There are several types of parallelepipeds:
· cuboid is a parallelepiped with all faces - rectangles;
A right parallelepiped is a parallelepiped that has 4 side faces- parallelograms;
· An oblique box is a box 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 diagonal parallelepiped. The lengths of three edges of a cuboid that have a common vertex are called measurements.
Properties
· The parallelepiped is symmetrical about the midpoint of its diagonal.
Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided by it in half; in particular, all the diagonals of the parallelepiped intersect at one point and bisect it.
Opposite faces of a parallelepiped are parallel and equal.
The square of the length of the diagonal 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 \u003d R o * h, where R o is the perimeter of the base, h is the height
· Square full surface S p \u003d S b + 2S o, where S o is the area of \u200b\u200bthe base
· Volume V=S o *h
cuboid
· Lateral surface area S b \u003d 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 \u003d 2 (ab + bc + ac)
· Volume V=abc, where a, b, c are the dimensions of the cuboid.
· Lateral surface area S=6*h 2 , where h is the height of the cube edge
34. Tetrahedron is a regular polyhedron, has 4 faces that are regular triangles. Vertices at the tetrahedron 4 , converges to each vertex 3 ribs, but total ribs 6 . The tetrahedron is also a pyramid.
The triangles that make up a tetrahedron are called faces (AOC, OSV, ACB, AOB), their sides --- edges (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 singled out and called it basis, and three others --- side faces.
The tetrahedron is called right if all its faces are equilateral triangles. At the same time, a regular tetrahedron and a regular triangular pyramid- it's not the same thing.
At regular tetrahedron all dihedral angles at edges and all trihedral angles at vertices are equal.
35. Correct prism
A prism is a polyhedron in which two faces (bases) lie in parallel planes, and all 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 side faces of the prism are parallelograms. The corresponding sides of the bases of the prism are equal and parallel. A straight prism is called, in which the lateral edge is perpendicular to the plane of the base, other prisms are called inclined. At the base of a regular prism lies regular polygon. In such a prism, all faces are equal rectangles.
The surface of a 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 base planes.
Side surface area S b prism is called the sum of the areas of its side faces. Full surface area S n of a prism is called the sum of the areas of all its faces. S n = S b + 2 S,Where S is the base area of the prism, S b – lateral surface area.
36. A polyhedron that has one face, called basis, is a polygon,
and the other faces are triangles with a common vertex, is called pyramid
.
Faces other than the base are called side.
The common vertex of the side faces is called top of the pyramid.
The edges that connect the top of the pyramid with the top of the base are called side.
The height of the pyramid
called the 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.
apothem side face of a regular pyramid is called the height of this face, drawn from the top of the pyramid.
Plane, parallel to base pyramid, cuts it off into a similar pyramid and truncated pyramid.
Properties of regular pyramids
- Side ribs the correct pyramid - are equal.
- The side 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;
lateral ribs form equal angles with the base plane.
If the side faces are inclined to the base plane at one angle, then
Height is projected to the center of the inscribed circle;
the heights of the side faces are equal;
The area of the lateral surface is equal to half the product of the perimeter of the base and the height of the lateral face
37. The function y=f(x), where x belongs to the set natural numbers, is called a function of natural argument or numerical sequence. Designate it y=f(n), or (y n)
Sequences can be specified in various ways, verbally, this is how a sequence is specified prime numbers:
2, 3, 5, 7, 11 etc
It is considered that the sequence is given analytically if the formula of its n-th member is given:
1, 4, 9, 16, …, n2, …
2) y n = C. Such a sequence is called constant or stationary. For example:
2, 2, 2, 2, …, 2, …
3) y n \u003d 2 n. For example,
2, 2 2 , 2 3 , 2 4 , …, 2n , …
A sequence is said to be bounded from above if all its members are at most some number. In other words, a sequence can be called bounded if there is such a number M 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 said to be bounded from below if all of its members are greater than some number. If a sequence is bounded both above and below, it is said to be bounded.
A sequence is said to be increasing if each successive term is greater than the previous one.
A sequence is called decreasing if each successive term 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 members of this sequence on a real line, then we will notice that, in the second case, the members of the sequence condense around one point, and in the first case this is not the case. In such cases, we say that the sequence y n diverges, and the sequence x n converges.
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 some number.
In this case, we can write:
If the modulo quotient of the progression is less than one, 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 Theorem: If a sequence converges monotonically, then it is bounded.
The limit of a stationary sequence is equal to any member of the sequence.
Properties:
1) The sum limit is equal to the sum of the limits
2) The limit of the product is equal to the product of the limits
3) The limit of the quotient is equal to the quotient of the limits
4) Constant multiplier can be taken out of the limit sign
Question 38
the sum of an 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 (the denominator of the progression), where b 1 ≠0, q ≠0.
The sum of an infinite geometric progression is the limit number to which the progression sequence converges.
In other words, no matter how long the geometric progression is, the sum of its members is not more than a certain number and is practically equal to this number. It is called the sum of a geometric progression.
Not every geometric progression has such a limiting sum. It can only be in such a progression, the denominator of which is a fractional number less than 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 of the base.
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
It will be useful for high school students to learn how to solve USE tasks to find the volume and other unknown parameters of a rectangular parallelepiped. Experience previous years confirms the fact that such tasks are quite difficult for many graduates.
At the same time, high school students with any level of training should understand how to find the volume or area of a rectangular parallelepiped. Only in this case they will be able to count on getting competitive scores based on the results of passing the unified state exam in mathematics.
Key points to remember
- The parallelograms that make up the parallelepiped are its faces, their sides are edges. The vertices of these figures are considered to be the vertices of the polyhedron itself.
- All diagonals of a cuboid are equal. Since this is a straight polyhedron, the side faces are rectangles.
- Since a parallelepiped is a prism with a parallelogram at its base, this figure has all the properties of a prism.
- The side edges of a rectangular parallelepiped are perpendicular to the base. Therefore, they are its heights.
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Translated from Greek parallelogram means plane. A parallelepiped is a prism whose base is a parallelogram. There are five types of parallelogram: oblique, straight, and cuboid. The cube and the 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 are the properties of a parallelepiped?
- The faces of a parallelepiped lying on opposite sides are parallel to each other and equal to each other.
- If you draw diagonals from one vertex to another, then the intersection point of these diagonals will divide them in half.
- The sides of a parallelepiped lying at the same angle to the base will be equal. In other words, the angles of the codirectional sides will be equal to each other.
What are the types of parallelepiped?
Now let's figure out what parallelepipeds are. As mentioned above, there are several types of this figure: a straight, rectangular, oblique parallelepiped, as well as a cube and a rhombohedron. How do they differ from each other? It's all about the planes that form them and the angles that they form.
Let's take a closer look at each of the listed types of parallelepiped.
- As the name suggests, a slanted box has slanted faces, namely those faces that are not at an angle of 90 degrees with respect to the base.
- But for a right parallelepiped, the angle between the base and the face is just ninety degrees. It is for this reason that this type of parallelepiped has such a name.
- If all the faces of the parallelepiped are the same squares, then this figure can be considered a cube.
- The rectangular parallelepiped got its name because of the planes that form it. If they are all rectangles (including the base), then it is a cuboid. This type of parallelepiped is not so common. In Greek, rhombohedron means face or base. This is the name of a three-dimensional figure, in which the 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. You can choose the base of your choice. However, as a rule, a rectangle is used as the base.