Cone and its elements. How to find the generatrix of a cone? Cone its types and properties

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 - a 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 guide 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.

) - a body in Euclidean space 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 that has a limited volume and is 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). If the base of a cone is a polygon, such a cone is a pyramid.

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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.
Cone angle- the angle between two opposite generatrices (the angle at the apex of the cone, inside the cone).
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, or conical layer.

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.

V = 1 3 S H , (\displaystyle V=(1 \over 3)SH,)

Where S- base area, H- height. Thus, all cones resting on a given base (of finite area) 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

2 π (1 − cos ⁡ α 2) , (\displaystyle 2\pi \left(1-\cos (\alpha \over 2)\right),) where α is the opening angle of the cone.

The lateral surface area of such a cone is equal to

S = π R l , (\displaystyle S=\pi Rl,)

and the total surface area (that is, the sum of the areas of the lateral surface and base)

S = π R (l + R), (\displaystyle S=\pi R(l+R),) Where R- base radius, l = R 2 + H 2 (\displaystyle l=(\sqrt (R^(2)+H^(2))))- length of the generatrix.

The volume of a circular (not necessarily straight) cone is equal to

V = 1 3 π R 2 H . (\displaystyle V=(1 \over 3)\pi R^(2)H.)

For a truncated cone (not necessarily straight and circular), the volume is equal to:

V = 1 3 (H S 2 − h S 1) , (\displaystyle V=(1 \over 3)(HS_(2)-hS_(1)),)

where S 1 and S 2 are the areas of the upper (closest to the top) and lower bases, respectively, h And H- distances from the plane of the upper and lower base, respectively, to the top.

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).

Cone equation

Equations defining the lateral surface of a right circular cone with an opening angle of 2Θ, a vertex at the origin and an axis coinciding with the axis Oz :

In a spherical coordinate system with coordinates ( r, φ, θ) :

θ = Θ. (\displaystyle \theta =\Theta.)

In a cylindrical coordinate system with coordinates ( r, φ, z) :

z = r ⋅ ctg ⁡ Θ (\displaystyle z=r\cdot \operatorname (ctg) \Theta ) or r = z ⋅ tan ⁡ Θ . (\displaystyle r=z\cdot \operatorname (tg) \Theta .)

In a Cartesian coordinate system with coordinates (x, y, z) :

z = ± x 2 + y 2 ⋅ cot ⁡ Θ . (\displaystyle z=\pm (\sqrt (x^(2)+y^(2)))\cdot \operatorname (ctg) \Theta .) This equation in canonical form is written as

where are the constants a, With determined by proportion c / a = cos ⁡ Θ / sin ⁡ Θ . (\displaystyle c/a=\cos \Theta /\sin \Theta .) This shows that the lateral surface of a right circular cone is a surface of the second order (it is called conical surface). In general, a second-order conical surface rests on an ellipse; in a suitable Cartesian coordinate system (axis Oh And OU parallel to the axes of the ellipse, the vertex of the cone coincides with the origin, the center of the ellipse lies on the axis Oz) its equation has the form

x 2 a 2 + y 2 b 2 − z 2 c 2 = 0 , (\displaystyle (\frac (x^(2))(a^(2)))+(\frac (y^(2))( b^(2)))-(\frac (z^(2))(c^(2)))=0,)

and a/c And b/c equal to the semi-axes of the ellipse. In the most general case, when a cone rests on an arbitrary flat surface, it can be shown that the equation of the lateral surface of the cone (with its vertex at the origin) is given by the equation f (x , y , z) = 0 , (\displaystyle f(x,y,z)=0,) where is the function f (x , y , z) (\displaystyle f(x,y,z)) is homogeneous, that is, satisfying the condition f (α x , α y , α z) = α n f (x , y , z) (\displaystyle f(\alpha x,\alpha y,\alpha z)=\alpha ^(n)f(x,y ,z)) for any real number α.

Scan

A right circular cone as a body of revolution is formed by a right triangle rotating around one of the legs, where h- the height of the cone from the center of the base to the top - is the leg of a right triangle around which rotation occurs. Second leg of a right triangle r- radius at the base of the cone. The hypotenuse of a right triangle is l- forming a cone.

Only two quantities can be used to create a cone scan r And l. Base radius r defines the circle of the base of the cone in the development, and the sector of the lateral surface of the cone is determined by the generatrix of the lateral surface l, which is the radius of the sector of the lateral surface. Sector angle φ (\displaystyle \varphi ) in the development of the lateral surface of the cone is determined by the formula:

φ = 360° ( r/l) .

In this lesson we will get acquainted with such a figure as a cone. Let's study the elements of a cone and the types of its sections. And we will find out which figure the cone has many properties in common with.

Fig.1. Cone-shaped objects

In the world, a huge number of things are shaped like a cone. Often we don't even notice them. Road cones warning of road works, roofs of castles and houses, ice cream cones - all these objects are shaped like a cone (see Fig. 1).

Rice. 2. Right Triangle

Consider an arbitrary right triangle with legs and (see Fig. 2).

Rice. 3. Straight circular cone

By rotating a given triangle around one of the legs (without loss of generality, let it be a leg), the hypotenuse will describe the surface, and the leg will describe the circle. Thus, a body will be obtained that is called a right circular cone (see Fig. 3).

Rice. 4. Types of cones

Since we are talking about a straight circular cone, apparently there is both an indirect and a non-circular one? If the base of a cone is a circle, but the vertex is not projected into the center of this circle, then such a cone is called inclined. If the base is not a circle, but an arbitrary figure, then such a body is also sometimes called a cone, but, of course, not circular (see Fig. 4).

Thus, we again come to the analogy already familiar to us from working with cylinders. In fact, a cone is something like a pyramid, it’s just that the pyramid has a polygon at the base, and the cone (which we will consider) has a circle (see Fig. 5).

The segment of the axis of rotation (in our case this is the leg) enclosed inside the cone is called the axis of the cone (see Fig. 6).

Rice. 5. Cone and pyramid

Rice. 6. - cone axis

Rice. 7. Base of the cone

The circle formed by the rotation of the second leg () is called the base of the cone (see Fig. 7).

And the length of this leg is the radius of the base of the cone (or, more simply, the radius of the cone) (see Fig. 8).

Rice. 8. - cone radius

Rice. 9. - top of the cone

The vertex of an acute angle of a rotating triangle lying on the axis of rotation is called the vertex of a cone (see Fig. 9).

Rice. 10. - cone height

The height of a cone is a segment drawn from the top of the cone perpendicular to its base (see Fig. 10).

Here you may have a question: how then does the segment of the axis of rotation differ from the height of the cone? In fact, they coincide only in the case of a straight cone; if you look at an inclined cone, you will notice that these are two completely different segments (see Fig. 11).

Rice. 11. Height in an inclined cone

Let's go back to the straight cone.

Rice. 12. Generators of the cone

The segments connecting the vertex of the cone with the points of the circle of its base are called generators of the cone. By the way, all the generatrices of a right cone are equal to each other (see Fig. 12).

Rice. 13. Natural cone-like objects

Translated from Greek, konos means “pine cone.” In nature there are enough objects that have the shape of a cone: spruce, mountain, anthill, etc. (see Fig. 13).

But we are used to the fact that the cone is straight. It has equal generatrices, and its height coincides with the axis. We called such a cone a straight cone. In school geometry courses, straight cones are usually considered, and by default any cone is considered a right circular one. But we have already said that there are not only straight cones, but also inclined ones.

Rice. 14. Perpendicular section

Let's return to straight cones. “Cut” the cone with a plane perpendicular to the axis (see Fig. 14).

What figure will be on the cut? Of course it's a circle! Let us remember that the plane runs perpendicular to the axis, and therefore parallel to the base, which is a circle.

Rice. 15. Inclined section

Now let's gradually tilt the section plane. Then our circle will begin to gradually turn into an increasingly elongated oval. But only until the section plane collides with the base circle (see Fig. 15).

Rice. 16. Types of sections using the example of a carrot

Those who like to explore the world experimentally can verify this with the help of a carrot and a knife (try cutting slices from a carrot at different angles) (see Fig. 16).

Rice. 17. Axial section of the cone

The section of a cone by a plane passing through its axis is called the axial section of the cone (see Fig. 17).

Rice. 18. Isosceles triangle - sectional figure

Here we get a completely different sectional figure: a triangle. This triangle is isosceles (see Fig. 18).

In this lesson we learned about the cylindrical surface, types of cylinder, elements of a cylinder and the similarity of a cylinder to a prism.

The generatrix of the cone is 12 cm and inclined to the plane of the base at an angle of 30 degrees. Find the axial cross-sectional area of the cone.

Solution

Let us consider the required axial section. This is an isosceles triangle in which the sides are 12 degrees and the base angle is 30 degrees. Then you can proceed in different ways. Or you can draw the height, find it (half of the hypotenuse, 6), then the base (according to the Pythagorean theorem), and then the area.

Rice. 19. Illustration for the problem

Or immediately find the angle at the vertex - 120 degrees - and calculate the area as the half-product of the sides and the sine of the angle between them (the answer will be the same).

Geometry. Textbook for grades 10-11. Atanasyan L.S. and others. 18th ed. - M.: Education, 2009. - 255 p.
Geometry 11th grade, A.V. Pogorelov, M.: Education, 2002
Workbook on geometry 11th grade, V.F. Butuzov, Yu.A. Glazkov

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Homework

Definition. Top of the cone is the point (K) from which the rays originate.

Definition. Cone base is the plane formed by the intersection of a flat surface and all the rays emanating from the top of the cone. A cone can have bases such as circle, ellipse, hyperbola and parabola.

Definition. Generatrix of the cone(L) is any segment that connects the vertex of the cone with the boundary of the base of the cone. The generatrix is a segment of the ray emerging from the vertex of the cone.

Formula. Generator length(L) of a right circular cone through the radius R and height H (via the Pythagorean theorem):

Definition. Guide cone is a curve that describes the contour of the base of the cone.

Definition. Side surface cone is the totality of all the constituents of the cone. That is, the surface that is formed by the movement of the generatrix along the cone guide.

Definition. Surface The cone consists of the side surface and the base of the cone.

Definition. Height cone (H) is a segment that extends from the top of the cone and is perpendicular to its base.

Definition. Axis cone (a) is a straight line passing through the top of the cone and the center of the base of the cone.

Definition. Taper (C) cone is the ratio of the diameter of the base of the cone to its height. In the case of a truncated cone, this is the ratio of the difference in the diameters of the cross sections D and d of the truncated cone to the distance between them: where R is the radius of the base, and H is the height of the cone.

Which emanate from one point (the top of the cone) and which pass through a flat surface.

It happens that a cone is a part of a body that has a limited volume and is obtained by combining each segment that connects the vertex and points of a flat surface. The latter, in this case, is base of the cone, and the cone is said to rest on this base.

When the base of a cone is a polygon, it is already pyramid .

Circular cone- this is a body consisting of a circle (the base of the cone), a point that does not lie in the plane of this circle (the top of the cone and all segments that connect the top of the cone with the points of the base).

The segments that connect the vertex of the cone and the points of the base circle are called forming a cone. The surface of the cone consists of a base and a side surface.

The lateral surface area is correct n-a carbon pyramid inscribed in a cone:

S n =½P n l n,

Where Pn- the perimeter of the base of the pyramid, and l n- apothem.

By the same principle: for the lateral surface area of a truncated cone with base radii R 1, R 2 and forming l we get the following formula:

S=(R 1 +R 2)l.

Straight and oblique circular cones with equal base and height. These bodies have the same volume:

Properties of a cone.

When the area of the base has a limit, it means that the volume of the cone also has a limit and is equal to the third part of the product of the height and the area of the base.

Where S- base area, H- height.

Thus, each cone that rests on this base and has a vertex that is located on a plane parallel to the base has equal volume, since their heights are the same.

The center of gravity of each cone with a volume having a limit is located at a quarter of the height from the base.
The solid angle at the vertex of a right circular cone can be expressed by the following formula: