Arbitrary parallelepiped. Rectangular parallelepiped – Knowledge Hypermarket

There are several types of parallelepipeds:

· Rectangular parallelepiped- is a parallelepiped, all of whose faces are - rectangles;

· A right parallelepiped is a parallelepiped that has 4 side faces - parallelograms;

· An inclined parallelepiped is a parallelepiped whose side faces are not perpendicular to the bases.

Essential elements

Two faces of a parallelepiped that do not have a common edge are called opposite, and those that have a common edge are called adjacent. Two vertices of a parallelepiped that do not belong to the same face are called opposite. Line segment, connecting opposite vertices is called diagonally parallelepiped. The lengths of three edges of a rectangular parallelepiped having a common vertex are called measurements.

Properties

· The parallelepiped is symmetrical about the middle of its diagonal.

· Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided in half by it; in particular, all diagonals of a parallelepiped intersect at one point and are bisected by it.

· Opposite faces of a parallelepiped are parallel and equal.

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

Basic formulas

Right parallelepiped

· Lateral surface area S b =P o *h, where P o is the perimeter of the base, h is the height

· Total surface area S p =S b +2S o, where S o is the base area

· Volume V=S o *h

Rectangular parallelepiped

· Lateral surface area S b =2c(a+b), where a, b are the sides of the base, c is the side edge of the rectangular parallelepiped

· Total surface area S p =2(ab+bc+ac)

· Volume V=abc, where a, b, c are the dimensions of a rectangular parallelepiped.

· Lateral surface area S=6*h 2, where h is the height of the cube edge

34. Tetrahedron- regular polyhedron, has 4 faces that are regular triangles. Vertices of a tetrahedron 4 , converges to every vertex 3 ribs, and total ribs 6 . Also, a tetrahedron is a pyramid.

The triangles that make up a tetrahedron are called faces (AOS, OSV, ACB, AOB), their sides --- ribs (AO, OC, OB), and the vertices --- vertices (A, B, C, O) tetrahedron. Two edges of a tetrahedron that do not have common vertices are called opposite... Sometimes one of the faces of the tetrahedron is isolated and called basis, and the other three --- side faces.

The tetrahedron is called correct, if all its faces are equilateral triangles. Moreover, a regular tetrahedron and a regular triangular pyramid are not the same thing.

U regular tetrahedron all dihedral angles at the edges and all trihedral angles at the vertices are equal.

35. Correct prism

A prism is a polyhedron whose two faces (bases) lie in parallel planes, and all the edges outside these faces are parallel to each other. The faces other than the bases are called side faces, and their edges are called side edges. All side edges are equal to each other as parallel segments bounded by two parallel planes. All lateral faces of the prism are parallelograms. The corresponding sides of the bases of the prism are equal and parallel. A prism whose side edge is perpendicular to the plane of the base is called a straight prism; other prisms are called inclined. At the base of a regular prism is a regular polygon. All faces of such a prism are equal rectangles.

The surface of the prism consists of two bases and a side surface. The height of a prism is a segment that is a common perpendicular to the planes in which the bases of the prism lie. The height of the prism is the distance H between the planes of the bases.

Lateral surface area S b of a prism is the sum of the areas of its lateral faces. Total surface area S n of a prism is the sum of the areas of all its faces. S n = S b + 2 S,Where S– area of ​​the base of the prism, S b – lateral surface area.

36. A polyhedron that has one face, called basis, – polygon,
and the other faces are triangles with a common vertex, called pyramid .

Faces other than the base are called lateral.
The common vertex of the lateral faces is called the top of the pyramid.
The edges connecting the top of the pyramid with the vertices of the base are called lateral.
Pyramid height is called a perpendicular drawn from the top of the pyramid to its base.

The pyramid is called correct, if its base is a regular polygon and its height passes through the center of the base.

Apotheme the lateral face of a regular pyramid is the height of this face drawn from the vertex of the pyramid.

A plane parallel to the base of the pyramid cuts it off into a similar pyramid and truncated pyramid.

Properties of regular pyramids

• The lateral edges of a regular pyramid are equal.
• The lateral faces of a regular pyramid are isosceles triangles equal to each other.

If all side edges are equal, then

·height is projected to the center of the circumscribed circle;

The side ribs form equal angles with the plane of the base.

If the side faces are inclined to the plane of the base at the same angle, then

·height is projected to the center of the inscribed circle;

· the heights of the side faces are equal;

·the area of ​​the side surface is equal to half the product of the perimeter of the base and the height of the side face

37. The function y=f(x), where x belongs to the set of natural numbers, is called a function of a natural argument or a number sequence. It is denoted by y=f(n), or (y n)

Sequences can be specified in various ways, verbally, this is how a sequence of prime numbers is specified:

2, 3, 5, 7, 11, etc.

A sequence is considered to be given analytically if the formula for its nth term is given:

1, 4, 9, 16, …, n 2, …

2) y n = C. Such a sequence is called constant or stationary. For example:

2, 2, 2, 2, …, 2, …

3) y n =2 n . For example,

2, 2 2, 2 3, 2 4, …, 2 n, …

A sequence is said to be bounded above if all its terms are not greater than a certain number. In other words, a sequence can be called bounded if there is a number M such that the inequality y n is less than or equal to M. The number M is called the upper bound of the sequence. For example, the sequence: -1, -4, -9, -16, ..., - n 2 ; limited from above.

Similarly, a sequence can be called bounded below if all its terms are greater than a certain number. If a sequence is bounded both above and below it is called bounded.

A sequence is called increasing if each subsequent term is greater than the previous one.

A sequence is called decreasing if each subsequent member is less than the previous one. Increasing and decreasing sequences are defined by one term - monotonic sequences.

Consider two sequences:

1) y n: 1, 3, 5, 7, 9, …, 2n-1, …

2) x n: 1, ½, 1/3, 1/ 4, …, 1/n, …

If we depict the terms of this sequence on the number line, we will notice that, in the second case, the terms of the sequence are condensed around one point, but in the first case this is not the case. In such cases, the sequence y n is said to diverge and the sequence x n to converge.

The number b is called the limit of the sequence y n if any pre-selected neighborhood of the point b contains all members of the sequence, starting from a certain number.

In this case we can write:

If the quotient of a progression is less than one in modulus, then the limit of this sequence, as x tends to infinity, is equal to zero.

If the sequence converges, then only to one limit

If the sequence converges, then it is bounded.

Weierstrass's theorem: If a sequence converges monotonically, then it is bounded.

The limit of a stationary sequence is equal to any term of the sequence.

Properties:

1) The amount limit is equal to the sum of the limits

2) The limit of a product is equal to the product of the limits

3) The limit of the quotient is equal to the quotient of the limits

4) The constant factor can be taken beyond the limit sign

Question 38
sum of infinite geometric progression

Geometric progression- a sequence of numbers b 1, b 2, b 3,.. (members of the progression), in which each subsequent number, starting from the second, is obtained from the previous one by multiplying it by a certain number q (denominator of the progression), where b 1 ≠0, q ≠0.

Sum of an infinite geometric progression is the limiting number to which the sequence of progression converges.

In other words, no matter how long a geometric progression is, the sum of its terms is not more than a certain number and is practically equal to this number. This is called the sum of a geometric progression.

Not every geometric progression has such a limiting sum. It can only be for a progression whose denominator is a fractional number less than 1.

Translated from Greek, parallelogram means plane. A parallelepiped is a prism with a parallelogram at its base. There are five types of parallelogram: oblique, straight and cuboid. The cube and rhombohedron also belong to the parallelepiped and are its variety.

Before moving on to the basic concepts, let's give some definitions:

• The diagonal of a parallelepiped is a segment that unites the vertices of the parallelepiped that are opposite each other.
• If two faces have a common edge, then we can call them adjacent edges. If there is no common edge, then the faces are called opposite.
• Two vertices that do not lie on the same face are called opposite.

What properties does a parallelepiped have?

1. The faces of a parallelepiped lying on opposite sides are parallel to each other and equal to each other.
2. If you draw diagonals from one vertex to another, then the intersection point of these diagonals will divide them in half.
3. The sides of the parallelepiped lying at the same angle to the base will be equal. In other words, the angles of the co-directed sides will be equal to each other.

What types of parallelepiped are there?

Now let's figure out what kind of parallelepipeds there are. As mentioned above, there are several types of this figure: straight, rectangular, inclined parallelepiped, as well as cube and rhombohedron. How do they differ from each other? It's all about the planes that form them and the angles they form.

Let's look in more detail at each of the listed types of parallelepiped.

• As is already clear from the name, an inclined parallelepiped has inclined faces, namely those faces that are not at an angle of 90 degrees in relation to the base.
• But for a right parallelepiped, the angle between the base and the edge is exactly ninety degrees. It is for this reason that this type of parallelepiped has such a name.
• If all the faces of the parallelepiped are identical squares, then this figure can be considered a cube.
• A rectangular parallelepiped received this name because of the planes that form it. If they are all rectangles (including the base), then this is a cuboid. This type of parallelepiped is not found very often. Translated from Greek, rhombohedron means face or base. This is the name given to a three-dimensional figure whose faces are rhombuses.

Basic formulas for a parallelepiped

The volume of a parallelepiped is equal to the product of the area of ​​the base and its height perpendicular to the base.

The area of ​​the lateral surface will be equal to the product of the perimeter of the base and the height.
Knowing the basic definitions and formulas, you can calculate the base area and volume. The base can be chosen at your discretion. However, as a rule, a rectangle is used as the base.

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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 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

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 Figure 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 right 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 straight 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.