Formula for expressing the length of a diagonal through the sides of a parallelepiped. Geometric figures

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 parallelepiped bases, 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 .

Arbitrary 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

Since all faces of a parallelepiped are parallelograms, then line AD is parallel to line BC, and line is parallel to line . It follows that the planes of the faces under consideration are parallel.

From the fact that the faces of a parallelepiped are parallelograms, it follows that AB, , CD are both parallel and equal. From this we conclude that the face is combined by parallel translation along the edge AB with the face. Therefore, these faces are equal.

2 ) Let's take two diagonals of the parallelepiped (Fig. 5), for example, and , and draw additional straight lines and . AB and respectively are equal and parallel to the edge DC, therefore they are equal and parallel to each other; As a result, the figure is a parallelogram in which the straight lines and are the diagonals, and in a parallelogram the diagonals are divided in half at the point of intersection. Similarly, we can prove that the other two diagonals intersect at one point and are bisected by that point. The point of intersection of each pair of diagonals lies in the middle of the diagonal. Thus, all four diagonals of the parallelepiped intersect at one point O and are bisected by this point. Thus, the point of intersection of the diagonals of a parallelepiped is its center of symmetry.

Theorem:

The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions.

Proof:

This emerges from the spatial theorem of Pythagoras. If is the diagonal of a rectangular parallelepiped , then are its projections onto three pairwise perpendicular lines (Fig. 6). Hence, .

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 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 =