Lecture: Cone. Base, height, lateral surface, generatrix, development

Cone- this is a body that consists of a circle, which is located at the base, from a point equidistant from all points on the circle, as well as from straight lines connecting this point (vertex) with all points lying on the circle.

A few questions earlier, we looked at the pyramid. So the cone is special case a pyramid with a circle at its base. Almost all the properties of a pyramid apply to a cone.

How can you get a cone? Remember the last question and how we got the cylinder. Now take an isosceles triangle and rotate it around its axis - you will get a cone.

Generators of the cone- these are segments enclosed between the points of the circle and the vertex of the cone. The generators of the cone are equal to each other.

To find the length of the generatrix, you should use the formula:

If all the generators are connected to each other, we can get lateral surface cone Its general surface consists of a side surface and a base in the form of a circle.

The cone has height. To obtain it, it is enough to lower the perpendicular from the top directly to the center of the base.

To find the lateral surface area, use the formula:

To find the total surface area of a cone, use the following formula.

Cone. Frustum

Conical surface is the surface formed by all straight lines passing through each point of a given curve and a point outside the curve (Fig. 32).

This curve is called guide , straight – forming , dot - top conical surface.

Straight circular conical surface is the surface formed by all straight lines passing through each point of a given circle and a point on a straight line that is perpendicular to the plane of the circle and passes through its center. In what follows we will briefly call this surface conical surface (Fig. 33).

Cone (straight circular cone ) is a geometric body bounded by a conical surface and a plane that is parallel to the plane of the guide circle (Fig. 34).

Rice. 32 Fig. 33 Fig. 34

A cone can be considered as a body obtained by rotating a right triangle around an axis containing one of the legs of the triangle.

The circle enclosing a cone is called its basis . The vertex of a conical surface is called top cone The segment connecting the vertex of a cone with the center of its base is called height cone The segments forming a conical surface are called forming cone Axis of a cone is a straight line passing through the top of the cone and the center of its base. Axial section called the section passing through the axis of the cone. Side surface development of a cone is a sector whose radius equal to length generatrix of the cone, and the length of the arc of the sector is equal to the circumference of the base of the cone.

The correct formulas for a cone are:

Where R– base radius;

H- height;

l– length of the generatrix;

S base– base area;

S side

S full

V– volume of the cone.

Truncated cone called the part of the cone enclosed between the base and the cutting plane parallel to the base of the cone (Fig. 35).

A truncated cone can be considered as a body obtained by rotating a rectangular trapezoid around an axis containing the side of the trapezoid perpendicular to the bases.

The two circles enclosing a cone are called its reasons . Height of a truncated cone is the distance between its bases. The segments forming the conical surface of a truncated cone are called forming . A straight line passing through the centers of the bases is called axis truncated cone. Axial section called the section passing through the axis of a truncated cone.

For a truncated cone the correct formulas are:

(8)

Where R– radius of the lower base;

r– radius of the upper base;

H– height, l – length of the generatrix;

S side– lateral surface area;

S full– total surface area;

V– volume of a truncated cone.

Example 1. The cross section of the cone parallel to the base divides the height in a ratio of 1:3, counting from the top. Find the lateral surface area of a truncated cone if the radius of the base and the height of the cone are 9 cm and 12 cm.

Solution. Let's make a drawing (Fig. 36).

To calculate the area of the lateral surface of a truncated cone, we use formula (8). Let's find the radii of the bases About 1 A And About 1 V and forming AB.

Consider similar triangles SO2B And SO 1 A, similarity coefficient, then

From here

Since then

The lateral surface area of a truncated cone is equal to:

Answer: .

Example 2. A quarter circle of radius is folded into a conical surface. Find the radius of the base and the height of the cone.

Solution. The quadrant of the circle is the development of the lateral surface of the cone. Let's denote r– radius of its base, H – height. Let's calculate the lateral surface area using the formula: . It is equal to the area of a quarter circle: . We get an equation with two unknowns r And l(forming a cone). In this case, the generatrix is equal to the radius of the quarter circle R, which means we get the following equation: , from where Knowing the radius of the base and the generator, we find the height of the cone:

Answer: 2 cm, .

Example 3. A rectangular trapezoid with an acute angle of 45 O, a smaller base of 3 cm and an inclined side equal to , rotates around a side perpendicular to the bases. Find the volume of the resulting body of revolution.

Solution. Let's make a drawing (Fig. 37).

As a result of rotation, we obtain a truncated cone; to find its volume, we calculate the radius of the larger base and height. In the trapeze O 1 O 2 AB we will conduct AC^O 1 B. B we have: this means that this triangle is isosceles A.C.=B.C.=3 cm.

Answer:

Example 4. A triangle with sides 13 cm, 37 cm and 40 cm rotates around an external axis, which is parallel to the larger side and located at a distance of 3 cm from it (the axis is located in the plane of the triangle). Find the surface area of the resulting body of revolution.

Solution . Let's make a drawing (Fig. 38).

The surface of the resulting body of revolution consists of the lateral surfaces of two truncated cones and the lateral surface of a cylinder. In order to calculate these areas, it is necessary to know the radii of the bases of the cones and the cylinder ( BE And O.C.), forming cones ( B.C. And A.C.) and cylinder height ( AB). The only unknown is CO. this is the distance from the side of the triangle to the axis of rotation. We'll find DC. The area of triangle ABC on one side is equal to the product of half the side AB and the altitude drawn to it DC, on the other hand, knowing all the sides of the triangle, we calculate its area using Heron’s formula.

Definitions:
Definition 1. Cone
Definition 2. Circular cone
Definition 3. Cone height
Definition 4. Straight cone
Definition 5. Right circular cone
Theorem 1. Generators of the cone
Theorem 1.1. Axial section of the cone

Volume and area:
Theorem 2. Volume of a cone
Theorem 3. Area of the lateral surface of a cone

Frustum :
Theorem 4. Section parallel to the base
Definition 6. Truncated cone
Theorem 5. Volume of a truncated cone
Theorem 6. Lateral surface area of a truncated cone

Definitions
A body bounded on the sides by a conical surface taken between its top and the plane of the guide, and the flat base of the guide formed by a closed curve, is called a cone.

Basic Concepts
A circular cone is a body that consists of a circle (base), a point not lying in the plane of the base (vertex) and all segments connecting the vertex to the points of the base.

A straight cone is a cone whose height contains the center of the base of the cone.

Consider any line (curve, broken or mixed) (for example, l), lying in a certain plane, and an arbitrary point (for example, M) not lying in this plane. All possible straight lines connecting point M to all points of a given line l, form surface called canonical. Point M is the vertex of such a surface, and the given line l - guide. All straight lines connecting point M to all points of the line l, called forming. A canonical surface is not limited by either its vertex or its guide. It extends indefinitely in both directions from the top. Let now the guide be a closed convex line. If the guide is a broken line, then the body, bounded on the sides by a canonical surface taken between its top and the plane of the guide, and a flat base in the plane of the guide, is called a pyramid.
If the guide is a curved or mixed line, then the body bounded on the sides by a canonical surface taken between its top and the plane of the guide, and a flat base in the plane of the guide, is called a cone or
Definition 1 . A cone is a body consisting of a base - flat figure, bounded by a closed line (curved or mixed), a vertex - a point that does not lie in the plane of the base, and all segments connecting the vertex with all possible points of the base.
All straight lines passing through the vertex of the cone and any of the points of the curve bounding the figure of the base of the cone are called generators of the cone. Most often in geometric problems, the generatrix of a straight line means a segment of this straight line, enclosed between the vertex and the plane of the base of the cone.
The base of a limited mixed line is a very rare case. It is indicated here only because it can be considered in geometry. The case with a curved guide is more often considered. Although, both the case with an arbitrary curve and the case with a mixed guideline are of little use and it is difficult to derive any patterns from them. Among the cones, the right circular cone is studied in the course of elementary geometry.

It is known that a circle is a special case of a closed curved line. A circle is a flat figure bounded by a circle. Taking the circle as a guide, we can define a circular cone.
Definition 2 . A circular cone is a body that consists of a circle (base), a point not lying in the plane of the base (vertex) and all segments connecting the vertex to the points of the base.
Definition 3 . The height of a cone is the perpendicular descended from the top to the plane of the base of the cone. You can select a cone, the height of which falls at the center of the flat figure of the base.
Definition 4 . A straight cone is a cone whose height contains the center of the base of the cone.
If we combine these two definitions, we get a cone, the base of which is a circle, and the height falls at the center of this circle.
Definition 5 . A right circular cone is a cone whose base is a circle, and its height connects the top and the center of the base of this cone. Such a cone is obtained by rotating a right triangle around one of its legs. Therefore, a right circular cone is a body of revolution and is also called a cone of revolution. Unless otherwise stated, for brevity in what follows we simply say cone.
So here are some properties of the cone:
Theorem 1. All generators of the cone are equal. Proof. The height of the MO is perpendicular to all straight lines of the base, by definition, a straight line perpendicular to the plane. Therefore, the triangles MOA, MOB and MOS are rectangular and equal on two legs (MO is the general one, OA=OB=OS are the radii of the base. Therefore, the hypotenuses, i.e., the generators, are also equal.
The radius of the base of the cone is sometimes called cone radius. The height of the cone is also called cone axis, therefore any section passing through the height is called axial section. Any axial section intersects the base in diameter (since the straight line along which the axial section and the plane of the base intersect passes through the center of the circle) and forms an isosceles triangle.
Theorem 1.1. The axial section of the cone is an isosceles triangle. So triangle AMB is isosceles, because its two sides MB and MA are generators. Angle AMB is the angle at the vertex of the axial section.

Obtained by combining all rays emanating from one point ( peaks cone) and passing through a flat surface. Sometimes a cone is a part of such a body obtained by combining all the segments connecting the vertex and points of a flat surface (the latter in this case is called basis cone, and the cone is called leaning on this basis). This is the case that will be considered below, unless otherwise stated. If the base of the cone is a polygon, the cone becomes a pyramid.

"== Related definitions ==

The segment connecting the vertex and the boundary of the base is called generatrix of the cone.
The union of the generators of a cone is called generatrix(or side) cone surface. The forming surface of the cone is a conical surface.
A segment dropped perpendicularly from the vertex to the plane of the base (as well as the length of such a segment) is called cone height.
If the base of a cone has a center of symmetry (for example, it is a circle or an ellipse) and the orthogonal projection of the vertex of the cone onto the plane of the base coincides with this center, then the cone is called direct. In this case, the straight line connecting the top and the center of the base is called cone axis.
Oblique (inclined) cone - a cone whose orthogonal projection of the vertex onto the base does not coincide with its center of symmetry.
Circular cone- a cone whose base is a circle.
Straight circular cone(often simply called a cone) can be obtained by rotating a right triangle around a line containing the leg (this line represents the axis of the cone).
A cone resting on an ellipse, parabola or hyperbola is called respectively elliptical, parabolic And hyperbolic cone(the last two have infinite volume).
The part of the cone lying between the base and a plane parallel to the base and located between the top and the base is called truncated cone.

Properties

If the area of the base is finite, then the volume of the cone is also finite and equal to one third of the product of the height and the area of the base. Thus, all cones resting on a given base and having a vertex located on a given plane parallel to the base have equal volume, since their heights are equal.
The center of gravity of any cone with a finite volume lies at a quarter of the height from the base.
The solid angle at the vertex of a right circular cone is equal to

Where - opening angle cone (that is, double the angle between the axis of the cone and any straight line on its lateral surface).

The lateral surface area of such a cone is equal to

where is the radius of the base, is the length of the generatrix.

The volume of a circular cone is equal to

The intersection of a plane with a right circular cone is one of the conic sections (in non-degenerate cases - an ellipse, parabola or hyperbola, depending on the position of the cutting plane).

Generalizations

In algebraic geometry cone is an arbitrary subset of a vector space over a field, for which for any

- (Latin conus; Greek konos). A body bounded by a surface formed by the inversion of a straight line, one end of which is motionless (the vertex of the cone), and the other moves along the circumference of a given curve; looks like a sugar loaf. Dictionary of foreign words,... ... Dictionary of foreign words of the Russian language

CONE- (1) in elementary geometry, a geometric body limited by a surface formed by the movement of a straight line (generating a cone) through a fixed point (top of the cone) along a guide (base of the cone). The formed surface is enclosed between... Big Polytechnic Encyclopedia

- (straight circular) geometric body formed by rotation of a right triangle around one of the legs. The hypotenuse is called the generator; fixed leg height; a circle described by a rotating leg with a base. Lateral surface K.... ... Encyclopedia of Brockhaus and Efron

- (straight circular K.) a geometric body formed by the rotation of a right triangle about one of the legs. The hypotenuse is called the generator; fixed leg height; a circle described by a rotating leg with a base. Side surface …

- (straight circular) geometric body formed by rotation of a right triangle about one of the legs. The hypotenuse is called the generator; fixed leg height; a circle described by a rotating leg with a base. Lateral surface K... encyclopedic Dictionary F. Brockhaus and I.A. Efron

- (Latin conus, from Greek konos) (mathematics), 1) K., or conical surface, the geometric locus of straight lines (generators) of space connecting all points of a certain line (guide) with a given point (vertex) of space.… … Great Soviet Encyclopedia

A truncated cone is obtained if a smaller cone is cut off from the cone with a plane parallel to the base (Fig. 8.10). A truncated cone has two bases: “lower” - the base of the original cone - and “upper” - the base of the cut off cone. According to the theorem on the section of a cone, the bases of a truncated cone are similar.

The altitude of a truncated cone is the perpendicular drawn from a point of one base to the plane of another. All such perpendiculars are equal (see section 3.5). Height is also called their length, i.e. the distance between the planes of the bases.

The truncated cone of revolution is obtained from the cone of revolution (Fig. 8.11). Therefore, its bases and all its sections parallel to them are circles with centers on the same straight line - on the axis. A truncated cone of revolution is obtained by rotating a rectangular trapezoid around its side perpendicular to the bases, or by rotating

isosceles trapezoid around the axis of symmetry (Fig. 8.12).

Lateral surface of a truncated cone of revolution

This is its part of the lateral surface of the cone of revolution from which it is derived. The surface of a truncated cone of revolution (or its full surface) consists of its bases and its lateral surface.

8.5. Images of cones of revolution and truncated cones of revolution.

A straight circular cone is drawn like this. First, draw an ellipse representing the circle of the base (Fig. 8.13). Then they find the center of the base - point O and draw a vertical segment PO, which depicts the height of the cone. From point P, tangent (reference) lines are drawn to the ellipse (practically this is done by eye, applying a ruler) and segments RA and PB of these lines are selected from point P to points of tangency A and B. Please note that segment AB is not the diameter of the base cone, and the triangle ARV is not the axial section of the cone. The axial section of the cone is a triangle APC: segment AC passes through point O. Invisible lines are drawn with strokes; The segment OP is often not drawn, but only mentally outlined in order to depict the top of the cone P directly above the center of the base - point O.

When depicting a truncated cone of revolution, it is convenient to first draw the cone from which the truncated cone is obtained (Fig. 8.14).

8.6. Conic sections. We have already said that the plane intersects the lateral surface of the cylinder of rotation along an ellipse (section 6.4). Also, the section of the lateral surface of a cone of rotation by a plane that does not intersect its base is an ellipse (Fig. 8.15). Therefore, an ellipse is called a conic section.

Conic sections also include other well-known curves - hyperbolas and parabolas. Let us consider an unbounded cone obtained by extending the lateral surface of the cone of revolution (Fig. 8.16). Let us intersect it with a plane a that does not pass through the vertex. If a intersects all the generators of the cone, then in the section, as already said, we obtain an ellipse (Fig. 8.15).

By rotating the OS plane, you can ensure that it intersects all the generatrices of the cone K, except one (to which the OS is parallel). Then in the cross section we get a parabola (Fig. 8.17). Finally, rotating the plane OS further, we transfer it to such a position that a, intersecting part of the generatrices of the cone K, no longer intersects infinite set its other constituents and parallel to two of them (Fig. 8.18). Then in the section of the cone K with the plane a we obtain a curve called a hyperbola (more precisely, one of its “branch”). Thus, a hyperbola, which is the graph of a function, is a special case of a hyperbola - an equilateral hyperbola, just as a circle is a special case of an ellipse.

Any hyperbolas can be obtained from equilateral hyperbolas using projection, in the same way as an ellipse is obtained by parallel projection of a circle.

To obtain both branches of the hyperbola, it is necessary to take a section of a cone that has two “cavities,” that is, a cone formed not by rays, but by straight lines containing the generatrices of the lateral surfaces of the cone of revolution (Fig. 8.19).

Conic sections were studied by ancient Greek geometers, and their theory was one of the peaks of ancient geometry. The most complete study of conic sections in antiquity was carried out by Apollonius of Perga (III century BC).

There are a number of important properties that combine ellipses, hyperbolas and parabolas into one class. For example, they exhaust the “non-degenerate”, i.e., curves that are not reducible to a point, line or pair of lines, which are defined on the plane in Cartesian coordinates by equations of the form

Conic sections play an important role in nature: bodies move in gravitational fields in elliptical, parabolic and hyperbolic orbits (remember Kepler's laws). The remarkable properties of conic sections are often used in science and technology, for example, in the manufacture of certain optical instruments or searchlights (the surface of the mirror in a searchlight is obtained by rotating the arc of a parabola around the axis of the parabola). Conical sections can be observed as the boundaries of the shadow of round lampshades (Fig. 8.20).