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 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 denote 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 that are parallelograms. Hexagon.
The parallelograms that make up the parallelepiped are faces this parallelepiped, the sides of these parallelograms are parallelepiped edges, and the vertices of the parallelograms are peaks parallelepiped. Each face of a parallelepiped is parallelogram.
As a rule, any 2nd opposite faces are distinguished and called them the bases of the parallelepiped, and the remaining faces side faces of the parallelepiped. The edges of the parallelepiped that do not belong to the bases are side ribs.
The 2 faces of a cuboid that share an edge are related, and those that do not have common edges - opposite.
A segment that connects 2 vertices that do not belong to the 1st face is the diagonal of the parallelepiped.
The lengths of the edges of a cuboid that are not parallel are linear dimensions (measurements) 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 grounds.
A cuboid with all 3 dimensions equal in magnitude is cube. Each of the faces of the cube is equal squares .
Arbitrary parallelepiped. The volume and ratios in a skew box are mostly defined using vector algebra. The volume of the box is equal to the absolute value of the mixed product of 3 vectors, which are determined by the 3 sides of the box (which come from the same 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 the given 3 vectors is equal to the square of their mixed product.
Properties of a parallelepiped.
- The parallelepiped is symmetrical about the midpoint of its diagonal.
- Any segment with ends that belong to the surface of the parallelepiped and which passes through the midpoint 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.
- Opposite faces of a parallelepiped are parallel and have equal dimensions.
- The square of the length of the diagonal of a cuboid is
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. Usually, math tutor first gives the appropriate definitions for the prism, and then transfers them to the 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 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 a 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 proved 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 side 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 2cm and 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