Formulas for planimetry and stereometry for the exam. Handbook with the basic facts of stereometry

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Some definitions:

Polyhedron is a geometric body bounded by a finite number of plane polygons, any two of which, having a common side, do not lie in the same plane. In this case, the polygons themselves are called faces, their sides are the edges of the polyhedron, and their vertices are the vertices of the polyhedron.
The figure formed by all the faces of a polyhedron is called its surface ( full surface), and the sum of the areas of all its faces is (full) surface area.
is a polyhedron with six faces that are equal squares. The sides of the squares are called the edges of the cube, and the vertices are called the vertices of the cube.
is a polyhedron that has six faces and each of them is a parallelogram. The sides of the parallelograms are called the edges of the parallelepiped, and their vertices are called the vertices of the parallelepiped. The two sides of a parallelepiped are called opposite, if they do not have a common edge, and those having a common edge are called related. Sometimes any two opposite faces of the parallelepiped are selected and called grounds, then the rest of the faces side faces, and their sides, connecting the vertices of the bases of the parallelepiped, are its side ribs.
Right parallelepiped- this is a parallelepiped whose side faces are rectangles. is a parallelepiped whose faces are all rectangles. Note that every cuboid is a cuboid, but not every cuboid is a cuboid.
opposite. A line segment connecting opposite vertices of a parallelepiped is called diagonal parallelepiped. A parallelepiped has only four diagonals.
Prism ( n-coal) is a polyhedron whose two faces are equal n-gons, and the rest n faces are parallelograms. Equal n-gons are called grounds, and the parallelograms side faces of the prism- this is such a prism, in which the side faces are rectangles. correct n- carbon prism- this is a prism, in which all side faces are rectangles, and its bases are regular n-gons.
The sum of the areas of the side faces of the prism is called its lateral surface area(denoted S side). The sum of the areas of all the faces of the prism is called prism surface area(denoted S full).
Pyramid ( n-coal)- this is a polyhedron, which has one face - some n-gon, and the rest n faces - triangles with a common vertex; n-gon is called basis; triangles that have a common vertex are called side faces, and their common vertex is called top of the pyramid. The sides of the faces of a pyramid are called its ribs, and edges that meet at a vertex are called lateral.
The sum of the areas of the side faces of the pyramid is called side surface area of the pyramid(denoted S side). The sum of the areas of all the faces of the pyramid is called pyramid surface area(surface area is denoted S full).
correctn- coal pyramid- this is such a pyramid, the base of which is the correct n-gon, and all side edges are equal to each other. The side faces of a regular pyramid are isosceles triangles equal to each other.
The triangular pyramid is called tetrahedron if all its faces are congruent regular triangles. The tetrahedron is a special case of a regular triangular pyramid (i.e. not every regular triangular pyramid will be a tetrahedron).

Axioms of stereometry:

Through any three points that do not lie on the same line, there is only one plane.
If two points of a line lie in a plane, then all points of the line lie in that plane.
If two planes have a common point, then they have a common line on which all common points of these planes lie.

Consequences from the axioms of stereometry:

Theorem 1. There is only one plane through a line and a point not on it.
Theorem 2. There is only one plane through two intersecting lines.
Theorem 3. There is only one plane through two parallel lines.

Construction of sections in stereometry

To solve problems in stereometry, it is urgently necessary to be able to build sections of polyhedra (for example, a pyramid, a parallelepiped, a cube, a prism) in a drawing by a certain plane. Let's give a few definitions explaining what a section is:

cutting plane A pyramid (prism, parallelepiped, cube) is such a plane, on both sides of which there are points of this pyramid (prism, parallelepiped, cube).
cross section of a pyramid(prism, parallelepiped, cube) is a figure consisting of all points that are common to the pyramid (prism, parallelepiped, cube) and the cutting plane.
The cutting plane intersects the faces of the pyramid (parallelepiped, prism, cube) along segments, therefore section is a polygon lying in the secant plane, the sides of which are the indicated segments.

To construct a section of a pyramid (prism, parallelepiped, cube), it is possible and necessary to construct the intersection points of the secant plane with the edges of the pyramid (prism, parallelepiped, cube) and connect every two of them lying in one face. Note that the sequence of constructing the vertices and sides of the section is not essential. The construction of sections of polyhedra is based on two tasks for construction:

Lines of intersection of two planes.

To construct a line along which some two planes intersect α And β (for example, the secant plane and the plane of the face of the polyhedron), you need to build their two common points, then the line passing through these points is the line of intersection of the planes α And β .

Points of intersection of a line and a plane.

To construct a point of intersection of a line l and plane α draw the point of intersection of the line l and direct l 1 , along which the plane intersects α and any plane containing a line l.

Mutual arrangement of straight lines and planes in stereometry

Definition: In the course of solving problems in stereometry, two straight lines in space are called parallel if they lie in the same plane and do not intersect. If straight but And b, or AB And CD are parallel, we write:

Several theorems:

Theorem 1. Through any point in space that does not lie on a given line, there is only one line parallel to the given line.
Theorem 2. If one of two parallel lines intersects a given plane, then the other line intersects this plane.
Theorem 3(sign of parallel lines). If two lines are parallel to a third line, then they are parallel to each other.
Theorem 4(on the point of intersection of the diagonals of a parallelepiped). The diagonals of the parallelepiped intersect at one point and bisect that point.

There are three cases of mutual arrangement of a straight line and a plane in stereometry:

The line lies in the plane (each point of the line lies in the plane).
The line and the plane intersect (have a single common point).
A line and a plane do not have a single common point.

Definition: Line and plane are called parallel if they do not have common points. If straight but parallel to the plane β , then they write:

Theorems:

Theorem 1(a sign of parallelism of a straight line and a plane). If a line not lying in a given plane is parallel to some line lying in this plane, then it is parallel to the given plane.
Theorem 2. If the plane (in the figure - α ) passes through a straight line (in the figure - from), parallel to another plane (in the figure - β ), and intersects this plane, then the line of intersection of the planes (in the figure - d) is parallel to the given line:

If two distinct lines lie in the same plane, then they either intersect or are parallel. However, in space (i.e., in stereometry), a third case is also possible, when there is no plane in which two lines lie (in this case, they neither intersect nor are parallel).

Definition: The two lines are called interbreeding, if there is no plane in which they both lie.

Theorems:

Theorem 1(a sign of intersecting lines). If one of the two lines lies in a certain plane, and the other line intersects this plane at a point that does not belong to the first line, then these lines are skew.
Theorem 2. Through each of the two intersecting lines there is a single plane parallel to the other line.

Now we introduce the concept of the angle between skew lines. Let be a And b O in space and draw straight lines through it. a 1 and b 1 parallel to straight lines a And b respectively. Angle between skew lines a And b called the angle between the constructed intersecting lines a 1 and b 1 .

However, in practice the point O more often choose so that it belongs to one of the straight lines. This is usually not only elementary more convenient, but also more rational and correct in terms of constructing a drawing and solving a problem. Therefore, for the angle between skew lines, we give the following definition:

Definition: Let be a And b are two intersecting lines. Take an arbitrary point O on one of them (in our case, on a straight line b) and draw a line through it parallel to another of them (in our case a 1 parallel a). Angle between skew lines a And b is the angle between the constructed line and the line containing the point O(in our case, this is the angle β between straight lines a 1 and b).

Definition: The two lines are called mutually perpendicular(perpendicular) if the angle between them is 90°. Crossing lines can be perpendicular, as well as lines lying and intersecting in the same plane. If straight a perpendicular to the line b, then they write:

Definition: The two planes are called parallel, if they do not intersect, i.e. do not have common points. If two planes α And β parallel, then, as usual, write:

Theorems:

Theorem 1(sign of parallel planes). If two intersecting lines of one plane are respectively parallel to two lines of another plane, then these planes are parallel.
Theorem 2(on the property of opposite faces of a parallelepiped). Opposite faces of a parallelepiped lie in parallel planes.
Theorem 3(on the lines of intersection of two parallel planes by a third plane). If two parallel planes are intersected by a third, then their lines of intersection are parallel to each other.
Theorem 4. Segments of parallel lines located between parallel planes are equal.
Theorem 5(on the existence of a unique plane parallel to a given plane and passing through a point outside it). Through a point not lying in a given plane, there is only one plane parallel to the given one.

Definition: A line intersecting a plane is said to be perpendicular to the plane if it is perpendicular to every line in that plane. If straight a perpendicular to the plane β , then write, as usual:

Theorems:

Theorem 1. If one of two parallel lines is perpendicular to a third line, then the other line is also perpendicular to this line.
Theorem 2. If one of two parallel lines is perpendicular to a plane, then the other line is also perpendicular to that plane.
Theorem 3(on the parallelism of lines perpendicular to the plane). If two lines are perpendicular to the same plane, then they are parallel.
Theorem 4(a sign of perpendicularity of a straight line and a plane). If a line is perpendicular to two intersecting lines lying in a plane, then it is perpendicular to that plane.
Theorem 5(about a plane passing through a given point and perpendicular to a given line). Through any point in space there is only one plane perpendicular to the given line.
Theorem 6(about a straight line passing through a given point and perpendicular to a given plane). Through any point in space there is only one line perpendicular to the given plane.
Theorem 7(on the property of the diagonal of a rectangular parallelepiped). The square of the length of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of the lengths of its three edges that have a common vertex:

Consequence: All four diagonals of a rectangular parallelepiped are equal to each other.

Three perpendiculars theorem

Let the point BUT does not lie flat α . Let's pass through the point BUT straight line perpendicular to the plane α , and denote by the letter ABOUT the point of intersection of this line with the plane α . A perpendicular drawn from a point BUT to the plane α , is called a segment JSC, dot ABOUT called the base of the perpendicular. If JSC- perpendicular to the plane α , but M is an arbitrary point of this plane, different from the point ABOUT, then the segment AM is called a slope drawn from a point BUT to the plane α , and the point M- inclined base. Section OM- orthogonal projection (or, in short, projection) oblique AM to the plane α . Now we present a theorem that plays an important role in solving many problems.

Theorem 1 (about three perpendiculars): A straight line drawn in a plane and perpendicular to the projection of an inclined plane onto this plane is also perpendicular to the inclined plane itself. The converse is also true:

Theorem 2 (about three perpendiculars): A straight line drawn in a plane and perpendicular to an inclined one is also perpendicular to its projection on this plane. These theorems, for the notation from the drawing above, can be briefly formulated as follows:

Theorem: If from one point, taken outside the plane, a perpendicular and two oblique lines are drawn to this plane, then:

two oblique, having equal projections, are equal;
of the two inclined ones, the one whose projection is larger is larger.

Definitions of distances by objects in space:

The distance from a point to a plane is the length of the perpendicular drawn from that point to that plane.
The distance between parallel planes is the distance from an arbitrary point of one of the parallel planes to another plane.
The distance between a line and a plane parallel to it is the distance from an arbitrary point on the line to the plane.
The distance between skew lines is the distance from one of the skew lines to a plane passing through the other line and parallel to the first line.

Definition: In stereometry, orthogonal projection of a straight line a to the plane α is called the projection of this line onto a plane α if the straight line defining the design direction is perpendicular to the plane α .

Comment: As you can see from the previous definition, there are many projections. Other (except orthogonal) projections of a straight line onto a plane can be constructed if the straight line that determines the projection direction is not perpendicular to the plane. However, it is the orthogonal projection of a straight line onto a plane that we will encounter in problems in the future. And we will call the orthogonal projection simply a projection (as in the drawing).

Definition: The angle between a straight line that is not perpendicular to a plane and this plane is the angle between a straight line and its orthogonal projection onto a given plane (the angle AOA’ in the drawing above).

Theorem: The angle between a line and a plane is the smallest of all the angles that a given line forms with lines lying in a given plane and passing through the point of intersection of the line and the plane.

Definitions:

dihedral angle A figure is called a figure formed by two half-planes with a common boundary line and a part of the space for which these half-planes serve as a boundary.
Linear dihedral angle An angle is called, the sides of which are rays with a common origin on the edge of the dihedral angle, which are drawn in its faces perpendicular to the edge.

Thus, the linear angle of a dihedral angle is the angle formed by the intersection of the dihedral angle with a plane perpendicular to its edge. All linear angles of a dihedral angle are equal to each other. The degree measure of a dihedral angle is the degree measure of its linear angle.

A dihedral angle is called right (acute, obtuse) if its degree measure is 90° (less than 90°, more than 90°). In the future, when solving problems in stereometry, by a dihedral angle we will always understand that linear angle, the degree measure of which satisfies the condition:

Definitions:

A dihedral angle at an edge of a polyhedron is a dihedral angle whose edge contains the edge of the polyhedron, and the faces of the dihedral angle contain the faces of the polyhedron that intersect along the given edge of the polyhedron.
The angle between intersecting planes is the angle between straight lines drawn respectively in these planes perpendicular to their line of intersection through some of its points.
Two planes are said to be perpendicular if the angle between them is 90°.

Theorems:

Theorem 1(a sign of perpendicularity of planes). If one of the two planes passes through a line perpendicular to the other plane, then these planes are perpendicular.
Theorem 2. A line lying in one of two perpendicular planes and perpendicular to the line in which they intersect is perpendicular to the other plane.

Symmetry of figures

Definitions:

points M And M 1 are called symmetrical about a point O , if O is the midpoint of the segment MM 1 .
points M And M 1 are called symmetrical about a straight line l if straight l MM 1 and perpendicular to it.
points M And M 1 are called symmetrical about the plane α if the plane α passes through the middle of the segment MM 1 and is perpendicular to this segment.
Dot O(straight l, plane α ) is called center (axis, plane) of symmetry figure, if each point of the figure is symmetrical about a point O(straight l, plane α ) to some point of the same figure.
A convex polyhedron is called right, if all its faces are equal regular polygons and the same number of edges converge at each vertex.

Prism

Definitions:

Prism- a polyhedron, two faces of which are equal polygons lying in parallel planes, and the remaining faces are parallelograms that have common sides with these polygons.
Grounds - these are two faces that are equal polygons lying in parallel planes. On the drawing it is: ABCDE And KLMNP.
Side faces- all faces except bases. Each side face is necessarily a parallelogram. On the drawing it is: ABLK, BCML, CDNM, DEPN And EAKP.
Side surface- union of side faces.
Full surface- the union of the bases and the lateral surface.
Side ribs are the common sides of the side faces. On the drawing it is: AK, BL, CM, DN And EP.
Height- a segment connecting the bases of the prism and perpendicular to them. In the drawing, for example, KR.
Diagonal- a segment connecting two vertices of a prism that do not belong to the same face. In the drawing, for example, BP.
Diagonal plane is the plane passing through the lateral edge of the prism and the diagonal of the base. Other definition: diagonal plane- a plane passing through two side edges of the prism that do not belong to the same face.
Diagonal section- the intersection of the prism and the diagonal plane. A parallelogram is formed in the section, including, sometimes, its special cases - a rhombus, a rectangle, a square. In the drawing, for example, EBLP.
Perpendicular (orthogonal) section- the intersection of the prism and the plane perpendicular to its side edge.

Properties and formulas for a prism:

The bases of the prism are equal polygons.
The side faces of the prism are parallelograms.
The side edges of the prism are parallel and equal.
Prism Volume equal to the product of its height and the area of the base:

where: S base - the area of \u200b\u200bthe base (in the drawing, for example, ABCDE), h- height (in the drawing it is MN).

Total surface area of the prism is equal to the sum of the area of its lateral surface and twice the area of the base:

The perpendicular section is perpendicular to all side edges of the prism (in the drawing below, the perpendicular section is A 2 B 2 C 2 D 2 E 2).
The angles of a perpendicular section are the linear angles of the dihedral angles at the corresponding side edges.
A perpendicular (orthogonal) section is perpendicular to all side faces.
Volume of an inclined prism is equal to the product of the area of the perpendicular section and the length of the side rib:

where: S sec - the area of the perpendicular section, l- the length of the side rib (in the drawing below, for example, AA 1 or BB 1 and so on).

Lateral surface area of an arbitrary prism is equal to the product of the perimeter of the perpendicular section and the length of the side edge:

where: P sec - the perimeter of a perpendicular section, l is the length of the lateral edge.

Types of prisms in stereometry:

If the side edges are not perpendicular to the base, then such a prism is called oblique(pictured above). The bases of such a prism, as usual, are located in parallel planes, the side edges are not perpendicular to these planes, but parallel to each other. The side faces are parallelograms.
- a prism in which all lateral edges are perpendicular to the base. In a right prism, the side edges are the heights. The side faces of a straight prism are rectangles. And the area and perimeter of the base are equal, respectively, to the area and perimeter of the perpendicular section (for a straight prism, generally speaking, the entire perpendicular section is the same figure as the base). Therefore, the area of the lateral surface of a straight prism is equal to the product of the perimeter of the base and the length of the lateral edge (or, in this case, the height of the prism):

where: P base - the perimeter of the base of a straight prism, l- the length of the lateral edge, equal in a straight prism to the height ( h). The volume of a straight prism is found by the general formula: V = S main ∙ h = S main ∙ l.

Correct prism- a prism at the base of which lies a regular polygon (that is, one in which all sides and all angles are equal to each other), and the side edges are perpendicular to the planes of the base. Examples of correct prisms:

Properties of the correct prism:

The bases of a regular prism are regular polygons.
The side faces of a regular prism are equal rectangles.
The side edges of a regular prism are equal to each other.
The correct prism is straight.

Definition: Parallelepiped - It is a prism whose bases are parallelograms. In this definition, the key word is "prism". Thus, a parallelepiped is a special case of a prism, which differs from the general case only in that its base is not an arbitrary polygon, but a parallelogram. Therefore, all the above properties, formulas and definitions regarding the prism remain relevant for the parallelepiped. However, there are several additional properties characteristic of the parallelepiped.

Other properties and definitions:

Two faces of a parallelepiped that do not have a common edge are called opposite, and having a common edge - related.
Two vertices of a parallelepiped that do not belong to the same face are called opposite.
A line segment connecting opposite vertices is called diagonal parallelepiped.
The parallelepiped has six faces and all of them are parallelograms.
The opposite faces of the parallelepiped are equal and parallel in pairs.
The parallelepiped has four diagonals; they all intersect at one point, and each of them is bisected by that point.
If the four side faces of a parallelepiped are rectangles (and the bases are arbitrary parallelograms), then it is called direct(in this case, as with a straight prism, all side edges are perpendicular to the bases). All properties and formulas for a straight prism are relevant for a right parallelepiped.
The parallelepiped is called oblique if not all of its side faces are rectangles.
Volume of a straight or oblique box is calculated by the general formula for the volume of a prism, i.e. is equal to the product of the area of the base of the parallelepiped and its height ( V = S main ∙ h).
A right parallelepiped, in which all six faces are rectangles (i.e., in addition to the side faces, the bases are also rectangles), is called rectangular. For a cuboid, all the properties of a cuboid are relevant, as well as:
- d and his ribs a, b, c related by the ratio:

d 2 = a 2 + b 2 + c 2 .

- From the general formula for the volume of a prism, the following formula can be obtained for volume of a cuboid:

A rectangular parallelepiped all of whose faces are equal squares is called cube. Among other things, the cube is a regular quadrangular prism, and in general a regular polyhedron. For a cube, all the properties of a rectangular parallelepiped and the properties of regular prisms are valid, as well as:
- Absolutely all edges of a cube are equal to each other.
- cube diagonal d and the length of its edge a related by the ratio:

From the formula for the volume of a rectangular parallelepiped, one can obtain the following formula for cube volume:

Pyramid

Definitions:

Pyramid is a polyhedron whose base is a polygon and the remaining faces are triangles having a common vertex. According to the number of corners of the base, pyramids are triangular, quadrangular, and so on. The figure shows examples: quadrangular and hexagonal pyramids.

Base is a polygon to which the vertex of the pyramid does not belong. In the drawing, the base is BCDE.
Faces other than the base are called lateral. On the drawing it is: ABC, ACD, ADE And AEB.
The common vertex of the side faces is called top of the pyramid(precisely the top of the entire pyramid, and not just a top, like all other peaks). On the drawing it A.
The edges that connect the top of the pyramid with the top of the base are called lateral. On the drawing it is: AB, AC, AD And AE.
Denoting the pyramid, first they call its top, and then - the tops of the base. For a pyramid from a drawing, the designation will be as follows: ABCDE.

Heightpyramids called the perpendicular drawn from the top of the pyramid to its base. The length of this perpendicular is denoted by the letter H. In the drawing, the height is AG. Note: only if the pyramid is a regular quadrangular pyramid (as in the drawing), the height of the pyramid falls on the diagonal of the base. In other cases, this is not the case. In the general case, for an arbitrary pyramid, the point of intersection of the height and base can be anywhere.
Apothem - side edge height correct pyramid drawn from its top. In the drawing, for example, AF.
Diagonal section of a pyramid- section of the pyramid, passing through the top of the pyramid and the diagonal of the base. In the drawing, for example, ACE.

Another stereometric drawing with symbols for better memorization(in the figure, the correct triangular pyramid):

If all side edges ( SA, SB, SC, SD in the drawing below) the pyramids are equal, then:

A circle can be described near the base of the pyramid, and the top of the pyramid is projected into its center (point O). In other words, height (line SO), lowered from the top of such a pyramid to the base ( ABCD), falls into the center of the circumscribed circle around the base, i.e. at the point of intersection of the perpendicular midpoints of the base.
The side ribs form equal angles with the base plane (in the drawing below, these are the angles SAO, SBO, SCO, SDO).

Important: The opposite is also true, that is, if the side edges form equal angles with the base plane, or if a circle can be described near the base of the pyramid, and the top of the pyramid is projected into its center, then all the side edges of the pyramid are equal.

If the side faces are inclined to the base plane at one angle (the corners DMN, DKN, DLN in the drawing below are equal), then:

A circle can be inscribed at the base of the pyramid, and the top of the pyramid is projected into its center (point N). In other words, height (line DN), lowered from the top of such a pyramid to the base, falls into the center of the circle inscribed in the base, i.e. to the point of intersection of the bisectors of the base.
The heights of the side faces (apothems) are equal. On the drawing below DK, DL, DM- equal apothems.
The lateral surface area of such a pyramid equal to half the product of the perimeter of the base and the height of the side face (apothem).

where: P- perimeter of the base, a- apothem length.

Important: The opposite is also true, that is, if a circle can be inscribed in the base of the pyramid, and the top of the pyramid is projected into its center, then all side faces are inclined to the base plane at the same angle and the heights of the side faces (apothem) are equal.

Correct pyramid

Definition: The pyramid is called correct, if its base is a regular polygon, and the vertex is projected into the center of the base. Then it has the following properties:

All side edges of a regular pyramid are equal.
All side faces of a regular pyramid are inclined to the plane of the base at one angle.

Important note: As you can see, regular pyramids are one of those pyramids that include the properties described just above. Indeed, if the base of a regular pyramid is a regular polygon, then the center of its inscribed and circumscribed circles coincide, and the top of a regular pyramid is projected precisely into this center (by definition). However, it is important to understand that not only correct pyramids can have the properties mentioned above.

In a regular pyramid, all side faces are equal isosceles triangles.
In any regular pyramid, you can both inscribe a sphere and describe a sphere around it.
The area of the lateral surface of a regular pyramid is equal to half the product of the perimeter of the base and the apothem.

Formulas for volume and area of a pyramid

Theorem(on the volume of pyramids having equal heights and equal areas of bases). Two pyramids that have equal heights and equal areas of bases have equal volumes (of course, you probably already know the formula for the volume of a pyramid, well, or you see it a few lines below, and this statement seems obvious to you, but in fact, judging "on eye", then this theorem is not so obvious (see the figure below). By the way, this also applies to other polyhedra and geometric shapes: their appearance is deceptive, therefore, indeed, in mathematics you need to trust only formulas and correct calculations).

pyramid volume can be calculated using the formula:

where: S base is the area of the base of the pyramid, h is the height of the pyramid.

Lateral surface of the pyramid is equal to the sum of the areas of the side faces. For the area of the lateral surface of the pyramid, one can formally write the following stereometric formula:

where: S side - side surface area, S 1 , S 2 , S 3 - areas of side faces.

Full surface of the pyramid equal to the sum of the area of the lateral surface and the area of the base:

Definitions:

- the simplest polyhedron, the faces of which are four triangles, in other words, a triangular pyramid. For a tetrahedron, any of its faces can serve as a base. In total, a tetrahedron has 4 faces, 4 vertices and 6 edges.
The tetrahedron is called right if all its faces are equilateral triangles. For a regular tetrahedron:
1. All edges of a regular tetrahedron are equal.
2. All faces of a regular tetrahedron are equal to each other.
3. Perimeters, areas, heights and all other elements of all faces are respectively equal to each other.

The drawing shows a regular tetrahedron, while the triangles ABC, ADC, CBD, bad are equal. From the general formulas for the volume and areas of the pyramid, as well as knowledge from planimetry, it is not difficult to obtain formulas for volume and area of a regular tetrahedron(but- rib length):

Definition: When solving problems in stereometry, the pyramid is called rectangular, if one of the side edges of the pyramid is perpendicular to the base. In this case, this edge is the height of the pyramid. Below are examples of triangular and pentagonal rectangular pyramids. The picture on the left SA is an edge that is also a height.

Truncated pyramid

Definitions and properties:

truncated pyramid is called a polyhedron enclosed between the base of the pyramid and a cutting plane parallel to its base.
The figure obtained at the intersection of the cutting plane and the original pyramid is also called basis truncated pyramid. So, the truncated pyramid in the drawing has two bases: ABC And A 1 B 1 C 1 .
The side faces of the truncated pyramid are trapezoids. In the drawing, for example, AA 1 B1B.
Lateral edges of a truncated pyramid are called parts of the edges of the original pyramid, enclosed between the bases. In the drawing, for example, AA 1 .
The height of a truncated pyramid is a perpendicular (or the length of this perpendicular) drawn from some point in the plane of one base to the plane of the other base.
The truncated pyramid is called correct, if it is a polyhedron that is cut off by a plane parallel to the base correct pyramids.
The bases of a regular truncated pyramid are regular polygons.
The side faces of a regular truncated pyramid are isosceles trapezoids.
apothem a regular truncated pyramid is called the height of its lateral face.
The area of the lateral surface of a truncated pyramid is the sum of the areas of all its lateral faces.

Formulas for a truncated pyramid

The volume of the truncated pyramid is:

where: S 1 and S 2 - base areas, h is the height of the truncated pyramid. However, in practice, it is more convenient to search for the volume of a truncated pyramid as follows: you can complete the truncated pyramid to the pyramid, extending the side edges to the intersection. Then the volume of the truncated pyramid can be found as the difference between the volumes of the entire pyramid and the completed part. The lateral surface area can also be found as the difference between the lateral surface areas of the entire pyramid and the completed part. Lateral surface area of a regular truncated pyramid is equal to the half product of the sum of the perimeters of its bases and the apothem:

where: P 1 and P 2 - base perimeters correct truncated pyramid, but- apothem length. The total surface area of any truncated pyramid is obviously found as the sum of the areas of the bases and the lateral surface:

Pyramid and ball (sphere)

Theorem: Around the pyramid describe the scope when at the base of the pyramid lies an inscribed polygon (i.e., a polygon around which a sphere can be described). This condition is necessary and sufficient. The center of the sphere will be the point of intersection of the planes passing through the midpoints of the edges of the pyramid perpendicular to them.

Remark: It follows from this theorem that a sphere can be described both around any triangular and around any regular pyramid. However, the list of pyramids around which you can describe the sphere is not limited to these types of pyramids. In the drawing on the right, at a height SH need to pick a point ABOUT, equidistant from all vertices of the pyramid: SO = OB = OS = OD = OA. Then the point ABOUT is the center of the circumscribed sphere.

Theorem: You can in the pyramid inscribe a sphere when the bisector planes of the internal dihedral angles of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.

Comment: You obviously did not understand what you read the line above. However, it is important to remember that any regular pyramid is one in which a sphere can be inscribed. At the same time, the list of pyramids into which a sphere can be inscribed is not exhausted by the correct ones.

Definition: Bisector plane divides the dihedral angle in half, and each point of the bisector plane is equidistant from the faces forming the dihedral angle. The figure on the right plane γ is the bisector plane of the dihedral angle formed by the planes α And β .

The stereometric drawing below shows a ball inscribed in a pyramid (or a pyramid described near the ball), while the point ABOUT is the center of the inscribed sphere. This point ABOUT equidistant from all faces of the ball, for example:

OM = OO 1

pyramid and cone

In stereometry a cone is called inscribed in a pyramid, if their vertices coincide, and its base is inscribed in the base of the pyramid. Moreover, it is possible to inscribe a cone in a pyramid only when the apothems of the pyramid are equal to each other (a necessary and sufficient condition).

The cone is called inscribed near the pyramid when their vertices coincide, and its base is described near the base of the pyramid. Moreover, it is possible to describe a cone near the pyramid only when all the side edges of the pyramid are equal to each other (a necessary and sufficient condition).

Important property:

pyramid and cylinder

The cylinder is said to be inscribed in a pyramid, if one of its bases coincides with the circle of a plane inscribed in the section of the pyramid, parallel to the base, and the other base belongs to the base of the pyramid.

The cylinder is said to be circumscribed near the pyramid, if the top of the pyramid belongs to one of its bases, and its other base is described near the base of the pyramid. Moreover, it is possible to describe a cylinder near the pyramid only when there is an inscribed polygon at the base of the pyramid (a necessary and sufficient condition).

Sphere and ball

Definitions:

Sphere- a closed surface, the locus of points in space equidistant from a given point, called the center of the sphere. A sphere is also a body of revolution formed by the rotation of a semicircle around its diameter. sphere radius is called a segment connecting the center of the sphere with any point of the sphere.
Chordoy sphere is a segment that connects two points on the sphere.
diameter sphere is called a chord passing through its center. The center of a sphere divides any of its diameters into two equal segments. Any sphere diameter with a radius R is 2 R.
Ball- geometric body; the set of all points in space that are at a distance not greater than a specified distance from a certain center. This distance is called ball radius. A ball is formed by rotating a semicircle around its fixed diameter. Note: the surface (or boundary) of a sphere is called a sphere. It is possible to give the following definition of a ball: a geometric body is called a ball, consisting of a sphere and a part of the space bounded by this sphere.
Radius, chord And diameter ball are called the radius, chord and diameter of the sphere, which is the boundary of this ball.
The difference between a ball and a sphere is similar to the difference between a circle and a circle. A circle is a line, and a circle is also all points inside this line. A sphere is a shell, and a ball is also all the points inside this shell.
The plane passing through the center of the sphere (ball) is called diametral plane.
A section of a sphere (ball) by a diametral plane is called great circle (big circle).

Theorems:

Theorem 1(on the section of a sphere by a plane). The section of a sphere by a plane is a circle. Note that the assertion of the theorem remains true even if the plane passes through the center of the sphere.
Theorem 2(on the section of a sphere by a plane). The section of a ball by a plane is a circle, and the base of the perpendicular drawn from the center of the ball to the section plane is the center of the circle obtained in the section.

The largest circle, from among those that can be obtained in a section of a given ball by a plane, lies in a section passing through the center of the ball ABOUT. It's called the big circle. Its radius is equal to the radius of the sphere. Any two great circles intersect in the diameter of the ball AB. This diameter is also the diameter of the intersecting great circles. Through two points of a spherical surface located at the ends of the same diameter (in Fig. A And B), you can draw an infinite number of great circles. For example, an infinite number of meridians can be drawn through the poles of the Earth.

Definitions:

Tangent plane to sphere is called a plane that has only one common point with the sphere, and their common point is called the point of contact of the plane and the sphere.
Tangent plane to the ball is called the tangent plane to the sphere, which is the boundary of this ball.
Any line lying in the tangent plane of the sphere (ball) and passing through the point of contact is called tangent to a straight line to a sphere (ball). By definition, the tangent plane has only one common point with the sphere, therefore, the tangent line also has only one common point with the sphere - the point of contact.

Theorems:

Theorem 1(sign of the tangent plane to the sphere). A plane perpendicular to the radius of the sphere and passing through its end lying on the sphere touches the sphere.
Theorem 2(on the property of the tangent plane to the sphere). The tangent plane to the sphere is perpendicular to the radius drawn to the point of contact.

Polyhedra and the sphere

Definition: In stereometry, a polyhedron (such as a pyramid or prism) is called inscribed in the scope if all its vertices lie on a sphere. In this case, the sphere is called circumscribed near a polyhedron (pyramids, prisms). Similarly: the polyhedron is called inscribed in a ball if all its vertices lie on the boundary of this ball. In this case, the ball is said to be inscribed near the polyhedron.

Important property: The center of the sphere circumscribed about the polyhedron is at a distance equal to the radius R spheres, from each vertex of the polyhedron. Here are examples of polyhedra inscribed in the sphere:

Definition: The polyhedron is called described about the sphere (ball), if the sphere (ball) touches all polyhedron faces. In this case, the sphere and the ball are called inscribed in the polyhedron.

Important: The center of a sphere inscribed in a polyhedron is at a distance equal to the radius r spheres, from each of the planes containing the faces of the polyhedron. Here are examples of polyhedra described near the sphere:

Volume and surface area of a sphere

Theorems:

Theorem 1(about the area of the sphere). The area of a sphere is:

where: R is the radius of the sphere.

Theorem 2(about the volume of the ball). The volume of a sphere with a radius R calculated by the formula:

Ball segment, layer, sector

In stereometry ball segment called the part of the ball cut off by the cutting plane. In this case, the ratio between the height, the radius of the base of the segment and the radius of the ball:

where: h− segment height, r− segment base radius, R− ball radius. The area of the base of the spherical segment:

The area of the outer surface of the spherical segment:

Full surface area of the ball segment:

Ball segment volume:

In stereometry spherical layer The part of a sphere enclosed between two parallel planes is called. The area of the outer surface of the spherical layer:

where: h is the height of the spherical layer, R− ball radius. Full surface area of the spherical layer:

where: h is the height of the spherical layer, R− ball radius, r 1 , r 2 are the radii of the bases of the spherical layer, S 1 , S 2 are the areas of these bases. The volume of a spherical layer is most simply found as the difference between the volumes of two spherical segments.

In stereometry ball sector called the part of the ball, consisting of a spherical segment and a cone with a vertex in the center of the ball and a base coinciding with the base of the spherical segment. Here it is assumed that the ball segment is less than half the ball. Full surface area of the spherical sector:

where: h is the height of the corresponding spherical segment, r is the radius of the base of the spherical segment (or cone), R− ball radius. The volume of the spherical sector is calculated by the formula:

Definitions:

In some plane, consider a circle with center O and radius R. Through each point of the circle we draw a line perpendicular to the plane of the circle. Cylindrical surface the figure formed by these lines is called, and the lines themselves are called forming a cylindrical surface. All generators of the cylindrical surface are parallel to each other, since they are perpendicular to the plane of the circle.

Straight circular cylinder or simply cylinder called a geometric body bounded by a cylindrical surface and two parallel planes that are perpendicular to the generators of the cylindrical surface. Informally, you can think of a cylinder as a straight prism with a circle at the base. This will help to easily understand and, if necessary, derive formulas for the volume and area of the lateral surface of the cylinder.
Side surface of the cylinder the part of the cylindrical surface located between the cutting planes that are perpendicular to its generatrix is called, and the parts (circles) cut off by the cylindrical surface on parallel planes are called cylinder bases. The bases of the cylinder are two equal circles.
Cylinder generatrix called a segment (or the length of this segment) of the generatrix of a cylindrical surface, located between parallel planes in which the bases of the cylinder lie. All generators of the cylinder are parallel and equal to each other, and also perpendicular to the bases.
Cylinder axis called a segment connecting the centers of the circles that are the bases of the cylinder.
cylinder height called a perpendicular (or the length of this perpendicular), drawn from some point in the plane of one base of the cylinder to the plane of the other base. In a cylinder, the height is equal to the generatrix.
Cylinder radius is called the radius of its bases.
The cylinder is called equilateral if its height is equal to the diameter of the base.
A cylinder can be obtained by rotating a rectangle around one of its sides by 360°.
If the cutting plane is parallel to the axis of the cylinder, then the section of the cylinder is a rectangle, two sides of which are generators, and the other two are the chords of the bases of the cylinder.
Axial section A cylinder is a section of a cylinder by a plane passing through its axis. The axial section of a cylinder is a rectangle, two sides of which are generators of the cylinder, and the other two are the diameters of its bases.
If the cutting plane is perpendicular to the axis of the cylinder, then a circle is formed in the section equal to the bases. In the drawing below: on the left - axial section; in the center - a section parallel to the axis of the cylinder; on the right - a section parallel to the base of the cylinder.

Cylinder and prism

A prism is said to be inscribed in a cylinder if its bases are inscribed in the bases of the cylinder. In this case, the cylinder is said to be circumscribed about a prism. The height of the prism and the height of the cylinder in this case will be equal. All side edges of the prism will belong to the side surface of the cylinder and coincide with its generators. Since by a cylinder we mean only a straight cylinder, only a straight prism can also be inscribed in such a cylinder. Examples:

A prism is said to be circumscribed about a cylinder, if its bases are described near the bases of the cylinder. In this case, the cylinder is said to be inscribed in a prism. The height of the prism and the height of the cylinder in this case will also be equal. All side edges of the prism will be parallel to the generatrix of the cylinder. Since by a cylinder we mean only a straight cylinder, such a cylinder can only be inscribed in a straight prism. Examples:

Cylinder and sphere

A sphere (ball) is called inscribed in a cylinder if it touches the bases of the cylinder and each of its generators. In this case, the cylinder is called circumscribed about a sphere (ball). A sphere can be inscribed in a cylinder only if it is an equilateral cylinder, i.e. its base diameter and height are equal. The center of the inscribed sphere will be the middle of the axis of the cylinder, and the radius of this sphere will coincide with the radius of the cylinder. Example:

The cylinder is said to be inscribed in a sphere, if the circles of the bases of the cylinder are sections of the sphere. A cylinder is said to be inscribed in a sphere if the bases of the cylinder are sections of the sphere. In this case, the ball (sphere) is called inscribed near the cylinder. A sphere can be described around any cylinder. The center of the described sphere will also be the middle of the axis of the cylinder. Example:

Based on the Pythagorean theorem, it is easy to prove the following formula relating the radius of the circumscribed sphere ( R), cylinder height ( h) and radius of the cylinder ( r):

Volume and area of the lateral and full surfaces of the cylinder

Theorem 1(about the area of the lateral surface of a cylinder): The area of the lateral surface of a cylinder is equal to the product of the circumference of its base and the height:

where: R is the radius of the base of the cylinder, h- his high. This formula is easily derived (or proven) based on the formula for the lateral surface area of a straight prism.

Full surface area of the cylinder, as usual in stereometry, is the sum of the areas of the lateral surface and the two bases. The area of each base of the cylinder (i.e. just the area of a circle) is calculated by the formula:

Therefore, the total surface area of the cylinder S full cylinder is calculated by the formula:

Theorem 2(about the volume of a cylinder): The volume of a cylinder is equal to the product of the area of the base and the height:

where: R And h are the radius and height of the cylinder, respectively. This formula is also easily derived (proved) based on the formula for the volume of a prism.

Theorem 3(Archimedes): The volume of a sphere is one and a half times less than the volume of the cylinder described around it, and the surface area of such a ball is one and a half times less than the total surface area of the same cylinder:

Cone

Definitions:

A cone (more precisely, a circular cone) called the body, which consists of a circle (called cone base), a point not lying in the plane of this circle (called the top of the cone) and all possible segments connecting the top of the cone with the points of the base. Informally, you can perceive the cone as a regular pyramid, which has a circle at the base. This will help to easily understand, and if necessary, derive formulas for the volume and area of the lateral surface of the cone.

The segments (or their lengths) connecting the top of the cone with the points of the circle of the base are called forming a cone. All generators of a right circular cone are equal to each other.
The surface of a cone consists of the base of the cone (the circle) and the side surface (composed of all possible generators).
The union of the generators of a cone is called generatrix (or side) surface of the cone. The generatrix of a cone is a conical surface.
The cone is called direct if the line connecting the vertex of the cone with the center of the base is perpendicular to the plane of the base. In what follows, we will consider only the right cone, calling it simply the cone for brevity.
Visually, a straight circular cone can be imagined as a body obtained by rotating a right triangle around its leg as an axis. In this case, the lateral surface of the cone is formed by the rotation of the hypotenuse, and the base is formed by the rotation of the leg, which is not an axis.
cone radius called the radius of its base.
cone height called the perpendicular (or its length), lowered from its top to the plane of the base. For a right cone, the base of the height coincides with the center of the base. The axis of a right circular cone is a straight line containing its height, i.e. a straight line passing through the center of the base and the top.
If the cutting plane passes through the axis of the cone, then the section is an isosceles triangle, the base of which is the diameter of the base of the cone, and the sides are the generatrix of the cone. Such a cut is called axial.

If the cutting plane passes through the inner point of the height of the cone and is perpendicular to it, then the section of the cone is a circle, the center of which is the point of intersection of the height and this plane.
Height ( h), radius ( R) and the length of the generatrix ( l) of a right circular cone satisfy the obvious relation:

Volume and area of the lateral and full surfaces of the cone

Theorem 1(on the area of the lateral surface of the cone). The area of the lateral surface of the cone is equal to the product of half the circumference of the base and the generatrix:

where: R is the radius of the base of the cone, l is the length of the generatrix of the cone. This formula is easily derived (or proven) based on the formula for the lateral surface area of a regular pyramid.

Full surface area of the cone is the sum of the lateral surface area and the base area. The area of the base of the cone (i.e. just the area of the circle) is: S base = πR 2. Therefore, the total surface area of the cone S full cone is calculated by the formula:

Theorem 2(on the volume of a cone). The volume of a cone is equal to one third of the base area multiplied by the height:

where: R is the radius of the base of the cone, h- his high. This formula is also easily derived (proved) based on the formula for the volume of the pyramid.

Definitions:

A plane parallel to the base of a cone and intersecting the cone cuts off a smaller cone from it. The rest is called truncated cone.

The base of the original cone and the circle obtained in the section of this cone by a plane are called grounds, and the segment connecting their centers - truncated cone height.
The straight line passing through the height of the truncated cone (i.e. through the centers of its bases) is its axis.
The part of the lateral surface of the cone that bounds the truncated cone is called its side surface, and the segments of the generatrix of the cone located between the bases of the truncated cone are called its generating.
All generators of a truncated cone are equal to each other.
A truncated cone can be obtained by rotating a rectangular trapezoid through 360° around its side perpendicular to the bases.

Formulas for a truncated cone:

The volume of a truncated cone is equal to the difference between the volumes of a full cone and a cone cut off by a plane parallel to the base of the cone. The volume of a truncated cone is calculated by the formula:

where: S 1 = π r 1 2 and S 2 = π r 2 2 - areas of bases, h is the height of the truncated cone, r 1 and r 2 - radii of the upper and lower bases of the truncated cone. However, in practice, it is still more convenient to look for the volume of a truncated cone as the difference between the volumes of the original cone and the cut off part. The lateral surface area of a truncated cone can also be found as the difference between the lateral surface areas of the original cone and the cut off part.

Indeed, the area of the lateral surface of a truncated cone is equal to the difference between the areas of the lateral surfaces of a full cone and a cone cut off by a plane parallel to the base of the cone. Lateral surface area of a truncated cone calculated by the formula:

where: P 1 = 2π r 1 and P 2 = 2π r 2 - perimeters of the bases of a truncated cone, l- the length of the generatrix. Total surface area of a truncated cone, obviously, is found as the sum of the areas of the bases and the lateral surface:

Please note that the formulas for the volume and area of the lateral surface of a truncated cone are derived from formulas for similar characteristics of a regular truncated pyramid.

Cone and sphere

A cone is said to be inscribed in a sphere(ball), if its vertex belongs to the sphere (the boundary of the ball), and the circumference of the base (the base itself) is a section of the sphere (ball). In this case, the sphere (ball) is called circumscribed near the cone. A sphere can always be described around a right circular cone. The center of the circumscribed sphere will lie on a straight line containing the height of the cone, and the radius of this sphere will be equal to the radius of the circle circumscribed about the axial section of the cone (this section is an isosceles triangle). Examples:

A sphere (ball) is called inscribed in a cone, if the sphere (ball) touches the base of the cone and each of its generators. In this case, the cone is called inscribed near the sphere (ball). A sphere can always be inscribed in a right circular cone. Its center will lie at the height of the cone, and the radius of the inscribed sphere will be equal to the radius of the circle inscribed in the axial section of the cone (this section is an isosceles triangle). Examples:

Cone and pyramid

A cone is called inscribed in a pyramid (a pyramid is described near a cone) if the base of the cone is inscribed in the base of the pyramid, and the vertices of the cone and pyramid coincide.
A pyramid is called inscribed in a cone (a cone is described near a pyramid) if its base is inscribed in the base of the cone, and the side edges are generators of the cone.
The heights of such cones and pyramids are equal to each other.

Note: More details about how in solid geometry a cone fits into a pyramid or is described near a pyramid have already been discussed in

How to successfully prepare for the CT in Physics and Mathematics?

In order to successfully prepare for the CT in Physics and Mathematics, among other things, three critical conditions must be met:

Study all the topics and complete all the tests and tasks given in the study materials on this site. To do this, you need nothing at all, namely: to devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that the CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to quickly and without failures solve a large number of problems on various topics and varying complexity. The latter can only be learned by solving thousands of problems.
Learn all formulas and laws in physics, and formulas and methods in mathematics. In fact, it is also very simple to do this, there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty, solve most of the digital transformation at the right time. After that, you will only have to think about the most difficult tasks.
Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to solve both options. Again, on the DT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, it is also necessary to be able to properly plan time, distribute forces, and most importantly fill out the answer form correctly, without confusing either the numbers of answers and problems, or your own name. Also, during the RT, it is important to get used to the style of posing questions in tasks, which may seem very unusual to an unprepared person on the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result on the CT, the maximum of what you are capable of.

Found an error?

If you, as it seems to you, found an error in the training materials, then please write about it by mail. You can also write about the error on the social network (). In the letter, indicate the subject (physics or mathematics), the name or number of the topic or test, the number of the task, or the place in the text (page) where, in your opinion, there is an error. Also describe what the alleged error is. Your letter will not go unnoticed, the error will either be corrected, or you will be explained why it is not a mistake.

\((\color(red)(\textbf(Fact 1. About parallel lines)))\)
\(\bullet\) Two lines in space are parallel if they lie in the same plane and do not intersect.
\(\bullet\) There is only one plane passing through two parallel lines.
\(\bullet\) If one of two parallel lines intersects a plane, then the other line also intersects this plane.
\(\bullet\) If the line \(a\) is parallel to the line \(b\) , which in turn is parallel to the line \(c\) , then \(a\parallel c\) .
\(\bullet\) Let the plane \(\alpha\) and \(\beta\) intersect along the line \(a\) , the planes \(\beta\) and \(\pi\) intersect along the line \(b \) , the planes \(\pi\) and \(\alpha\) intersect along the line \(p\) . Then if \(a\parallel b\) , then \(p\parallel a\) (or \(p\parallel b\) ):

\((\color(red)(\textbf(Fact 2. About the parallelism of a line and a plane)))\)
\(\bullet\) There are three types of mutual arrangement of a line and a plane:
1. the line has two common points with the plane (that is, it lies in the plane);
2. the line has exactly one common point with the plane (that is, it intersects the plane);
3. the line has no common points with the plane (that is, it is parallel to the plane).
\(\bullet\) If a line \(a\) , not lying in the plane \(\pi\) , is parallel to some line \(p\) , lying in the plane \(\pi\) , then it is parallel to the given plane.

\(\bullet\) Let the line \(p\) be parallel to the plane \(\mu\) . If the plane \(\pi\) passes through the line \(p\) and intersects the plane \(\mu\) , then the line of intersection of the planes \(\pi\) and \(\mu\) is the line \(m\) - parallel to the line \(p\) .

\((\color(red)(\textbf(Fact 3. About parallel planes)))\)
\(\bullet\) If two planes have no common points, then they are called parallel planes.
\(\bullet\) If two intersecting lines from one plane are respectively parallel to two intersecting lines from another plane, then such planes will be parallel.

\(\bullet\) If two parallel planes \(\alpha\) and \(\beta\) are intersected by a third plane \(\gamma\) , then the intersection lines of the planes are also parallel: \[\alpha\parallel \beta, \ \alpha\cap \gamma=a, \ \beta\cap\gamma=b \Longrightarrow a\parallel b\]

\(\bullet\) Segments of parallel lines enclosed between parallel planes are equal to: \[\alpha\parallel \beta, \ a\parallel b \Longrightarrow A_1B_1=A_2B_2\]

\((\color(red)(\textbf(Fact 4. About intersecting lines)))\)
\(\bullet\) Two straight lines in space are called intersecting if they do not lie in the same plane.
\(\bullet\) Sign:
Let the line \(l\) lie in the plane \(\lambda\) . If the line \(s\) intersects the plane \(\lambda\) at a point \(S\) not lying on the line \(l\) , then the lines \(l\) and \(s\) intersect.

\(\bullet\) algorithm for finding the angle between skew lines \(a\) and \(b\):

Step 2. In the plane \(\pi\) find the angle between the lines \(a\) and \(p\) (\(p\parallel b\) ). The angle between them will be equal to the angle between the skew lines \(a\) and \(b\) .

\((\color(red)(\textbf(Fact 5. About the perpendicularity of a line and a plane)))\)
\(\bullet\) A line is said to be perpendicular to a plane if it is perpendicular to any line in that plane.
\(\bullet\) If two lines are perpendicular to a plane, then they are parallel.
\(\bullet\) Sign: if a line is perpendicular to two intersecting lines lying in a given plane, then it is perpendicular to this plane.

\((\color(red)(\textbf(Fact 6. About distances)))\)
\(\bullet\) In order to find the distance between parallel lines, you need to drop a perpendicular from any point of one line to another line. The length of the perpendicular is the distance between these lines.
\(\bullet\) In order to find the distance between a plane and a line parallel to it, you need to drop a perpendicular to this plane from any point on the line. The length of the perpendicular is the distance between this line and the plane.
\(\bullet\) In order to find the distance between parallel planes, you need to lower the perpendicular to the other plane from any point of one plane. The length of this perpendicular is the distance between the parallel planes.
\(\bullet\) algorithm for finding the distance between skew lines \(a\) and \(b\):
Step 1. Through one of the two intersecting lines \(a\) draw a plane \(\pi\) parallel to the other line \(b\) . How to do it: draw the plane \(\beta\) through the line \(b\) so that it intersects the line \(a\) at the point \(P\) ; draw a line through the point \(P\) \(p\parallel b\) ; then the plane passing through \(a\) and \(p\) is the plane \(\pi\) .
Step 2. Find the distance from any point of the line \(b\) to the plane \(\pi\) . This distance is the distance between the skew lines \(a\) and \(b\) .

\((\color(red)(\textbf(Fact 7. About the Three Perpendicular Theorem (TTP))))\)
\(\bullet\) Let \(AH\) be the perpendicular to the plane \(\beta\) . Let \(AB, BH\) be an oblique and its projection onto the plane \(\beta\) . Then the line \(x\) in the plane \(\beta\) will be perpendicular to the oblique if and only if it is perpendicular to the projection: \[\begin(aligned) &1. AH\perp \beta, \AB\perp x\quad \Rightarrow\quad BH\perp x\\ &2. AH\perp \beta, \ BH\perp x\quad\Rightarrow\quad AB\perp x\end(aligned)\]

Note that the line \(x\) need not pass through the point \(B\) . If it does not pass through the point \(B\) , then a line \(x"\) is constructed passing through the point \(B\) and parallel to \(x\) . If, for example, \(x"\perp BH\ ) , then so is \(x\perp BH\) .

\((\color(red)(\textbf(Fact 8. About the angle between a line and a plane, as well as the angle between planes)))\)
\(\bullet\) The angle between an oblique line and a plane is the angle between this line and its projection onto the given plane. Thus, this angle takes values from the interval \((0^\circ;90^\circ)\) .
If the line lies in a plane, then the angle between them is considered equal to \(0^\circ\) . If the line is perpendicular to the plane, then, based on the definition, the angle between them is \(90^\circ\) .
\(\bullet\) To find the angle between an oblique line and a plane, it is necessary to mark some point \(A\) on this line and draw a perpendicular \(AH\) to the plane. If \(B\) is the point of intersection of the line with the plane, then \(\angle ABH\) is the desired angle.

\(\bullet\) In order to find the angle between the planes \(\alpha\) and \(\beta\) , you can use the following algorithm:
Mark an arbitrary point \(A\) in the plane \(\alpha\) .
Draw \(AH\perp h\) , where \(h\) is the line of intersection of the planes.
Draw \(AB\) perpendicular to the plane \(\beta\) .
Then \(AB\) is a perpendicular to the plane \(\beta\) , \(AH\) is oblique, hence \(HB\) is a projection. Then by TTP \(HB\perp h\) .
Therefore, \(\angle AHB\) is the linear angle of the dihedral angle between the planes. The degree measure of this angle is the degree measure of the angle between the planes.

Note that we got a right triangle \(\triangle AHB\) (\(\angle B=90^\circ\) ). As a rule, it is convenient to find \(\angle AHB\) from it.

\((\color(red)(\textbf(Fact 9. About perpendicularity of planes)))\)
\(\bullet\) Sign: if a plane passes through a line perpendicular to another plane, then it is perpendicular to this plane. \

\(\bullet\) Note that since an infinite number of planes can be drawn through the line \(a\), there are an infinite number of planes perpendicular to \(\beta\) (and passing through \(a\) ).

In order to adequately solve the exam in mathematics, first of all, it is necessary to study the theoretical material, which introduces numerous theorems, formulas, algorithms, etc. At first glance, it may seem that this is quite simple. However, finding a source in which the theory for the Unified State Examination in mathematics is presented in an easy and understandable way for students with any level of preparation is, in fact, a rather difficult task. School textbooks cannot always be kept at hand. And finding the basic formulas for the exam in mathematics can be difficult even on the Internet.

Why is it so important to study theory in mathematics, not only for those who take the exam?

Because it broadens your horizons. The study of theoretical material in mathematics is useful for anyone who wants to get answers to a wide range of questions related to the knowledge of the world. Everything in nature is ordered and has a clear logic. This is precisely what is reflected in science, through which it is possible to understand the world.
Because it develops the intellect. Studying reference materials for the exam in mathematics, as well as solving various problems, a person learns to think and reason logically, to formulate thoughts correctly and clearly. He develops the ability to analyze, generalize, draw conclusions.

We invite you to personally evaluate all the advantages of our approach to the systematization and presentation of educational materials.