Stereometry Mathematics Formulas Problems & Axioms

Basic theorems, axioms, and definitions of Stereometry

Introductory definitions and axioms of Stereometry

Some definitions of stereometry are:

1. A polyhedron is a geometric body bounded by a finite number of flat polygons, any two of which, having a common side, do not lie in the same plane. Moreover, the polygons themselves are called faces, their sides are the edges of the polyhedron, and their vertices are the vertices of the polyhedron.
2. The figure formed by all faces of a polyhedron is called its surface ( full surface ), and the sum of the areas of all its faces is called the area of ​​the (full) surface .
3. A cube is a polyhedron having six faces that are equal squares. The sides of the squares are called the edges of the cube, and the vertices are the vertices of the cube.
4. A parallelepiped is a polyhedron with six faces and each of them is a parallelogram. The sides of a parallelogram are called the edges of a parallelepiped, and their vertices are called the vertices of a parallelepiped. Two faces of a parallelepiped are called opposite if they do not have a common edge, and those having a common edge are called adjacent . Sometimes any two opposite faces of a parallelepiped are highlighted and called bases , then the other faces are side faces , and their sides connecting the top of the bases of the parallelepiped are its side edges .
5. A straight parallelepiped is a parallelepiped whose side faces are rectangles. A rectangular parallelepiped is a parallelepiped whose faces are all rectangles. Note that every rectangular parallelepiped is a straight parallelepiped, but not every straight parallelepiped is rectangular.
6. Two vertices of the parallelepiped that do not belong to the same face are called opposite . The segment connecting the opposite vertices of the parallelepiped is called the diagonal of the parallelepiped. The parallelepiped has only four diagonals.
7. A prism ( n- angle) is a polyhedron whose two faces are equal n -counters, and the other n faces are parallelograms. Equal n -gons are called bases , and parallelograms are called side faces of a prism . A straight prism is a prism whose side faces are rectangles. A regular n -prism is a prism for which all lateral faces are rectangles, and its bases are regular n -counters.
8. The sum of the areas of the lateral faces of the prism is called the area of ​​its lateral surface (denoted by S _side ). The sum of the areas of all faces of a prism is called the surface area of ​​the prism (denoted by S _full ).
9. A pyramid ( n- angular) is a polyhedron with one face, some n -gon, and the other n faces, triangles with a common vertex; An n -gon is called a base ; triangles with a common vertex are called lateral faces , and their common vertex is called the vertex of the pyramid . The sides of the pyramid are called its edges , and the edges converging at the vertex are called lateral .
10. The sum of the areas of the lateral sides of the pyramid is called the lateral surface area of ​​the pyramid (denoted by S _side ). The sum of the areas of all faces of the pyramid is called the surface area of ​​the pyramid (the surface area is denoted by S _full ).
11. A regular n -pound pyramid is such a pyramid, the base of which is a regular n -gon, and all side edges are equal to each other. A regular pyramid has side faces that are isosceles triangles equal to each other.
12. A triangular pyramid is called a tetrahedron if all its faces are equal regular triangles. The tetrahedron is a special case of a regular triangular pyramid (that is, not every regular triangular pyramid will be a tetrahedron).

Axioms of stereometry:

1. Through any three points not lying on one straight line, there passes a single plane.
2. If two points of a line lie in a plane, then all points of a line lie in this plane.
3. If two planes have a common point, then they have a common line on which all common points of these planes lie.

Consequences from stereometry axioms:

• Theorem 1. A single plane passes through a line and not lying on it.
• Theorem 2. A single plane passes through two intersecting straight lines.
• Theorem 3. A single plane passes through two parallel lines.

Construction of sections in stereometry

To solve problems in stereometry, it is urgently necessary to build a cross section of polyhedra (for example, a pyramid, a parallelepiped, a cube, a prism) on the figure with a certain plane. We give several definitions explaining what a section is:

• The secant plane of the pyramid (prism, parallelepiped, cube) is such a plane, on either side of which there are points of a given pyramid (prism, parallelepiped, cube).
• A section of a pyramid (prism, parallelepiped, cube) is a figure consisting of all points that are common to the pyramid (prism, parallelepiped, cube) and a cutting plane.
• The cutting plane intersects the faces of the pyramid (parallelepiped, prism, cube) along segments, so the section is a polygon lying in a sectioning plane, the sides of which are the specified segments.

To build a section of a pyramid (a prism, a parallelepiped, a cube), we can and need to build the points of intersection of the cutting plane with the edges of the pyramid (a prism, a parallelepiped, a cube) and connect every two of them lying on the same face. Note that the sequence of constructing vertices and sides of the section is not significant. The construction of polyhedron cross sections is based on two construction tasks:

1. Intersection lines of two planes.

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

2. The intersection points of the line and the plane.

To construct the point of intersection of the line l and the plane α, we need to construct the point of intersection of the line land the line l ₁ along which the plane α and any plane containing the line l intersect .

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 the lines a and b , or AB and CD are parallel, then write:

Some theorems:

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

There are three cases of the relative position of the line and the plane in stereometry:

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

Definition: A line and a plane are called parallel if they do not have common points. If the line a is parallel to the plane β, then write:

Theorems:

• Theorem 1 (a sign of parallelism of a line and a plane). If a straight line that is not lying in this plane is parallel to any straight line lying in this plane, then it is parallel to this plane.
• Theorem 2. If a plane (in the figure – α ) passes through a straight line (in the figure – c ) 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 this straight line:

If two different straight lines lie in the same plane, then they either intersect or are parallel. However, in space (i.e., in stereometry) the third case is also possible, when there is no plane in which two straight lines lie (they do not intersect and are not parallel).

Definition: Two straight lines are called crossed if there is no plane in which they both lie.

Theorems:

• Theorem 1 (a sign of crossing lines). If one of the two straight lines lies in a certain plane, and the other straight line intersects this plane at a point that does not belong to the first straight line, then these straight lines intersect.
• Theorem 2. Through each of the two crossed lines there passes a single plane parallel to the other straight line.

Now we introduce the concept of angle between crossed straight lines. Let a and b be two skew lines. Take an arbitrary point O in space and draw through it straight lines a ₁ and b ₁ , parallel to straight lines a and b, respectively. The angle between the crossing lines a and b is the angle between the constructed intersecting lines a ₁ and b ₁ .

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

Definition: Let a and b be two skew lines. Take an arbitrary point O on one of them (in our case, on the straight line b ) and draw through it a straight line parallel to the other of them (in our case, a _{1 is} parallel to a ). The angle between the crossed straight lines a and b is the angle between the constructed straight line and the straight line containing the point O (in our case it is the angle β between the straight lines a ₁ and b ).

Definition: Two straight lines are called mutually perpendicular (perpendicular) if the angle between them is 90 °. Both the crossed straight lines and the straight lines lying and intersecting in the same plane can be perpendicular. If the line a is perpendicular to the line b , then write:

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

Theorems:

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

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

Theorems:

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

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

Three perpendicular theorem

Let point A not lie on the plane α . Draw a straight line through point A , perpendicular to the plane α , and denote by the letter O the point of intersection of this line with the plane α . The perpendicular drawn from point A to the plane α is called the segment AO , the point O is called the base of the perpendicular. If AO is perpendicular to the plane α , and M is an arbitrary point of this plane, different from point O , then the segment AM is called inclined, drawn from the pointAnd to the plane α , and the point M – the base of the inclined. The segment OM is the orthogonal projection (or, in short, the projection) of the inclined AM on the plane α . Now we give a theorem that plays an important role in solving many problems.

Theorem 1 (on three perpendiculars): A straight line drawn in a plane and perpendicular to the projection inclined on this plane is perpendicular to the most inclined one. The converse is also true:

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

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

• two oblique, having equal projections, are equal;
• of the two inclined is greater than the one whose projection is greater.

Determination of distances by objects in space:

• The distance from the point to the plane is the length of the perpendicular drawn from this point to this 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 straight line and a plane parallel to it is the distance from an arbitrary point of a straight line to a plane.
• The distance between the crossing lines is the distance from one of the crossing lines to the plane passing through the other straight line and parallel to the first straight line.

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

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

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

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

Dihedral angle

Definitions:

• A dihedral angle is 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.
• A linear angle of a dihedral angle is an angle whose sides are rays with a common origin on the edge of the dihedral angle, which are held in its faces perpendicular to the edge.

Thus, the linear angle of the dihedral angle is the angle formed by the intersection of the dihedral angle by a plane perpendicular to its edge. All linear angles of the 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 a right angle (acute, obtuse) if its degree measure is 90 ° (less than 90 °, more than 90 °). In the future, when solving problems in stereometry, under the dihedral angle we will always understand that linear angle, the degree measure of which satisfies the condition:

Definitions:

• The dihedral angle at the 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 this edge of the polyhedron.
• The angle between intersecting planes is the angle between the lines drawn, respectively, in these planes perpendicular to their intersection line through some of its points.
• Two planes are called perpendicular if the angle between them is 90 °.

Theorems:

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

Symmetry shapes

Definitions:

1. The points M and M ₁ are called symmetric with respect to the point O if O is the midpoint of the segment MM ₁ .
2. The points M and M ₁ are called symmetric with respect to the straight line l if the straight line l passes through the midpoint of the segment MM ₁ and is perpendicular to it.
3. The points M and M ₁ are called symmetric with respect to the plane α if the plane α passes through the midpoint of the segment MM ₁ and is perpendicular to this segment.
4. The point O (line l , plane α ) is called the center (axis, plane) of the symmetry of the figure if each point of the figure is symmetric about the point O (line l , plane α ) to a certain point of the same figure.
5. A convex polyhedron is called correct if all its faces are regular polygons equal to each other and the same number of edges converges at each vertex.

Prism

Definitions:

1. A prism is a polyhedron whose two faces are equal polygons lying in parallel planes, and the other faces are parallelograms having common sides with these polygons.
2. Bases are two faces that are equal polygons lying in parallel planes. The drawing is: ABCDE and KLMNP .
3. Side faces – all faces, except for the bases. Each side face is necessarily a parallelogram. The drawing is: ABLK , BCML , CDNM , DEPN and EAKP .
4. Side surface – the union of the side faces.
5. Full surface – combining the bases and the lateral surface.
6. The side edges are common sides of the side faces. The drawing is: AK , BL , CM , DN and EP .
7. Height – the segment connecting the base of the prism and perpendicular to them. In the drawing, for example, KR .
8. A diagonal is a segment connecting two vertices of a prism that do not belong to the same face. In the drawing, this is, for example, BP .
9. The diagonal plane is the plane passing through the side edge of the prism and the diagonal of the base. Another definition: a diagonal plane is a plane passing through two side edges of a prism that do not belong to the same face.
10. Diagonal section – the intersection of the prism and the diagonal plane. A parallelogram is formed in the cross section, including, sometimes, its special cases – a rhombus, a rectangle, and a square. In the drawing, for example, this is the EBLP .
11. The perpendicular (orthogonal) section is the intersection of the prism and the plane perpendicular to its lateral edge.

Properties and formulas for 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.
• The volume of the prism is equal to the product of its height by the area of ​​the base:

where: S _D is the area of ​​the base (in the drawing this is, for example, ABCDE ),  h is the height (in the drawing it is MN ).

• The total surface area of ​​the prism is equal to the sum of its lateral surface area and the double base area:

• 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 the perpendicular section are the linear angles of the dihedral angles at the respective side edges.
• Perpendicular (orthogonal) section perpendicular to all lateral faces.
• The volume of the inclined prism is equal to the product of the area of ​​the perpendicular section by the length of the side edge:

where: S _Sich is the area of ​​perpendicular section, l is the length of the side edge (in the drawing below it is, for example, AA ₁ or BB ₁ and so on).

• The area of ​​the side surface of an arbitrary prism is equal to the product of the perimeter of a perpendicular section by the length of the side edge:

where: P _Sech – the perimeter of the 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 inclined (shown 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 straight prism is a prism in which all lateral edges are perpendicular to the base. In a straight prism, the side edges are heights. The side faces of a straight prism are rectangles. And the area and the perimeter of the base are equal, respectively, to the area and perimeter of the perpendicular section (for a straight prism, generally speaking, the perpendicular section is entirely the same figure as the base). Therefore, the lateral surface area of ​​a straight prism is equal to the product of the base perimeter and the length of the lateral edge (or, in this case, the height of the prism):

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

• A regular prism is a prism at the base of which a regular polygon lies (that is, one whose 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:

1. The bases of a regular prism are regular polygons.
2. The side faces of a regular prism are equal rectangles.
3. The side edges of a regular prism are equal.
4. The correct prism is straight.

Parallelepiped

Definition: A parallelepiped is a prism whose base is a parallelogram. In this definition, the key word is “prism”. Thus, the parallelepiped is a special case of a prism, which differs from the general case only in that it has not an arbitrary polygon at its base, but a parallelogram. Therefore, all the above properties, formulas and definitions relating to the prism remain relevant for the parallelepiped. However, there are several additional properties that are typical for the parallelepiped.

Other properties and definitions:

• Two faces of the parallelepiped that do not have a common edge are called opposite , and those that have a common edge are adjacent .
• Two vertices of the parallelepiped that do not belong to the same face are called opposite .
• The segment connecting the opposite vertices is called the diagonal of the parallelepiped.
• A parallelepiped has six faces, and all of them are parallelograms.
• The opposite faces of the parallelepiped are pairwise equal and parallel.
• The parallelepiped has four diagonals; they all intersect at one point, and each of them is divided by this point in half.
• If the four side faces of the parallelepiped are rectangles (and the bases are arbitrary parallelograms), then it is called straight (in this case, as with a straight prism, all lateral edges are perpendicular to the bases). All properties and formulas for a straight prism are relevant for a right parallelepiped.
• A parallelepiped is called oblique , if not all its side faces are rectangles.
• The volume of a straight or inclined parallelepiped is calculated using the general formula for a prism volume, i.e. equal to the product of the area of ​​the base of the parallelepiped and its height ( V = S _base ∙ h ).
• A straight parallelepiped, in which all six faces — rectangles (that is, except for the side faces, are also rectangles), is called rectangular . For a rectangular parallelepiped, all properties of a right parallelepiped are relevant, as well as:
  • The diagonal of the rectangular parallelepiped d and its edges a , b , c are connected by the relation:

d ² = a ² + b ² + c ² .

• • From the general formula for the volume of a prism, you can get the following formula for the volume of a rectangular parallelepiped :

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

• From the formula for the volume of a rectangular parallelepiped, you can get the following formula for the volume of the cube :

Pyramid

Definitions:

• The pyramid is a polyhedron whose base is a polygon, and the other faces are triangles with a common vertex. According to the number of corners of the base, pyramids are distinguished triangular, quadrangular, and so on. The figure shows examples: quadrilateral and hexagonal pyramids.

• The base is a polygon that does not belong to the top of the pyramid. In the drawing, the base is BCDE .
• The edges that are different from the base, called the side . The drawing is: ABC , ACD , ADE and AEB .
• The common vertex of the side faces is called the vertex of the pyramid (namely, the vertex of the whole pyramid, and not just the vertex, like all the other vertices). The drawing is A .
• The ribs connecting the top of the pyramid with the vertices of the base, called the side . The drawing is: AB , AC , AD and AE .
• Denoting the pyramid, first call its top, and then – the top of the base. For the pyramid from the drawing the designation will be: ABCDE .

• The height of the pyramid is called perpendicular, drawn from the top of the pyramid at its base. The length of this perpendicular is designated 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, it is not. In the general case, for an arbitrary pyramid, the point of intersection of height and base can be anywhere.
• Apothem – the height of the side face of a regular pyramid, drawn from its top. In the drawing, this is, for example, AF .
• A diagonal section of the pyramid is a section of the pyramid passing through the top of the pyramid and the diagonal of the base. In the drawing this is, for example, ACE .

Another stereometric drawing with symbols for better memorization (in the figure is a regular triangular pyramid):

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

• A circle can be described near the base of the pyramid, with the top of the pyramid being projected into its center (point O ). In other words, the height (segment SO ), lowered from the top of such a pyramid to the base ( ABCD ), falls into the center of the circumcircle, i.e. at the intersection point of the perpendicular perpendiculars of the base.
• The side edges form equal angles with the base plane (in the drawing below these are SAO , SBO , SCO , SDOangles ).

Important: The reverse 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 angles DMN , DKN , DLN  in the drawing are equal below), then:

• A circle can be inscribed at the base of the pyramid, with the top of the pyramid being projected into its center (point N ). In other words, the height (segment 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. at the intersection of the bisectors of the base.
• The heights of the side faces (apothems) are equal. In the drawing below, DK , DL , DM are equal apothems.
• The lateral surface area of ​​such a pyramid is equal to half the product of the base perimeter by the height of the lateral face (apothem).

where: P is the base perimeter, a is the apothem length.

Important: The reverse is also true, that is, if a circle can be inscribed at 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 one angle and the heights of the side faces (apothems) are equal.

Correct pyramid

Definition: A 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 lateral edges of a regular pyramid are equal.
• All side faces of a regular pyramid are inclined to the base plane at one angle.

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

• In a regular pyramid, all side faces are equal isosceles triangles.
• In any regular pyramid, you can either enter a sphere or describe a sphere near it.
• The lateral surface area of ​​a regular pyramid is equal to half the product of the base perimeter and apothem.

Formulas for the volume and area of ​​the pyramid

Theorem  (on the volume of pyramids having equal heights and equal areas of bases). Two pyramids that have equal heights and equal base areas have equal volumes (of course, you probably already know the formula for the volume of the pyramid, well, or see it several lines below, and you think this statement is obvious, but actually, judging “ eye “, this theorem is not so obvious (see the figure below). This also applies to other polyhedra and geometric figures: their appearance is deceptive, therefore, in mathematics, you need to trust only formulas and correct calculations.

• The volume of the pyramid can be calculated by the formula:

where: S _osn – the area of ​​the base of the pyramid, h – the height of the pyramid.

• The lateral surface of the pyramid is equal to the sum of the areas of the lateral faces. For the lateral surface area of ​​the pyramid, you can formally write the following stereometric formula:

where: S _side – the area of ​​the side surface, S ₁ , S ₂ , S ₃ – the area of ​​the side faces.

• The total surface of the pyramid is equal to the sum of the lateral surface area and the base area:

Tetrahedron

Definitions:

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

The drawing shows a regular tetrahedron, with the triangles ABC , ADC , CBD , BAD being equal. From the general formulas for the volume and area of ​​the pyramid, as well as knowledge from planimetry, it is not difficult to obtain formulas for the volume and area of ​​a regular tetrahedron ( a is the edge length):

Rectangular pyramid

Definition: When solving problems in stereometry, a 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. In the figure to the left, SA is an edge that is also a height.

 

Truncated pyramid

Definitions and properties:

• A truncated pyramid is a polyhedron enclosed between the base of the pyramid and the cutting plane parallel to its base.
• The shape obtained at the intersection of the section plane and the original pyramid is also called the base of thetruncated 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 trapeziums. The drawing is, for example, AA ₁ B ₁ B .
• The side edges of the truncated pyramid are the parts of the edges of the original pyramid that are enclosed between the bases. In the drawing, for example, AA ₁ .
• The height of the truncated pyramid is called perpendicular (or the length of this perpendicular), drawn from some point of the plane of one base to the plane of another base.
• A truncated pyramid is called regular if it is a polyhedron that is clipped by a plane parallel to the base of the regularpyramid.
• The bases of a regular truncated pyramid are regular polygons.
• The side faces of a regular truncated pyramid are isosceles trapezium.
• The height of its side face is called the apothem of a regular truncated pyramid.
• The area of ​​the side surface of the truncated pyramid is the sum of the areas of all its side faces.

Truncated Pyramid Formulas

The volume of the truncated pyramid is:

where: S ₁ and S ₂ are the base areas, h is the height of the truncated pyramid. However, in practice, it is more convenient to look for the volume of the truncated pyramid like this: you can complete the truncated pyramid to the pyramid by 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 searched for as the difference between the lateral surface areas of the entire pyramid and the completed part. The lateral surface area of ​​a regular truncated pyramid is equal to the half-sum of the perimeters of its bases and apothem:

where: P ₁ and P ₂ are the perimeters of the base of a regular truncated pyramid, and is the length of the apothem. The total surface area of ​​any truncated pyramid is obviously found as the sum of the base areas and the lateral surface:

Pyramid and ball (sphere)

Theorem: Around the pyramid one can describe a sphere when the inscribed polygon lies at the base of the pyramid (that is, a polygon near which one can describe a sphere). 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.

Note: From this theorem it follows that a sphere can be described both near any triangular and about any regular pyramid. However, the list of pyramids around which you can describe the scope is not limited to these types of pyramids. In the drawing to the right, at the height of SH, it is necessary to choose a point O equidistant from all vertices of the pyramid: SO = OB = OС = OD = OA . Then the point O is the center of the described ball.

Theorem: A sphere can be inscribed into a pyramid 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.

Note: You obviously did not understand what you read in the line above. However, the main thing to remember is that any correct pyramid is one into which a sphere can be inscribed . Moreover, the list of pyramids in which the sphere can be entered is not exhausted by the correct ones.

Definition: The bisector plane divides the dihedral angle in half, and each point of the bisector plane is equidistant from the faces that form the dihedral angle. In the figure to the right, the plane γ is the bisector of the dihedral angle formed by the planes α and β .

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

OM = OO ₁

 

Pyramid and cone

To the table of contents …

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).

A cone is described as described 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 lateral edges of the pyramid are equal to each other (a necessary and sufficient condition).

Important property: The heights of such cones and pyramids are equal to each other.

Pyramid and cylinder

A cylinder is called inscribed in a pyramid if its one base coincides with a circle inscribed into the pyramid section by a plane parallel to the base, and the other base belongs to the base of the pyramid.

A cylinder is described as described near the pyramid if the top of the pyramid belongs to its one base, 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:

1. A sphere is a closed surface, the locus of points in space that are equidistant from a given point, called the center of the sphere . A sphere is also a body of rotation formed by rotating a semicircle around its diameter. The radius of a sphere is a segment that connects the center of the sphere with a point of the sphere.
2. The chord of a sphere is a segment connecting two points of a sphere.
3. The diameter of the sphere is called the chord passing through its center. The center of a sphere divides any diameter of it into two equal segments. Any diameter of a sphere of radius R is 2 R .
4. The ball is a geometric body; the set of all points of space that are at a distance not greater than the given one from some center. This distance is called the radius of the ball . The ball is formed by rotating a semicircle near its fixed diameter. Note that the surface (or boundary) of a ball is called a sphere. It is possible to give such a definition of a ball: a ball is a geometric body consisting of a sphere and a part of the space bounded by this sphere.
5. The radius , chord and diameter of the ball are called the radius, chord and diameter of the sphere, which is the boundary of the ball.
6. 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 within this line. A sphere is a shell, and a ball is also all points inside this shell.
7. A plane passing through the center of a sphere (ball) is called the diametral plane .
8. The cross section of a sphere (ball) with a diametrical plane is called a large circle ( large circle ).

Theorems:

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

The largest circle from among those which can be obtained in a section of a sphere plane, lies in a section passing through the ball center O . He then called the big circle. Its radius is equal to the radius of the ball. Any two large circles intersect along the diameter of the ball AB . This diameter is also the diameter of intersecting large circles. Through two points of a spherical surface located at the ends of the same diameter (in Fig. A and B ), an infinite number of large circles can be drawn. For example, an infinite number of meridians can be drawn through the poles of the Earth.

Definitions:

1. A tangent plane to a sphere is a plane that has only one common point with the sphere, and their common point is called the tangency point of the plane and the sphere.
2. The tangent plane to the ball is called the tangent plane to the sphere, which is the boundary of this ball.
3. Any straight line lying in the tangent plane of a sphere (ball) and passing through a tangency point is called a tangent 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 tangency point.

Theorems:

• Theorem 1 (a sign of a tangent plane to a sphere). The 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 a tangent plane to a sphere). The tangent plane to the sphere is perpendicular to the radius drawn to the point of tangency.

Polyhedra and Sphere

Definition:  In stereometry, a polyhedron (for example, a pyramid or a prism) is called inscribed in a sphere if all its vertices lie on a sphere. In this case, the sphere is called described near a polyhedron (pyramid, prism). Similarly: a polyhedron is called inscribed in a ball if all its vertices lie on the boundary of this ball. Moreover, the ball is described as described near the polyhedron.

Important property: The center of the sphere described near the polyhedron is at a distance equal to the radius R of the sphere from each vertex of the polyhedron. Let us give examples of polyhedra inscribed in the sphere:

Definition: A polyhedron is described as described near a sphere (ball) if the sphere (ball) touches all faces of the polyhedron. In this case, the sphere and ball are called inscribed in a polyhedron.

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

Volume and surface area of ​​the ball

Theorems:

• Theorem 1 (on the area of ​​a sphere). The area of ​​the sphere is:

where: R is the radius of the sphere.

• Theorem 2 (on the volume of the ball). The volume of the ball of radius R is calculated by the formula:

Ball segment, layer, sector

Ball segment

In stereometry, a spherical segment is a part of a ball, cut off by a cutting plane. The ratio between the height, the radius of the base of the segment and the radius of the ball:

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

The external surface area of ​​the ball segment:

The total surface area of ​​the ball segment:

The volume of the ball segment:

Ball layer

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

where: h is the height of the ball layer, R is the radius of the ball. The total surface area of ​​the ball layer:

where: h is the height of the spherical layer, R is the radius of the ball, 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 the ball layer is easiest to look for as the difference between the volumes of the two ball segments.

Ball sector

In stereometry, a spherical sector is a part of a sphere consisting of a spherical segment and a cone with the top in the center of the sphere and the base coinciding with the base of the spherical segment. This implies that the ball segment is less than half the ball. The total surface area of ​​the ball sector:

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

Cylinder

Definitions:

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

2. A straight circular cylinder or simply a cylinder is a geometric body bounded by a cylindrical surface and two parallel planes that are perpendicular to the forming cylindrical surface. Informally, the cylinder can be perceived as a straight prism, with a circle at its base. This will help to easily understand and, if necessary, derive formulas for the volume and side surface area of ​​the cylinder.
3. The side surface of a cylinder is a part of a cylindrical surface located between the cutting planes that are perpendicular to its generators, and the parts (circles) cut off by the cylindrical surface on parallel planes are called the bases of the cylinder . The bases of the cylinder are two equal circles.
4. Forming a cylinder is called a segment (or the length of this segment) forming a cylindrical surface located between parallel planes in which lie the base of the cylinder. All forming cylinder parallel and equal to each other, as well as perpendicular to the base.
5. The axis of the cylinder is the segment connecting the centers of the circles, which are the bases of the cylinder.
6. The height of a cylinder is called perpendicular (or the length of this perpendicular), drawn from some point of the plane of one base of the cylinder to the plane of another base. In the cylinder, the height is equal to the generator.
7. The radius of a cylinder is the radius of its base.
8. A cylinder is called equilateral if its height is equal to the diameter of the base.
9. The cylinder can be obtained by rotating a rectangle around one of its sides through 360 °.
10. If the cutting plane is parallel to the axis of the cylinder, then the section of the cylinder is a rectangle, the two sides of which are the generators, and the other two are the chords of the bases of the cylinder.
11. The axial section of the cylinder is called the section of the cylinder by the plane passing through its axis. The axial section of the cylinder is a rectangle, the two sides of which are forming the cylinder, and the other two are the diameters of its bases.
12. If the cutting plane is perpendicular to the axis of the cylinder, then a cross section forms a circle 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; right – section parallel to the base of the cylinder.

Cylinder and prism

A prism is called inscribed in a cylinder if its bases are inscribed in the bases of a cylinder. In this case, the cylinder is called described near the prism. The height of the prism and the height of the cylinder in this case will be equal. All the side edges of the prism will belong to the side surface of the cylinder and coincide with its forming. Since we understand only a straight cylinder as a cylinder, we can also include only a direct prism in such a cylinder. Examples:

A prism is described as described near a cylinder if its bases are described near the bases of the cylinder. In this case, the cylinder is called inscribed in a prism. The height of the prism and the height of the cylinder in this case will also be equal. All lateral edges of the prism will be parallel to the cylinder. Since we understand only a straight cylinder as a cylinder, such a cylinder can be entered only into 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 described as described near a sphere (ball). A sphere can be inscribed in a cylinder only if it is an equilateral cylinder, i.e. diameter of its base and height are equal to each other. 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:

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

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

Volume and area of ​​the side and full surfaces of the cylinder

Theorem 1 (on the area of ​​the side surface of the cylinder): The area of ​​the side surface of the cylinder is equal to the product of the length of the circumference of its base and the height:

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

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

Therefore, the area of ​​the full surface of the cylinder S is _{full. cylinder is} calculated by the formula:

Theorem 2 (on the volume of the cylinder): The volume of the cylinder is equal to the product of the area of ​​the base by 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 prism volume.

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

Cone

Definitions:

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

2. The segments (or their lengths) connecting the vertex of the cone with the points of the circle of the base are called cone-forming . All the generators of a right circular cone are equal to each other.
3. The surface of the cone consists of the base of the cone (circle) and the side surface (composed of all possible forming).
4. The union of the generatrix of a cone is called the generatrix (or side) of the cone . The cone forming surface is a conical surface.
5. A cone is called straight if the straight line connecting the vertex of the cone with the center of the base is perpendicular to the plane of the base. In the following, we will consider only a straight cone, calling it just a cone for brevity.
6. Visually, a straight circular cone can be thought of as a body obtained by rotating a right-angled 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 – by the rotation of the leg, which is not an axis.
7. The radius of a cone is the radius of its base.
8. The height of the cone is called the perpendicular (or its length), lowered from its top to the plane of the base. At a straight cone, the base of the height coincides with the center of the base. The axis of a right circular cone is called a straight line, containing its height, i.e. direct passing through the center of the base and the top.
9. If the cutting plane passes through the axis of the cone, then the cross section is an isosceles triangle, the base of which is the diameter of the base of the cone, and the sides – forming the cone. This section is called the axial .

10. 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 whose center is the point of intersection of the height and this plane.
11. The height ( h ), the radius ( R ) and the length of the generator ( l ) of a right circular cone satisfy the obvious relation:

Volume and area of ​​the side and full surfaces of the cone

Theorem 1 (on the lateral surface area of ​​a cone). The area of ​​the side surface of the cone is equal to the product of half the length of the base circumference by the generator:

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.

The area of ​​the full surface of the cone is the sum of the side surface area and the base area. The area of ​​the base of the cone (i.e., simply the area of ​​the circle) is equal to: S _base = πR ² . Therefore, the area of ​​the full surface of the cone S is _{full. the cone is} calculated by the formula:

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

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

Frustum

Definitions:

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

2. The base of the original cone and the circle resulting in the cross section of this cone by a plane are called bases , and the segment connecting their centers is the height of the truncated cone .
3. The straight line passing through the height of the truncated cone (i.e., through the centers of its bases) is its axis .
4. The part of the lateral surface of the cone, limiting the truncated cone, is called its lateral surface , and the segments forming the cone located between the bases of the truncated cone are called its constants .
5. All forming a truncated cone are equal to each other.
6. A truncated cone can be obtained by rotating a 360 ° rectangular trapezium around its side, perpendicular to the bases.

Truncated cone formulas:

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

where: S ₁ = π r ₁ ² and S ₂ = π r ₂ ² are the areas of the bases, h is the height of the truncated cone, r ₁ and r ₂ are the 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 the truncated cone as the difference between the volumes of the original cone and the cut-off part. The lateral surface area of ​​the truncated cone can also be searched for as the difference between the lateral surface areas of the original cone and the clipped part.

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

where: P ₁ = 2 π r ₁ and P ₂ = 2 π r ₂ – the perimeters of the bases of the truncated cone, l – the length of the generator. The total surface area of ​​the truncated cone is obviously found as the sum of the base areas and the lateral surface:

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

Cone and Sphere

A cone is called inscribed in a sphere (ball) if its vertex belongs to a sphere (border of the ball), and the base circle (the base itself) is a section of the sphere (ball). In this case, the sphere (ball) is described as described near the cone. Around a straight circular cone you can always describe a sphere. The center of the described 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 described near the axial section of the cone (this section is an isosceles triangle). Examples:

A sphere (ball) is called inscribed into a cone if the sphere (ball) touches the base of the cone and each of its generators. In this case, the cone is called described near the sphere (ball). In a straight circular cone you can always enter a sphere. 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 (pyramid – 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 into a cone (cone — described near the pyramid) if its base is inscribed at the base of the cone, and the side edges form the cone.
• The heights of such cones and pyramids are equal.

Note: More details on how a cone fits into a pyramid in stereometry or is described near a pyramid already mentioned in earlier here .