A truncated cone is a body obtained by rotation. Straight circular cone

A truncated cone is obtained if a smaller cone is cut off from the cone by 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 the cone, the bases of the truncated cone are similar.

The height of a truncated cone is the perpendicular dropped from a point of one base to the plane of another. All such perpendiculars are equal (see Sec. 3.5). The 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 one straight line - on the axis. A truncated cone of revolution is obtained by rotating rectangular trapezoid around her side perpendicular to the bases, or rotation

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

Lateral surface of a truncated cone of revolution

This is the part of the lateral surface of the cone of revolution belonging to it, from which it is obtained. 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.

Straight circular cone draw like this. First, an ellipse is drawn representing the circumference of the base (Fig. 8.13). Then they find the center of the base - point O and vertically draw a segment RO, which depicts the height of the cone. From point P, tangent (reference) straight 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 touch points A and B. Please note that segment AB is not the diameter of the base cone, and the triangle ARV is not an axial section of the cone. The axial section of the cone is the triangle APC: the segment AC passes through the 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.

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 side surface of a cylinder of revolution the plane intersects along an ellipse (section 6.4). Also, the section of the lateral surface of the cone of revolution by a plane that does not intersect its base is an ellipse (Fig. 8.15). Therefore, the ellipse is called a conic section.

Conic sections also include other well-known curves - hyperbolas and parabolas. 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 not passing through the vertex. If a intersects all the generators of the cone, then in the section, as already mentioned, we get an ellipse (Fig. 8.15).

By rotating the OS plane, it is possible to ensure that it intersects all generators of the cone K, except for one (which the OS is parallel to). Then in the section we get a parabola (Fig. 8.17). Finally, rotating the OS plane further, we transfer it to such a position that a, crossing part of the generators of the cone K, does not intersect an 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 "branches"). So, the hyperbola, which is the graph of a function special case hyperbolas are isosceles hyperbolas, just as a circle is a special case of an ellipse.

Any hyperbolas can be obtained from isosceles by projection, just as an ellipse is obtained by parallel projection of a circle.

To obtain both branches of the hyperbola, one must take a section of a cone that has two "cavities", that is, a cone formed not by rays, but by straight lines containing generatrixes of the lateral surface of the cone of revolution (Fig. 8.19).

Conic sections were studied by the ancient Greek geometers, and their theory was one of the pinnacles of ancient geometry. Most full study conic sections in ancient times 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 "non-degenerate", i.e., not reducible to a point, a straight line, or a pair of straight lines, curves that are defined on a plane in Cartesian coordinates equations of the form

Conic sections play an important role in nature: bodies move along elliptical, parabolic and hyperbolic orbits in a gravitational field (remember Kepler's laws). The remarkable properties of conic sections are often used in science and technology, for example, in the manufacture of some optical instruments or searchlights (the surface of a 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 from round lampshades (Fig. 8.20).

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

"== Related definitions ==

The line segment that connects 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 generatrix of a cone is a conical surface.
A segment dropped perpendicularly from the vertex to the plane of the base (and also the length of such a segment) is called cone height.
If the base of the 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. The line connecting the vertex and the center of the base is called cone axis.
oblique (inclined) cone - a cone in which the orthogonal projection of the vertex to the base does not coincide with its center of symmetry.
circular cone A cone whose base is a circle.
Straight circular cone(often referred to simply as 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 based on an ellipse, parabola or hyperbola is called respectively elliptical, parabolic and hyperbolic cone(the last two have infinite volume).
The part of a cone that lies between the base and a plane parallel to the base and between the apex and base is called truncated cone.

Properties

If the area of the base is finite, then the volume of the cone is also finite and 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 the same volume, since their heights are equal.
The center of gravity of any cone with 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, twice 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

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 secant plane).

Generalizations

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

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

CONE- (1) in elementary geometry, a geometric body bounded by a surface formed by the movement of a straight line (cone generatrix) through a fixed point (cone apex) along a guide (cone base). The formed surface enclosed between ... Great Polytechnic Encyclopedia

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

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

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

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

Cone (from the Greek "konos")- Pine cone. The cone has been familiar to people since ancient times. In 1906, the book "On the Method", written by Archimedes (287-212 BC), was discovered, in this book a solution is given 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, with the help of this principle received formulas for calculating the volume of a pyramid and a cone.

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

A cone is called straight if the line that connects the vertex 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 drawn from its top to the plane of its base. For a right cone, the base of the height coincides with the center of the base. The axis of a right cone is a 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 axis of the cone intersects the cone in a circle, and the lateral surface in a circle centered on the axis of the cone.

A plane perpendicular to the axis of the cone cuts off a smaller cone from it. The rest 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 the same volume, since their heights are equal.

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

S side \u003d πRl,

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

S con \u003d 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

V = 1/3 πR 2 H,

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

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

S side = π(R + r)l,

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

S con \u003d 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.

Volume truncated cone can be found like this:

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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"== Related definitions ==

The line segment that connects 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 generatrix of a cone is a conical surface.
A segment dropped perpendicularly from the vertex to the plane of the base (and also the length of such a segment) is called cone height.
If the base of the 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. The line connecting the vertex and the center of the base is called cone axis.
oblique (inclined) cone - a cone in which the orthogonal projection of the vertex to the base does not coincide with its center of symmetry.
circular cone A cone whose base is a circle.
Straight circular cone(often referred to simply as 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 based on an ellipse, parabola or hyperbola is called respectively elliptical, parabolic and hyperbolic cone(the last two have infinite volume).
The part of a cone that lies between the base and a plane parallel to the base and between the apex and base is called truncated cone.

Properties

If the area of the base is finite, then the volume of the cone is also finite and 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 the same volume, since their heights are equal.
The center of gravity of any cone with 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, twice 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

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 secant plane).

Generalizations

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

A truncated cone is a body obtained by rotation. Straight circular cone

Properties

Generalizations

see also

See what the "Direct circular cone" is in other dictionaries:

Properties

Generalizations

see also

See what "Cone (geometric figure)" is in other dictionaries: