A parallelepiped is a prism whose bases are parallelograms. In this case, all edges will be parallelograms.
Each parallelepiped can be considered as a prism in three different ways, since every two opposite faces can be taken as bases (in Fig. 5, faces ABCD and A"B"C"D", or ABA"B" and CDC"D", or BCB "C" and ADA"D").
The body in question has twelve edges, four equal and parallel to each other.
Theorem 3
. The diagonals of a parallelepiped intersect at one point, coinciding with the middle of each of them.
The parallelepiped ABCDA"B"C"D" (Fig. 5) has four diagonals AC", BD", CA", DB". We must prove that the midpoints of any two of them, for example AC and BD", coincide. This follows from the fact that the figure ABC"D", having equal and parallel sides AB and C"D", is a parallelogram.
Definition 7
. A right parallelepiped is a parallelepiped that is also a straight prism, that is, a parallelepiped whose side edges are perpendicular to the plane of the base.
Definition 8
. A rectangular parallelepiped is a right parallelepiped whose base is a rectangle. In this case, all its faces will be rectangles.
A rectangular parallelepiped is a straight prism, no matter which of its faces we take as the base, since each of its edges is perpendicular to the edges emerging from the same vertex, and will, therefore, be perpendicular to the planes of the faces defined by these edges. In contrast, a straight, but not rectangular, parallelepiped can be viewed as a right prism in only one way.
Definition 9
. The lengths of three edges of a rectangular parallelepiped, of which no two are parallel to each other (for example, three edges emerging from the same vertex), are called its dimensions. Two rectangular parallelepipeds having correspondingly equal dimensions are obviously equal to each other.
Definition 10
.A cube is a rectangular parallelepiped, all three dimensions of which are equal to each other, so that all its faces are squares. Two cubes whose edges are equal are equal. 
Definition 11
. An inclined parallelepiped in which all edges are equal to each other and the angles of all faces are equal or complementary is called a rhombohedron.
All faces of a rhombohedron are equal rhombuses. (Some crystals of great importance have a rhombohedron shape, for example, Iceland spar crystals.) In a rhombohedron you can find a vertex (and even two opposite vertices) such that all the angles adjacent to it are equal to each other.
Theorem 4
. The diagonals of a rectangular parallelepiped are equal to each other. The square of the diagonal is equal to the sum of the squares of the three dimensions.
In the rectangular parallelepiped ABCDA"B"C"D" (Fig. 6), the diagonals AC" and BD" are equal, since the quadrilateral ABC"D" is a rectangle (the straight line AB is perpendicular to the plane ECB"C", in which BC lies") .
In addition, AC" 2 =BD" 2 = AB2+AD" 2 based on the theorem about the square of the hypotenuse. But based on the same theorem AD" 2 = AA" 2 + +A"D" 2; hence we have:
AC" 2 = AB 2 + AA" 2 + A" D" 2 = AB 2 + AA" 2 + AD 2.
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 =
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 plane of the base.
A rectangular parallelepiped is a quadrangular prism, all of whose faces are rectangles. A cube is a type of quadrangular prism in which all faces and edges are equal to each other.
The features of a figure predetermine its properties. These include the following 4 statements:
It is simple to remember all the above properties, they are easy to understand and are derived logically based on the type and characteristics 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 for finding the area and volume of a geometric body.
The area of the bases is found in the same way 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, the base of which is a rectangle.
The formula for finding the lateral surface of a parallelepiped may also be needed in test tasks.
Examples of solving typical Unified State Exam tasks
Exercise 1.
Given: a rectangular parallelepiped with dimensions of 3, 4 and 12 cm.
Necessary find the length of one of the main diagonals of the figure.
Solution: 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 execution of task conditions.
Having examined the drawing made and remembering all the properties of the geometric body, we come to the only correct method of solution. Applying the 4th property of a parallelepiped, we obtain the following expression:
After simple calculations we get the expression b2=169, therefore b=13. The answer to the task has been found; you need to spend no more than 5 minutes searching for it and drawing it.
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