straight and inclined. As a rule, a math tutor first gives the appropriate definitions for a 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 side 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 right parallelepiped is nothing more than a type of straight prism, the side edge of which coincides with the height. Definitions of such concepts as face, edge and vertex, which are common to the entire family of polyhedra, are preserved. 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 connecting two vertices of a polyhedron and not lying on any of its faces.
Diagonal section - a section of a parallelepiped passing through its diagonal and the diagonal of its base.
Properties of an inclined parallelepiped:
1) All its faces are parallelograms, and the opposite faces are equal parallelograms.
2)
The diagonals of a parallelepiped intersect at one point and bisect 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 half of the paralleleped with its diagonal section and divide it separately into 3 pyramids. Their bases must lie on different faces of the original parallelepiped. A mathematics tutor will find application of this property in analytical geometry. It is used to derive the volume of a pyramid through a 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 a parallelepiped is equal to the product of the cross-sectional area and the lateral 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 thing for a parallelepiped. However, when working with an average-level student (the formula is not useful to a weak student), it is advisable for the teacher to act exactly the opposite. Leave the prism alone and carry out a careful 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:
Math tutors don’t often work on problems involving inclined parallelepipeds. The likelihood of them appearing on the Unified State Exam is quite low, and the didactics are indecently poor. A more or less decent problem on the volume of an inclined parallelepiped raises serious problems associated with determining the location of point H - the base of its height. In this case, the math tutor can be advised to cut the parallelepiped to one of its six pyramids (which are discussed in property No. 3), try to find its volume and multiply it by 6.
If the side edge of a 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 to each other 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 inclined parallelepiped it is known that , and in ABCD the base is a rhombus with a side of 2 cm and an angle . Determine the volume of the parallelepiped.
Mathematics tutor, Alexander Kolpakov
or (equivalently) a polyhedron with six faces that are parallelograms. Hexagon.
The parallelograms that make up a parallelepiped are edges of this parallelepiped, the sides of these parallelograms are edges of a parallelepiped, and the vertices of parallelograms are peaks parallelepiped. In a parallelepiped, each face is parallelogram.
As a rule, any 2 opposite faces are identified and called bases of the parallelepiped, and the remaining faces - lateral faces of the parallelepiped. The edges of the parallelepiped that do not belong to the bases are lateral ribs.
2 faces of a parallelepiped that have a common edge are adjacent, and those that do not have common edges - opposite.
A segment that connects 2 vertices that do not belong to the 1st face is parallelepiped diagonal.
The lengths of the edges of a rectangular parallelepiped that are not parallel are linear dimensions (measurements) 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 equal is cube. Each of the faces of the cube is equal squares .
Any parallelepiped. The volume and ratios in an inclined parallelepiped are mainly determined using vector algebra. The volume of a 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 the same vertex). The relationship between the lengths of the sides of the parallelepiped and the angles between them shows the statement that the Gram determinant of the given 3 vectors is equal to the square of their mixed product.
Properties of a parallelepiped.
- The parallelepiped is symmetrical about the middle of its diagonal.
- Any segment with ends that belong to the surface of a parallelepiped and which passes through the middle of its diagonal is divided by it into two equal parts. All diagonals of the parallelepiped intersect at the 1st point and are divided by it into two equal parts.
- The opposite faces of the parallelepiped are parallel and have equal dimensions.
- The square of the length of the diagonal of a rectangular parallelepiped is equal to
In this lesson, everyone will be able to study the topic “Rectangular parallelepiped”. At the beginning of the lesson, we will repeat what arbitrary and straight parallelepipeds are, remember the properties of their opposite faces and diagonals of the parallelepiped. Then we'll look at what a cuboid is and discuss its basic 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 ABV 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. The opposite faces of a parallelepiped are parallel and equal.
(the shapes are equal, that is, they can be combined by overlapping)
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 a parallelepiped intersect at one point and are bisected 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 in half by this point (Fig. 2).
Rice. 2 The diagonals of a parallelepiped intersect and are divided in half by the intersection point.
3. There are three quadruples of equal and parallel edges of a 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 side 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 the side faces contain rectangles. And the bases contain arbitrary parallelograms. Let us denote ∠BAD = φ, the angle φ can be any.
Rice. 3 Right parallelepiped
So, a right 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 edges are perpendicular to the base. The bases are rectangles.
The parallelepiped ABCDA 1 B 1 C 1 D 1 is 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°, i.e. the base is a rectangle.
Rice. 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 cuboid.
So, cuboid is a parallelepiped whose side edges are perpendicular to the base. The base of a cuboid is a rectangle.
1. In a rectangular parallelepiped, 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 lateral faces of a rectangular parallelepiped are rectangles.
3. All dihedral angles of a rectangular parallelepiped are right.
Let us consider, for example, the dihedral angle of a rectangular parallelepiped with edge AB, i.e., the dihedral angle between planes ABC 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 dihedral angle under consideration can also be denoted as follows: ∠A 1 ABD.
Let's take point A on edge AB. AA 1 is perpendicular to edge AB in the plane АВВ-1, AD is perpendicular to edge AB in the plane ABC. This means that ∠A 1 AD is the linear angle of a given dihedral angle. ∠A 1 AD = 90°, which means that the dihedral angle at edge AB is 90°.
∠(ABB 1, ABC) = ∠(AB) = ∠A 1 ABD= ∠A 1 AD = 90°.
Similarly, it is proved that any dihedral angles of a rectangular parallelepiped are right.
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 a cuboid are the measurements of the cuboid. They are sometimes called length, width, height.
Given: ABCDA 1 B 1 C 1 D 1 - rectangular parallelepiped (Fig. 5).
Prove: .
Rice. 5 Rectangular parallelepiped
Proof:
Straight line CC 1 is perpendicular to plane ABC, and therefore to straight line AC. This means that the triangle CC 1 A is right-angled. According to the Pythagorean theorem:
Consider the right triangle ABC. According to the Pythagorean theorem:
But BC and AD are opposite sides of the rectangle. So BC = AD. Then:
Because , A , That. Since CC 1 = AA 1, this is what needed to be proven.
The diagonals of a rectangular parallelepiped are equal.
Let us denote the dimensions of the parallelepiped ABC as a, b, c (see Fig. 6), then AC 1 = CA 1 = B 1 D = DB 1 =
