Polyhedra
Polyhedron is a body whose surface consists of a finite number of flat polygons. The polyhedron is called convex, if it lies on one side of the plane of each of the planar polygons on its surface. a common part such a plane and the surface of a convex polygon is called edge.
The figure below shows a non-convex polyhedron on the left; in the figure on the right - convex.
The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called edges of the polyhedron, and the vertices of the faces - vertices of the polyhedron.
Prism
prism a polyhedron is called, which consists of two flat polygons lying in different planes and combined by parallel translation, and all segments connecting the corresponding points of these polygons (see figure). The polygons are called prism fundamentals, and the segments connecting the corresponding vertices - side edges of the prism.Designations: .
Side surface The prism is made up of parallelograms. Each of them has two sides that are the corresponding sides of the base, and the other two are adjacent side ribs. The bases of the prism are equal and lie in parallel planes. The side edges of the prism are parallel and equal. Prism height called the distance between the planes of its bases.
A segment connecting two vertices of a prism that do not belong to the same face is called prism diagonal. (In the figure - height, and - diagonals.)
Diagonal sections- these are sections of the prism by planes passing through two side edges that do not belong to the same face (see figures).
The prism is called straight if its side edges are perpendicular to the bases. Otherwise, the prism is called oblique.
The side faces of a straight prism are rectangles, the height of a straight prism is equal to the side edge, the diagonal sections are rectangles.
Side surface prism is called the sum of the areas of the side faces. Full surface of the prism equal to the sum of the lateral surface and the areas of the bases.
Theorem 1. The side surface of a straight prism is equal to the product of the perimeter of the base and the height, that is, the length of the side edge.
Perpendicular section of a prism we will call the section a plane perpendicular to the side edge of the prism (which means that this plane is perpendicular to all side edges of the prism).
Theorem 2. Lateral surface of an inclined prism is equal to the product the length of the side rib and the perimeter of the perpendicular section.
The figure shows a perpendicular section.
S b = H ⋅ P main;
S n = S b + 2 S main
S b = l ⋅ P ter;
S n = S b + 2 S main
Obviously, this theorem is also true in the case of a straight prism, because then the perpendicular section will be a section of a plane parallel to the planes of the bases of the prism.
Note that if a certain polygon is a perpendicular section of a prism, then its interior angles are the linear angles of the dihedral angles between the corresponding side faces.
In the case of a straight prism, the linear angles of the dihedral angles between the side faces are directly the corners of the base.
Example
The figure shows a straight prism.
- linear angle dihedral angle between the edges and .
The prism is called correct, If:
it is based on a regular polygon;
the prism is straight.
Parallelepiped
A parallelepiped is a prism, which is based on a parallelogram. All faces of a parallelepiped are parallelograms.
The faces of a parallelepiped that do not have common vertices are called opposite.
Theorem 1. Opposite faces of a parallelepiped are parallel and equal.
The parallelepiped remains a parallelepiped in all cases when we consider any of its faces as its basis (see figure).
Theorem 2. The diagonals of the parallelepiped intersect at one point and the intersection point is divided in half.
From this it follows that the point of intersection of the diagonals of the parallelepiped is its center of symmetry.
Please note: a right parallelepiped has four diagonals, equal in pairs to each other.
On the image ; .
This follows from the properties of obliques, since - level perpendicular to the base plane ABCD.
If two diagonals of a right parallelepiped coming out of neighboring vertices, then the largest of them is the one that is projected into the large diagonal of the base, that is, such a diagonal of the parallelogram that lies opposite the obtuse angle. So, if in the figure above we consider the angle ABC blunt, let's get, .
A right parallelepiped whose base is a rectangle is called rectangular parallelepiped(see picture).
All faces of a cuboid are rectangles that can be divided into three equal pairs. An arbitrary face of a rectangular parallelepiped can be considered its base. Given that in parallel projection an arbitrary parallelogram can be represented by an arbitrary parallelogram, the image of a rectangular parallelepiped does not differ in any way from the image of any right parallelepiped.
The lengths of non-parallel edges are called linear dimensions(measurements) of a rectangular parallelepiped.
Theorem 3. In a rectangular parallelepiped, all diagonals are equal. The square of a diagonal is equal to the sum of the squares of its three dimensions.
All dihedral angles of a cuboid are right angles.
A rectangular parallelepiped has three pairs of equal diagonal sections. Each of these sections is a rectangle (see figures).
Each pair of sections intersect along a straight line that passes through the intersection points of the diagonals of opposite faces. The segments between these points are parallel and equal to one of the edges of the cuboid.
A right triangle is a triangle that is formed by the diagonal of a rectangular parallelepiped, the diagonal of the side face and the side of the base (see figure). For example, .
A rectangular parallelepiped has a center of symmetry - this is the point of intersection of its diagonals.
It also has three planes of symmetry passing through the center of symmetry parallel to the faces.
A rectangular parallelepiped in which all edges are equal is called cube.
The plane of any diagonal section of a cube is its plane of symmetry. Thus, the cube has nine planes of symmetry.
In the figure, consider the relative position of some elements of a straight parallelepiped:
- the angle between the diagonal of the side face and the plane of the base ( - perpendicular, - inclined, CD- projection).
- the angle between the diagonal of the right parallelepiped and the plane of the base ( - perpendicular, - oblique, AU- projection).
- the angle of inclination of the diagonal of the side face ( AD- perpendicular, - oblique, - projection).
Let be a right parallelepiped (see figure), where ABCD- rhombus. We draw its section by a plane passing through the diagonal of the base BD and top.
In the section we get an isosceles triangle.
- linear angle of the dihedral angle between the base and section planes. by the property of the diagonals of a rhombus, - perpendicular, - oblique, SO- projection. According to the theorem of three perpendiculars: .
Pyramid
Pyramid a polyhedron is called, which consists of a flat polygon - the base of the pyramid, a point that does not lie in the plane of the base - the top of the pyramid, and all segments connecting the top of the pyramid with the points of the base. The segments connecting the top of the pyramid with the tops of the base are called side ribs.pyramid height- a perpendicular dropped from the top of the pyramid to the plane of the base.
The pyramid is called n-coal if its base is n-gon. The triangular pyramid is also called tetrahedron. Lateral face of the pyramid- triangle. One of its vertices is the top of the pyramid, and the opposite side is the side of the base of the pyramid.
On the image SO is the height of the pyramid. Then - the angle between the side edge and the plane of the base ( SO- perpendicular, SA- inclined, OA- projection).
From the base of the height of the pyramid (points A) draw a perpendicular to the side of the base (for example, AE). The base of this perpendicular (point F) connect with the top of the pyramid (point S). According to the theorem of three perpendiculars: . ( SO- perpendicular, SP- inclined, OF- projection, by construction.) Therefore, - linear angle of the dihedral angle between the plane of the side face ASE and base plane.
To solve pyramid problems, it is very important to find out where the base of its height is located.
1. If at least one of the following conditions is met:
all side edges of the pyramid are equal;
all side ribs are inclined to the base plane at the same angle;
all side edges form the same angles with the height of the pyramid;
all side edges are equidistant from the base of the height, then the base of the height of the pyramid is the center of the circle circumscribed around the base of the pyramid.
Side rib l, height H and radius R circumscribed around the base of the circle form a right triangle:
In this case, the side surface can be found by the formula , where l- length of the side edge, , ... - flat angles at the top.
2. If at least one of the following conditions is met:
all side faces are inclined to the base plane at the same angle;
all side faces have the same height;
the heights of the side faces form the same angles with the height of the pyramid;
the side faces are equidistant from the base of the height, then the base of the height lies in the center of the circle inscribed in the base of the pyramid.
On the image - rectangular, - radius of the inscribed circle in ABCDEF;
- the height of the pyramid, SP- side face height;
- linear angle of the dihedral angle between the side face and the plane of the base;
ABOUT- the center of the circle inscribed in the base, that is, the point of intersection of the bisectors ABCDEF.
In this case .
3. If the side edge is perpendicular to the plane of the base, then this edge is the height of the pyramid (see figures).
In this case And - angles of inclination of the side ribs SW And SC respectively to the base plane. is linear angle dihedral angle between side faces SAC And SBA.
4. If the side face is perpendicular to the base plane (see figure), then the height of the pyramid will be the height of this face (according to the theorem “If a straight line lying in one of two perpendicular planes is perpendicular to the line of their intersection, then it is perpendicular to the second plane” ).
5. If two side faces are perpendicular to the plane of the base, then the height of the pyramid is their common side edge.
Distances from the base of the height of the pyramid
The distance from the base of the height of the pyramid to the lateral edge is the perpendicular dropped from the point ABOUT on this edge (see figure). Please note: , but in the figure should not be straight: angles are not preserved during parallel design. OF- distance from the base of the height to the side edge SE;
ON- distance from the base of the height to the side face ASB(See below for more on this distance.)
, where is the angle between the edge SE and base plane.
Distance from the base of the height to the side face
Let , then by the theorem of three perpendiculars. Hence, AB perpendicular to the plane S.O.K.. Hence, if , then ON perpendicular to the plane ASB..
The pyramid is called correct, if its base is a regular polygon and the base of its height is the same as the center of the polygon. axis a regular pyramid is called a straight line containing its height. The side edges of a regular pyramid are equal, the side faces are equal isosceles triangles. The height of the side face drawn from the top of the pyramid is called apothem. It is the bisector and median of the side face, and is an isosceles triangle.
Theorem. The lateral surface of a regular pyramid is equal to the product of the pіvperimeter of the base and the apothem.
; ,
Where R- perimeter of the base, A- base side l- apothem length.
Regular triangular pyramid
At the base of a regular triangular pyramid lies equilateral triangle depicted by an arbitrary triangle (see figure).
The center is the point of intersection of its bisectors, which are both heights and medians. Medians in parallel projection are represented by medians. Therefore, we build two medians of the base. The point of their intersection is the base of the height of the pyramid. We represent the height, and then we connect the top of the pyramid with the tops of the base. We get side ribs.
In the figure: - the angle of inclination of the side rib to the plane of the base (the same for all ribs); - the angle of inclination of the side face to the plane of the base (the same for all faces).
Let .
Then ; ; ;
; ; .
Hence, .
; .
Plane of axial section ASD is the plane of symmetry of a regular triangular pyramid.
This plane is perpendicular to the plane of the base and the plane of the face BSC.
It is also interesting to note that the crossing edges of the pyramid ( SA And BC, SB And AC, SC And AB) are perpendicular. If , then ON is the distance from the base of the height not only to the anathema, but also to the side face BSC.
.
Regular quadrangular pyramid
At the base of a regular quadrangular pyramid lies a square, which is represented by an arbitrary parallelogram. Its center is the intersection point of the diagonals. This point is the base of the height of the pyramid.
Let the side of the square A(see picture).
Then ;
;
;
;
.
Note: , , that is .
Parallel design preserves parallelism.
; .
Distance from height base to side face:
; .
Regular hexagonal pyramid
At the heart of a regular hexagonal pyramid is a regular hexagon (see figure). Its center is the intersection point of the diagonals. This point is the base of the height of the pyramid.
Then ;
Let the side of a regular hexagon A.
;
;
.
; .
Truncated pyramid
Sheared pyramid a polyhedron is called, which will remain if a pyramid with the same vertex is separated from the pyramid by a plane that is parallel to the base.
Theorem. A plane parallel to the base of the pyramid and intersecting it cuts off a similar pyramid.
Please note: in order to correctly depict the cut pyramid, you must start with the image of the original full pyramid (see figure).
The bases of a truncated pyramid are similar polygons. Lateral faces - trapezoids. - the height of the truncated pyramid, the height of the side face - the angle of inclination of the side edge to the plane of the base (any), - the angle of inclination of the side face to the plane of the lower base.
Correct truncated pyramid- this is a truncated pyramid, which was taken out of a regular pyramid.
Its lateral ribs are equal and inclined to the plane of the base at the same angle. Its lateral faces are equal to the equilateral trapezium and are inclined to the plane of the lower base at the same angle. The heights of the side faces of the pyramid are called apothems.
The lateral surface of a regular truncated pyramid is equal to the product of half the sum of the perimeters of the bases and the apothem.
, Where P n and P c - the perimeters of the corresponding bases, l- apothem.
The figures show figures that can be very useful to consider when solving problems on a truncated pyramid.
;
.
;
- a rectangular trapezoid.
- the height of the truncated pyramid.
-
side edge height.
In the case when the truncated pyramid is regular, the segments OD and are the radii of the circumscribed circle, and OF and - radii of the inscribed circle for the lower and upper base, respectively.
Regular polyhedra
A convex polyhedron is called right, if its faces are regular polyhedra with the same number of sides, and the same number of edges coincide at each vertex of the polyhedron. There are five types of regular convex polyhedra: regular tetrahedron, cube, octahedron, dodecahedron, icosahedron.
1. A regular tetrahedron has faces - regular triangles; there are three edges at each vertex. A tetrahedron is a triangular pyramid, all edges of which are equal.
2. All faces of a cube are squares; there are three edges at each vertex. A cube is a rectangular parallelepiped with equal edges.
3. The faces of the octahedron are regular triangles. Each of its vertices has four edges.
4. In the dodecahedron, the faces are regular p "yatikutniks. Three edges coincide at each of its vertices.
5. The faces of the icosahedron are regular triangles. Each of its vertices has five edges.
The figures show examples of regular polyhedra with names.
Polyhedra are the simplest solids in space, just as polygons are the simplest figures in the plane. We see multifaceted forms every day: a matchbox, a book, a room, multi-storey building(with a horizontal roof) - rectangular parallelepipeds; milk bags-tetrahedra or also parallelepipeds; a faceted pencil, a nut give an idea of \u200b\u200bprisms (however, a parallelepiped is also a quadrangular prism). Many architectural structures or their details are pyramids or truncated pyramids - such forms have the famous Egyptian pyramids or the towers of the Kremlin. Many polyhedral shapes, such as the "house" in Fig. 1 and " round house» in fig. 2 do not have special names. From a purely geometric point of view, a polyhedron is a part of space bounded by flat polygons - faces. The sides and vertices of the faces are called the edges and vertices of the polyhedron itself. The faces form the so-called polyhedral surface. To exclude from consideration polyhedral figures of the type shown in Fig. 3, which are not usually called polyhedra, the following restrictions are usually imposed on a polyhedral surface:
1) each edge must be a common side of two, and only two, faces, called adjacent;
2) every two faces can be connected by a chain of successively adjacent faces;
3) for each vertex, the angles of the faces adjacent to this vertex must limit some polyhedral angle.
A polyhedron is called convex if it lies on one side of the plane of any of its faces. This condition is equivalent to each of the other two: 1) a segment with ends at any two points of the polyhedron lies entirely in the polyhedron, 2) the polyhedron can be represented as the intersection of several half-spaces.
For any convex polyhedron, the Euler formula is valid (see Topology), which establishes a connection between the number of vertices B, edges P, and faces G:
For non-convex polyhedra, this relation, generally speaking, is not true, for example, for the polyhedral surface shown in Fig. 2; , , That's why . The number is called the Euler characteristic of the polyhedron and can be equal to 
. The Euler characteristic shows, roughly speaking, how many "holes" a polyhedron has. Number of holes (or ).
The simplest classification by the number of vertices (angles, sides) for polyhedra is inefficient. The simplest polyhedra - four-vertex or tetrahedra - are always limited to four triangular faces. But already pentahedrons can be completely different types, for example: a quadrangular pyramid is limited by four triangles and one quadrangle (Fig. 4, a), and a triangular prism is limited by two triangles and three quadrangles (Fig. 4, b). Examples of five-vertex structures are a quadrangular pyramid and a triangular dihedron (Fig. 4c).
The most common polyhedra in the world around us, of course, have special names. So, - coal pyramid has a -gon at the base and lateral triangular faces converging at a common vertex of the triangles (Fig. 4a, where ); -angular prism is limited by two equal, parallel and equally located -gons - bases - and parallelograms - side faces connecting the corresponding sides of the bases (Fig. 4,b, where ).
An intermediate position between pyramids and prisms is occupied by truncated pyramids, obtained from pyramids by cutting off smaller pyramids by planes parallel to the bases (Fig. 5). Among the natural forms of crystals there are dihedra, or bipyramids, composed of two pyramids with common ground(Fig. 4c). Archimedes also considered -gonal antiprisms bounded by two parallel, but rotated relative to each other -gons and connecting them, as shown in Fig. 6, -triangles (with a large antiprism, it looks like a pioneer drum - Fig. 6).
Like polygons, polyhedra are also classified according to their degree of symmetry. Among the pyramids, the correct ones are distinguished: they have a regular polygon at the base, and the height - the perpendicular drawn from the top to the base plane - falls into the center of the base of the pyramid.
The analogue of a parallelogram is a parallelepiped; just like a parallelogram, a parallelepiped has a center of symmetry at which all four diagonals (segments connecting vertices that do not belong to the same face) intersect and bisect. Regular prisms at the bases have regular polygons arranged so that the line passing through their centers is perpendicular to the planes of the bases. The bases of a regular -angular antiprism must also be located, but only one base must be rotated through an angle relative to the other. All regular polyhedra have quite a lot of self-coincidences - rotations and symmetries that transform the polyhedron into itself. The totality of all self-combinations, considering and identical, forms the so-called group of symmetries of the polyhedron. According to symmetry groups in crystallography, single crystals are classified, which, as a rule, have a polyhedral shape.
The symmetry and regularity of the polyhedra discussed above are not entirely complete - they may have unequal faces, different polyhedral angles. The exceptions are three polyhedra: a regular tetrahedron is a regular triangular pyramid with equal edges, bounded by four regular triangles (Fig. 7, a); a cube, or a regular hexahedron, is a regular quadrangular prism with equal edges, limited by six squares (Fig. 7, b); finally, the octahedron is a regular quadrangular dihedron with equal edges, bounded by eight regular triangles (Fig. 7c); An octahedron can also be defined as a regular triangular antiprism with equal edges. Unlike arbitrary regular pyramids, prisms, dihedra and antiprisms, the tetrahedron, cube, octahedron are such that any two of their faces (and any two polyhedral angles) can be combined using some self-combination of the entire polyhedron. In addition, their polyhedral angles are regular, i.e. have equal plane and equal dihedral angles.
Similarly to regular polygons on the plane, one can also define regular polyhedra "in general": these are convex polyhedra bounded by equal regular polygons and having equal regular polyhedral angles. It turns out that in addition to the three types of regular polyhedra mentioned above - a regular tetrahedron, a cube and an octahedron - there are only two more types of regular polyhedra: a dodecahedron (dodecahedron) and an icosahedron (twenty-sided), limited respectively by 12 regular pentagons and 20 regular triangles - Fig. 8, a, b. These two polyhedra are interconnected in the same way as a cube and a tetrahedron (see Cube): the centers of the faces of the dodecahedron are the vertices of the icosahedron - fig. 9, and vice versa.
The very fact of the existence of only five really regular polyhedra is surprising - after all, there are infinitely many regular polygons on the plane.
All regular polyhedra were known back in Ancient Greece, and the final, XIII book of the famous "Beginnings" of Euclid is dedicated to them. These polyhedra are often also called the Platonic solids - in the idealistic picture of the world given by the great ancient Greek thinker Plato, four of them personified the four elements: the tetrahedron - fire, the cube - earth, the icosahedron - water and the octahedron - air; the fifth polyhedron, the dodecahedron, symbolized the entire universe - they began to call it in Latin quinta essentia (“fifth essence”). Apparently, it was not difficult to come up with the correct tetrahedron, cube, octahedron, especially since these forms have natural crystals, for example: a cube is a monocrystal of sodium chloride (NaCl), an octahedron is a single crystal of potassium alum 
. There is an assumption that the ancient Greeks obtained the shape of the dodecahedron by considering crystals of pyrite (sulphurous pyrite FeS). Having a dodecahedron, it is not difficult to build an icosahedron: as already mentioned, its vertices will be the centers of the twelve faces of the dodecahedron - fig. 9.
“My name will be slandered, many atrocities will be attributed to me. World Zionism will strive with all its might to destroy our Union so that Russia can never rise again. The spearhead of the struggle will be aimed at separating the border regions from Russia. Nationalism will raise its head with particular force. Many pygmy leaders will appear, traitors within their nations ... "
“Stalin is the center, the heart of everything that radiates from Moscow all over the world.”
French writer A. Barbusse
65 years ago, on March 5, 1953, the great people's leader Joseph Stalin passed away. The man who was able to revive the Russian empire in the form of the Soviet Union, who won the Second World War, who created powerful armed forces, a nuclear shield for our Motherland, the best science and education in the world.
In "democratic Russia", created in 1991-1993, he was declared a maniac and a bloody dictator. Why is Stalin so hated by various Westerners, liberals and small-town nationalists? The answer is simple. Stalin was a real people's leader who devoted his whole life to solving the global and national problems of Russian civilization and the Russian people. He forced the government and the Communist Party to serve the Motherland without sparing himself. And after his death, he left no wealth, no accounts in foreign banks, no palaces and villas, no stolen billions and gold. The Soviet superpower became his treasure.
Most importantly, Stalin showed the main path of the future great Russia(USSR) and all mankind - a society of the "golden age", a society of social justice, service and creation. A society where the ethics of conscience dominates, and a person is a creator, a creator, serves the Motherland and the people. Stalin showed an alternative path for the development of all mankind. The masters of the Western project and civilization are building an unjust world order - a global slave, slave-owning, caste civilization, where there is a handful of "masters of life and money", "chosen ones" who are allowed everything, and who have access to genuine knowledge, the most advanced achievements of science, technology, medicine.
And the rest of the people are immersed in the darkness of poverty, do not have access to normal education and healthcare, are constantly intoxicated with various drugs: tobacco, alcohol, heavier intoxicants, food surrogates, information-virtual illusions, etc. Their lifespan is deliberately reduced, spirituality , intellect and physical condition are suppressed, descend to the level of two-legged tools, cattle.
At the same time, Western "elites" are constantly developing and implementing plans to reduce the human "biomass". So that more resources remain "chosen", so that you can create a clean planet, without two-legged "viruses" that kill the Earth.
This is junk food, and putting people on drugs, with the suppression of normal immunity and the absence of normal programs for physical and spiritual development of people. This is the creation of a society of stress, where people spin like squirrels in a wheel, extracting resources for a “normal” life, but in reality they ruin the mental and physical health, sit down on stimulants and dope to temporarily forget. This is a consumer society, which destroys both the planet, its biosphere, and the person himself, as part of a common living system. A person is turned into a consumer animal, completely dependent on the “masters of life”. This is also a system aimed at destroying the reproduction of mankind - propaganda of abortions, contraceptives, ideas of childlessness, homosexual "marriages", various perversions (perverts do not give birth to children), virtual sex, sex robots are next in line, etc.
Under Stalin, a just state and society began to be built in the USSR, a society of service and creation, a society with the dominance of the ethics of conscience. Hence the most powerful spiritual impulse of the people, which made it possible not only to create a superpower, to win the most terrible war in the history of mankind, but also to eliminate all the consequences of the most severe world slaughter, to create socialist camp, which made it possible to resist Western world based on its colonies and semi-colonies. Popular support made it possible to build an independent National economy that provided everything you need Soviet people and even support allies, create the best armed forces in the world, eliminating the threat of a new open large-scale attack on the USSR-Russia for several generations (most of the inhabitants of Russia live in the world only thanks to this foundation), create the world's best science, education, a system that reveals creative, creative potential of children and youth and much more.
During the life of Joseph Vissarionovich, the common people idolized him. Songs were sung about him, monuments were erected to him, cities and towns were given his name. large enterprises. Stalin and his government accepted a ruined and devastated Russia, which went through the catastrophe of the former development project in 1917. The Bolsheviks (Russian communists), contrary to popular belief, had practically nothing to do with this catastrophe, they simply took power in the deceased " old Russia". Offered to the people new project- Soviet civilization, which was in the interests of the vast majority of the people. Managed to create a Soviet superpower - returned most of the lands lost during the years of unrest, defeated Japan and Germany, whom they lost royal Russia. Soviet Union included in its sphere of influence half the planet, including China. During the years of Stalin's rule, the national economy was rebuilt, which became more efficient than in the countries of the leaders of the capitalist world, they created advanced industries that only the most advanced powers had - aircraft building, shipbuilding, mechanical engineering, machine tool building, chemical industry, military-industrial complex, rocket science. They created nuclear weapons and created the foundation of the space industry. Unemployment was eliminated, education and health care became free and publicly available. Children from the poor peasant families, who had no chance under capitalism, became professors and marshals, aces pilots and ministers under socialism.
Under the leadership of Stalin, the Second World War was won, when the masters of the West allowed to take power in Europe German Nazis led by Hitler. The masters of the West were afraid of the Soviet project. Russia was becoming an alternative center of a new just world order. Sympathy of a large part of mankind, the best people The lands were on the side of the "sunny" Soviet civilization. As a result, in fact, the "European Union" was created, headed by Germany, and all its power - military-technical, demographic and economic - was thrown against the Soviet civilization, which challenged the dominance of the West over the planet. However, the Russian (Soviet) army defeated a strong and cruel enemy. Eastern and part Central Europe, including East Germany entered the sphere of influence of Moscow. The Soviet Union defeated militarist Japan, taking revenge for the disgrace Russo-Japanese War 1904-1905 and regained its influence on Far East. With our help, the Communists won in China and the Celestial Empire recognized the USSR as its “big brother”.
Stalin did not flinch in the face of the atomic threat from the United States, which conducted a bloody "test" nuclear weapons in Japan. Moscow had such powerful armed forces that the United States and England and their allies did not dare immediately after the end of World War II to start a “hot” Third world war(although there were plans). Soon Moscow created its atomic bomb and rapidly built up a first-class nuclear arsenal. The West started the "cold" World War III - the information-ideological, economic, secret war of special services, the war on the territory of other countries (Korean War, etc.).
Therefore, our enemies in the West and Russian Westernizers, who betrayed the USSR and the ideals of socialism, social justice, hate Stalin. They created a mass of black myths to slander the great national leader. However, the truth finds its way even in the atmosphere total lie. Therefore, the image of Stalin is now again popular among the Russian people. During his reign, people had faith in social justice, in the future of the people and the country. A powerful economic, scientific, technical, educational, cultural and military foundation was created, which allowed Russia to survive to this day.
Even an outspoken enemy of the Union and an implacable anti-communist, the famous British Prime Minister W. Churchill, speaking in the House of Commons on December 21, 1959, on the day of Stalin's 80th birthday, recognized his outstanding role in the world: “He was the most outstanding personality, impressing our changeable and cruel time of the period in which his life passed. Stalin was a man of extraordinary energy and unbending willpower, sharp, cruel, merciless in conversation, to whom even I, brought up here in the British Parliament, could not oppose anything. Stalin first of all possessed great feeling humor and sarcasm and the ability to accurately perceive thoughts. This force was so great in Stalin that he seemed unique among the leaders of all times and peoples. Stalin made the greatest impression on us. He possessed deep, devoid of any panic, logically meaningful wisdom. He was consummate master find a way out of the most hopeless situation in difficult moments of the journey. In addition, Stalin in the most critical moments, and also in moments of triumph was equally restrained and never succumbed to illusions.
The part of geometry studied so far is called planimetry - this part was about the properties of plane geometric shapes, that is, figures entirely located in a certain plane. But most of the objects around us are not flat. Any real object occupies some part of the space.
The branch of geometry that studies the properties of figures in space is called stereometry.
If the surfaces of geometric bodies are composed of polygons, then such bodies are called polyhedra.
The polygons that make up a polyhedron are called its faces. It is assumed that no two adjacent faces of the polyhedron lie in the same plane.
The sides of the faces are called edges, and the ends of the edges are the vertices of the polyhedron.
A segment connecting two vertices that do not belong to the same face is called the diagonal of the polyhedron.
Polyhedra are either convex or non-convex.
A convex polyhedron is characterized by the fact that it is located on one side of the plane of each of its faces. In the figure, a convex polyhedron is an octahedron. The octahedron has eight faces, all faces are regular triangles.
The figure shows a non-convex (concave) polygon. If we consider, for example, the plane of the triangle \(EDC\), then, obviously, part of the polygon is on one side, and part is on the other side of this plane.
For further definitions, we introduce the concept of parallel planes and parallel lines in space and perpendicularity of a line and a plane.
Two planes are said to be parallel if they have no common points.
Two lines in space are called parallel if they lie in the same plane and do not intersect.
The straight line is called perpendicular to the plane if it is perpendicular to any line in that plane.
Prism
Now we can introduce the definition of a prism.
\(n\)-gonal prism is called a polyhedron composed of two equal \(n\)- squares, lying in parallel planes, and \(n\)-parallelograms, which were formed by connecting the vertices of \(n\)-gons by segments of parallel lines.
Equal \(n\)-gons are called the bases of a prism.
The sides of polygons are called base edges.
Parallelograms are called side faces prisms.
Parallel lines are called lateral ribs prisms.
Prisms are straight and oblique.
If the bases of a straight prism are regular polygons, then such a prism is called regular.
For straight prisms, all side faces are rectangles. The side edges of a straight prism are perpendicular to the planes of its bases.
If from any point of one base to draw a perpendicular to another base of the prism, then this perpendicular is called the height of the prism.
In the figure, an inclined quadrangular prism, in which the height B 1 E is drawn.
In a straight prism, each of the side edges is the height of the prism.
The figure shows a right triangular prism. All side faces are rectangles, any side edge can be called the height of the prism. A triangular prism has no diagonals, since all vertices are connected by edges.
The figure shows a regular quadrangular prism. The bases of a prism are squares. All diagonals of a regular quadrangular prism are equal, intersect at one point and bisect at that point.
A quadrangular prism whose bases are parallelograms is called parallelepiped.
The above regular quadrangular prism can also be called straight parallelepiped.
If the bases of a right parallelepiped are rectangles, then this parallelepiped is rectangular.
The figure shows a rectangular parallelepiped. The lengths of three edges with a common vertex are called the dimensions of the cuboid.
For example, AB , AD and A A 1 can be called dimensions.
Since triangles ABC and AC C 1 are rectangular, then, therefore, the square of the length of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its measurements:
A C 1 2 = AB 2 + AD 2 + A A 1 2
If a section is drawn through the corresponding diagonals of the bases, then it is called diagonal section prisms.
In straight prisms, the diagonal sections are rectangles. Equal diagonal sections pass through equal diagonals.
The figure shows a regular hexagonal prism in which two different diagonal sections are drawn, which pass through diagonals with different lengths.
Basic formulas for calculations in direct prisms
1. Side surface S side. = P main. ⋅ H , where \(H\) is the height of the prism. For inclined prisms, the area of each side face is determined separately.
2. Complete surface S is complete. = 2 ⋅ S main. + S side. . This formula is valid for all prisms, not only for straight lines.
3. Volume V = S main. ⋅H. This formula is valid for all prisms, not only for straight lines.
Pyramid
\(n\)- coal pyramid- a polyhedron composed of \(n\)-gon at the base and \(n\)-triangles, which were formed by connecting the point of the top of the pyramid with all the vertices of the base polygon.
The \(n\)-gon is called the base of the pyramid.
Triangles are the side faces of the pyramid.
The common vertex of the triangles is the vertex of the pyramid.
The edges coming out of the top are the side edges of the pyramid.
The perpendicular from the top of the pyramid to the plane of the base is called the height of the pyramid.
Basic plan-summary on the topic:
"Introduction to stereometry" Grade 10
Basic concepts: point, line, plane, polyhedron, polyhedron face, opposite faces, adjacent faces, side faces, bases, polyhedron edge, polyhedron vertex, opposite vertices, diagonal, full surface, area full surface, convex polytope, non-convex polytope
Building tools: ruler without divisions, compasses, drawing square.
Stereometry (from the Greek "stereos" - spatial) - a section of geometry in which the properties of not only flat, but also spatial geometric shapes are studied.Basic figures (the simplest figures) in stereometry are points, lines and planes.
Axioms (statements accepted without proof)
1. There are infinitely many planes in space, and on each of them planimetry is fulfilled, i.e., the axioms of planimetry and their consequences are valid.
2. The signs of equality and similarity of triangles studied in planimetry are also valid for triangles lying in different planes.
Basic rules for drawing figures
The segment is shown as a segment
The middle of the segment is represented by the middle of its image
A point dividing a segment in relationm: nrepresented by a dot dividing its image in relation tom: n
Parallel lines (segments) are depicted as parallel lines (segments). Parallelism is preserved
An arbitrary triangle can be taken as an image of any triangle
Polyhedron – geometric body, bounded by a finite number of planar polygons, no two adjacent of which lie in the same plane. The polygons are calledfaces, their sides are ribs polyhedron, and their vertices arepeaks polyhedron.
Complete surface is a figure formed by all the faces of a polyhedron.
Total surface area (S full) is the sum of the areas of all faces.
Lateral surface area (S side) - the sum of the areas of the side faces.
Basic concepts: cube, cuboid, cuboid, cuboid, prism, cuboid, regular n-gonal prism, pyramid, regular pyramid, tetrahedron
| Cube is a polyhedron with six faces that are equal squares (Fig. 1). The sides of the squares are called ribs Cuba. The vertices of the squares are called peaks Cuba. | |
| Parallelepiped - this is a polyhedron with six faces and each of them is a parallelogram (Fig. 2). Opposite faces are faces that do not have a common edge. Related faces are faces that have a common edge. Opposite vertices are two vertices of the box that do not belong to the same face. Diagonal is a segment connecting opposite vertices (Fig. 3). Right parallelepiped is a parallelepiped whose lateral faces are rectangles. cuboid - this is a parallelepiped, in which all faces are rectangles (Fig. 4). | |
| Prism ( n -coal) is a polyhedron in which two faces are equal n-gons, and the rest n the faces are parallelograms (Fig. 5). Equal n-gons are called grounds , and the parallelograms lateral faces. straight prism - this is such a prism, in which the side faces are rectangles (Fig. 6). correct n - carbon prism - this is such a prism, in which the side faces are rectangles, and its bases are regular n-gons. | |
| Pyramid ( n -coal)- this is a polyhedron, in which one face is some n-gon, and the remaining n faces are triangles with a common vertex (Fig. 7). Base pyramid is called an n-gon. Side faces are triangles that have a common vertex. Vertex the pyramid is their common top. Ribs the pyramid is the sides of the faces of the pyramid. Side The edges of a pyramid are edges that converge at a vertex. Correct pyramid - this is such a pyramid, the base of which is a regular n-gon, and all side edges are equal to each other. Tetrahedron is a triangular pyramid. regular tetrahedron is a triangular pyramid if all its faces are regular triangles. |
