What is called the axis of the cone. Geometric bodies

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 P n- 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:

Where α - cone opening angle.

• The lateral surface area of ​​such a cone, formula:

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

S=πR(l+R),

Where R- radius of the base, l— length of the generatrix.

• Volume of a circular cone, formula:

• For a truncated cone (not just straight or circular), volume, formula:

Where S 1 And S 2- area of ​​the upper and lower bases,

h And H- distances from the plane of the upper and lower base to the top.

• The intersection of a plane with a right circular cone is one of the conic sections.

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, draw tangent (reference) lines to the ellipse (practically this is done by eye, applying a ruler) and select segments RA and PB of these lines 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 will transfer it to such a position that a, intersecting part of the generators of the cone K, does not intersect the infinite number of its other generators and is 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).

Cone (from Greek "konos")- Pine cone. The cone has been known to people since ancient times. In 1906, the book “On the Method”, written by Archimedes (287-212 BC), was discovered; this book gives a solution to the problem of the volume of the common part of intersecting cylinders. Archimedes says that this discovery belongs to the ancient Greek philosopher Democritus (470-380 BC), who, using this principle, obtained formulas for calculating the volume of a pyramid and a cone.

A cone (circular cone) is a body that consists of a circle - the base of the cone, a point not belonging to the plane of this circle - the vertex of the cone and all segments connecting the vertex of the cone and the points of the base circle. The segments that connect the vertex of the cone with the points of the base circle are called generators of the cone. The surface of the cone consists of a base and a side surface.

A cone is called straight if the straight line that connects the top of the cone with the center of the base is perpendicular to the plane of the base. A right circular cone can be considered as a body obtained by rotating a right triangle around its leg as an axis.

The height of a cone is the perpendicular descended from its top to the plane of the base. For a straight cone, the base of the height coincides with the center of the base. The axis of a right cone is the straight line containing its height.

The section of a cone by a plane passing through the generatrix of the cone and perpendicular to the axial section drawn through this generatrix is ​​called the tangent plane of the cone.

A plane perpendicular to the cone axis intersects the cone in a circle, and the lateral surface intersects a circle centered on the cone axis.

A plane perpendicular to the axis of the cone cuts off a smaller cone from it. The remaining part is called a truncated cone.

The volume of a cone is 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 lateral surface area of ​​the cone can be found using the formula:

S side = πRl,

The total surface area of ​​the cone is found by the formula:

S con = πRl + πR 2,

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

The volume of a circular cone is equal to

V = 1/3 πR 2 H,

where R is the radius of the base, H is the height of the cone

The lateral surface area of ​​a truncated cone can be found using the formula:

S side = π(R + r)l,

The total surface area of ​​a truncated cone can be found using the formula:

S con = πR 2 + πr 2 + π(R + r)l,

where R is the radius of the lower base, r is the radius of the upper base, l is the length of the generatrix.

The volume of a truncated cone can be found as follows:

V = 1/3 πH(R 2 + Rr + r 2),

where R is the radius of the lower base, r is the radius of the upper base, H is the height of the cone.

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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.

Introduction

Rice. 1. Objects from life that have the shape of a truncated ko-nu-sa

Where do you think new figures come from in geometry? It’s all very simple: a person in life has become with similar objects and comes, as if call them. Let's look at the cabinet on which the lions in the circus sit, a piece of carrots that are harvested when we're just about -a part of it, an active volcano and, for example, light from the fo-na-ri-ka (see Fig. 1).

Truncated cone, its elements and axial section

Rice. 2. Geo-met-ri-che-fi-gu-ry

We see that all these figures are of a similar shape - both from below and from above they are bounded by circles, but they narrow towards the top ( see Fig. 2).

Rice. 3. From the upper part of the co-nu-sa

It looks like a cone. Just not enough top-hush. We mentally imagine that we take a cone and remove the upper part from it with one swing of a sharp sword (see Fig. 3).

Rice. 4. Truncated cone

This is exactly our figure; it is called a truncated cone (see Fig. 4).

Rice. 5. Se-che-nie, parallel-os-no-va-niyu ko-nu-sa

Let a cone be given. Let's create a plane, a parallel plane of the axis of this co-nu-sa and a cross-cutting cone (see. Fig. 5).

It will split the cone into two bodies: one of them is a cone of smaller size, and the second is called a truncated cone ( see Fig. 6).

Rice. 6. Obtained bodies in a parallel section

Thus, a truncated cone is a part of the cone, connected between its main body and the parallel main body. but flat. As in the case of a cone, a truncated cone can have a circle as its base - in this case it is called a circle. If the original cone was straight, then the truncated cone is called straight. As in the case of ko-nu-sa-mi, we will look at the keys, but straight circular truncated ko-nu-s sy, if it is not specifically indicated that we are talking about an indirect truncated co-nu-se or in its basis there are no circles.

Rice. 7. Rotation of a rectangular trap

Our global theme is bodies of rotation. A truncated cone is not an exception! Let us remember that in order to obtain a co-nu-sa, we smo-mat-ri-va-li a rectangular triangle and rotate it around ka-te-ta? If the resulting cone is cut with a plane parallel to the axis, then there will be no straight line left from the triangle -mo-coal-trape-tion. Its rotation around the smaller side will give us a truncated cone. Let us note again that we are, obviously, talking only about a direct circular co-nu-se (see Fig. 7).

Rice. 8. Os-no-va-niya truncated-no-go ko-nu-sa

I’ll make a few preparations. The basis of the half-ko-nu-sa and the circle, half-cha-yu-shay in the section of the ko-nu-sa flat, on- they call os-no-va-ni-ya-mi truncated ko-nu-sa (lower and upper) (see Fig. 8).

Rice. 9. Ob-ra-zu-yu-schi truncated ko-nu-sa

From the cuttings of the ra-zu-yu-shih half of the co-nu-sa, connected between the os-but-va-ni-mi truncated-but- go ko-nu-sa, they call about-ra-zu-yu-schi-mi truncated-no-go ko-nu-sa. Since all educational outcomes are equal and all educational outcomes are from the same are equal, then the ob-ra-zu-yu truncated co-nu-sa are equal (do not confuse the truncated and the truncated!). From here follows the equality of the tra-pe-tion of the axis of the sec-tion (see Fig. 9).

From the axis of rotation, enclosed inside the truncated co-nu-sa, they call it the axis of the truncated axis ko-nu-sa. This re-cut, ra-zu-me-et-sya, unites the centers of its fundamentals (see Fig. 10).

Rice. 10. Axis of truncated ko-nu-sa

You-so-ta truncated ko-nu-sa is a per-pen-di-ku-lyar, pro-ve-den from the point of one of the os-no-va- niya to another base. Most often, in the quality of you, you have truncated its axis.

Rice. 11. Ose-voe se-che-nie truncated-no-go-ko-nu-sa

The axial section of a truncated co-nu-sa is the section passing through its axis. It has the form of a trapezoid, a little later we will show its equality (see Fig. 11).

Areas of the lateral and total surfaces of a truncated cone

Rice. 12. Cone with introduced symbols

Let's find the area of ​​the bo-co-voy on the top of the truncated ko-nu-sa. Let the bases of the truncated co-nu-sa have radii and , and let the ob-ra-zu-yu be equal (see Fig. 12).

Rice. 13. Designation of the ob-ra-zu-yu-shchei from-se-chen-no-th ko-nu-sa

Let's find the area of ​​the bo-ko-voy on top of the truncated co-nu-sa as the difference in the area of ​​bo-ko-voys on the top-but- ste-khod-no-go ko-nu-sa and from-se-chen-no-go. To do this, we denote through the formation of the ko-nu-sa (see Fig. 13).

Then is-ko-may.

Rice. 14. Similar triangles

All that's left is for you to figure it out.

Let us note that from po-do-biy tri-corn-ni-kov, from-to-yes (see Fig. 14).

It would be possible to express it by dividing it into the difference between the radii, but we don’t need this, because in the ex- pression it is precisely the fi- gu-ri-ru-et pro-iz-ve-de-nie. Substituting instead of it, we finally have: .

Now it’s not difficult to get a shape for a full surface area. To do this, add exactly the area of ​​the two circles of the bases: .

Task

Rice. 15. Illu-stration to for-da-che

Let the truncated cone be rotated by a rectangular trap around its height. The middle line of the trapezoid is equal to , and the larger side is equal to (see Fig. 15). Find the area of ​​the bo-co-voy on the top-no-sti of the truncated ko-nu-sa.

Solution

From the formula we know that .

The formation of the ko-nu-sa will be a large hundred-ro-on-going tra-pe-tion, that is, Ra-di-u-sy ko- well-sa - this is the basis of the tra-pe-tion. We can't find them. But we don’t need it: we only need their sum, and the sum of the bases of a trapezoid is twice as large as its midline, that is, it is equal to . Then .

Similarities between truncated cones and pyramids

Pay attention to the fact that when we talk about co-nu-se, we talk about it between him and pi -ra-mi-doy - the formulas were analogous. It’s the same here, because a truncated cone is very similar to a truncated pi-ra-mi-du, so the formulas for the area are large and complete top-not-stey truncated ko-nu-sa and pi-ra-mi-dy (and soon there will be formulas for volume) analog-lo-gic- us.

Task

Rice. 1. Illu-strat-tion to for-da-che

The ra-di-u-sy os-no-va-niy use-chen-no-go ko-nu-sa are equal to and , and the ob-ra-zu-yu-shchaya is equal to . Find the truncated co-nu-sa and the area of ​​its axis (see Fig. 1).