What can be said about the diagonals of a parallelepiped? Volume of a parallelepiped: basic formulas and examples of problems. About the introduced notations

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.

Task 2.

Given: an inclined parallelepiped with a side edge of 10 cm, a rectangle KLNM with dimensions of 5 and 7 cm, which is a cross section of the figure parallel to the specified edge.
Necessary find the lateral surface area of the quadrangular prism.
Solution: First you need to sketch the given.

To solve this task you need to use ingenuity. The figure shows that the sides KL and AD are unequal, as are the pair ML and DC. However, the perimeters of these parallelograms are obviously equal.

Consequently, the lateral area of the figure will be equal to the sectional area multiplied by edge AA1, since by condition the edge is perpendicular to the section. Answer: 240 cm2.

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 that 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

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