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Formulas of stereometry with devils. Buy a diploma of higher education inexpensively |
Some definitions:
Axioms of stereometry:
Consequences from the axioms of stereometry:
Construction of sections in stereometryTo 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:
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:
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 .
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 stereometryDefinition: 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 a and b, or AB and CD are parallel, we write: Several theorems:
There are three cases of mutual arrangement of a straight line and a plane in stereometry:
Definition: Line and plane are called parallel if they do not have common points. If straight a parallel to the plane β , then they write: Theorems:
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:
Now we introduce the concept of the angle between skew lines. Let 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 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:
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:
Consequence: All four diagonals of a rectangular parallelepiped are equal to each other. Three perpendiculars theoremLet 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 O 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 O called the base of the perpendicular. If a JSC- perpendicular to the plane α , a M is an arbitrary point of this plane, different from the point O, then the segment AM is called a slope drawn from a point BUT to the plane α , and the point M- inclined base. Line segment 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 oblique plane onto this plane is also perpendicular to the oblique 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 inclined ones are drawn to this plane, then:
Definitions of distances by objects in space:
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:
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:
Theorems:
Symmetry of figuresDefinitions:
PrismDefinitions:
Properties and formulas for a prism:
where: S base - the area of \u200b\u200bthe base (in the drawing, for example, ABCDE), h- height (in the drawing it is MN).
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).
where: P sec - the perimeter of a perpendicular section, l is the length of the lateral edge. Types of prisms in stereometry:
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.
Properties of the correct prism:
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:
d 2 = a 2 + b 2 + c 2 .
PyramidDefinitions:
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:
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:
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 pyramidDefinition: 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:
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.
Formulas for volume and area of a pyramidTheorem(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).
where: S base is the area of the base of the pyramid, h is the height of the pyramid.
where: S side - side surface area, S 1 , S 2 , S 3 - areas of side faces.
Definitions:
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(a- 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 pyramidDefinitions and properties:
Formulas for a truncated pyramidThe 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, a- 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 near which a sphere can be described is not limited to these types of pyramids. In the drawing on the right, at a height SH need to pick a point O, equidistant from all vertices of the pyramid: SO = OB = OS = OD = OA. Then the point O 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 O is the center of the inscribed sphere. This point O equidistant from all faces of the ball, for example: OM = OO 1 pyramid and coneIn 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 cylinderThe 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 ballDefinitions:
Theorems:
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 O. 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 large circles. For example, an infinite number of meridians can be drawn through the poles of the Earth. Definitions:
Theorems:
Polyhedra and the sphereDefinition: 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 sphereTheorems:
where: R is the radius of the sphere.
Ball segment, layer, sectorIn 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:
Cylinder and prismA 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 sphereA 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 cylinderTheorem 1(on 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: ConeDefinitions:
Volume and area of the lateral and full surfaces of the coneTheorem 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 proved) 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:
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 sphereA 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
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:
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