Definition of the cone. Cone as a geometric figure

Date: 21.09.2019

Obtained by combining all rays emanating from one point ( tops cone) and passing through a flat surface. Sometimes a cone is called 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 case 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 top and the base border is called generatrix of the cone.
The union of the generators of the 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 base plane (as well as 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 apex 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 axis of the cone.
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 called simply a cone) can be obtained by rotating a right-angled triangle around a straight line containing the leg (this straight line is 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 is equal to a 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 a finite volume lies at a quarter of the height from the base.
Solid angle at apex of a straight line circular cone is equal to

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

The lateral surface area of such a cone is

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 a vector space over a field, for which, for any

Properties

If the area of the base is finite, then the volume of the cone is also finite and is equal to a 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 a finite volume lies at a quarter of the height from the base.
The solid angle at the apex of the right circular cone is

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

The lateral surface area of such a cone is

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 a vector space over a field, for which, for any

- (Latin conus; Greek konos). A body bounded by a surface formed by the revolution of a straight line, one end of which is stationary (the top 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 (generating a cone) through a fixed point (apex of a cone) along a guide (base of a cone). The formed surface, enclosed between ... Big Polytechnic Encyclopedia

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

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

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

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

Class: 11 Lesson number 14 Date: ____________

Lesson topic: “Straight circular cone, its elements. Axial sections of the cone. Sections of the cone with a plane parallel to the base. Cone sweep "

The purpose of the lesson:

Introduce the concepts of a conical surface, a cone, elements of a cone (side surface, base, apex, generatrix, axis, height), the concept of a truncated cone;

Derive formulas for calculating the areas of the lateral and full surfaces of a cone and a truncated cone;

Teach students to solve problems on this topic.

Promote student creativity teaching material and their desire to improve themselves.

To bring up organization, discipline, responsibility for their work and the work of classmates.

Lesson type: learning new material.

Lesson equipment: interactive whiteboard, tables, cone models, material for making models: knitting needles, plane model (polystyrene), paper, glue, scissors, compasses, protractor, ruler.

Form of organization of student activities : G ruppa.

During the classes

1. Frontal work

Choose a cone from the proposed geometric shapes

Getting to know the tapered surface

Definition # 1 A conical surface is a surface formed by the movement of a straight line that passes through a given point and intersects a given flat line.

Straight line a - generator;

Flat line MN - guide.

Open conical surface

If the guide is closed, thenthe conical surface is closed.

Definition No. 2 Cone is called a body bounded by a closed conical surface and a plane intersecting it.

Acquaintance with the cone and its elements

A) Cone

SO a (SO =N, SO = h)

SO - cone height

SA - generator

S - the top of the cone

ABA curve -guide .

B) Let the rectangular rectangle SOA rotate around the leg SO; at full revolution, the hypotenuse AS describes a conical surface, leg OA describes a circle.

Such a body is calledcone of revolution ... (straight circular cone).

Straight circular cone

S - the top of the cone

SA - generator

SO = h - cone height

(cone axis - a)

The base of the cone is a circle (O; r)

O is the center of the foundation,

AO = OB = r - radius of the base of the circle

D SAB -axial section

a || b, b SO, a SO

Circle (o; r) ~ Circle (o1; r1)

The concept of a side (full) surface.

II. Group work (3-5 people)

(assignments are given to each group on a card)

Assignment on the topic "Cone"

1) Draw a cone. Determine all the elements of the cone from the picture.

2) Using the given cone model, construct a flat pattern of this cone. Determine the correspondence of the elements of the flat pattern of the cone, the drawing and the model of the cone.

3) Make a cone from a sheet of thick paper so that its full surface is: S110 cm2 with a base radius r3.1 cm.

Determine what tools you need for this, what calculations need to be done, what formulas will you have to remember, and what new ones to derive?

4) Fill out the work on site according to the plan:

A) What are your assigned responsibilities in the group in the process of completing assignments:

idea's generator;

constructor;

calculator;

designer;

manufacturer.

B) Describe the methods and approaches to solving the problem.

Necessary calculations for making a cone model. (Drawing. Formulas. Conclusion)

Making a cone.

5) The cone model is ready.

6) Create a formula to calculate the cross-sectional area parallel to the base of the cone and dividing the height of the cone in a ratio of 1: 3, counting from the top

7) Draw up a formula to calculate the cross-sectional area passing through the axis of the cone. What is the angle at the top of this section?

8) How can you get a truncated cone from your model? Calculate its total surface using tasks (6).

9) Create and solve three more problems on this topic.

Comment: the teacher acts as a consultant in solving problems, using prompting questions and relying on keywords.

One of the groups was given easier tasks:

№1. Fill in the blanks:

A straight line that forms a conical surface when moving is called ...;

The line crossed by the generator is called ... ..;

Taper of revolution - special case… When the base of the cone is .. and the base of the height is ..;

Section of the cone of revolution by a plane parallel to the base -…. Find the cross-sectional area.

If the axial section of the cone is an equilateral triangle, then the cone ... .. Make a drawing:

№2. Solve the problem by filling in the blanks.

In the sweep of the lateral surface of the cone, the central angle is 200 o... Find the angle between the generatrix and the base of the cone.

Given:BSB = 200 o, SA = L, ОВ = r

FindSAO

Solution:

1) a =360 o…..| cos x =…

2) 200 o=…

3) cosx=… , x -

A) ... generating;

B) ... guide;

B)… cone,…. Circle ..., center of the base

D)… circle,… section distance from the top of the cone;

D) ... called equilateral

B) 200 o= 360 o* cos x;

Home assignment.

Examine the truncated cone, solve problems No.

Lesson summary.

As a result of the work, the students

Themselves derived formulas for calculating the lateral and full surfaces of the cone

Drew a scan

We made the necessary calculations

Groups

L (cm)

9,2

3,1

21,1754

89,5528

110,7282

7,8

28,26

73,476

101,74

9,4

28,26

88,548

116,808

10,4

4,9

75,3914

160,0144

235,4058

Conducted research work,

We solved the problems

We constantly communicated with each other, learned to think and motivate our fellow workers.

We received not only the necessary knowledge, but also great pleasure.

Found out that the word "Cone" comes from the Greek word "xwnos", which meanscone.

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

Volume and areas:
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. Area of the lateral surface of a truncated cone

Definition
A body bounded from the sides by a conical surface, taken between its vertex 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 that does not lie in the plane of the base (top) and all segments connecting the top with points of the base.

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

Consider a line (curve, broken line, or blended) (for example, l), lying in some plane, and an arbitrary point (for example, M), not lying in this plane. All kinds of straight lines connecting point M with all points of this line l, form canonical surface... Point M is the vertex of such a surface, and the given line l - guide... All straight lines connecting point M with all points of the line l are called generators... A canonical surface is not limited to either its vertex or its guide. It extends indefinitely on either side of the summit. Now let the guide be a closed convex line. If the guide is a broken line, then the body, bounded from the sides by the canonical surface, taken between its top and the guide plane, and a flat base in the guide plane, is called a pyramid.
If the guide is a curved or mixed line, then the body, bounded laterally by the canonical surface taken between its top and the guide plane, and a flat base in the guide plane, 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 top of the cone and any of the points of the curve that bounds the shape of the base of the cone are called generators of the cone. Most often, in geometric problems, a 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.
A bottom with a limited mixed line is a very rare case. It is listed here only because it can be considered in geometry. The case with a curved guide is considered more often. Although, the case with an arbitrary curve, that the case with a mixed guide, is of little use and it is difficult to deduce any regularities in them. From the number of cones in the course of elementary geometry, a straight circular cone is studied.

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, you can define a circular cone.
Definition 2 ... A circular cone is a body that consists of a circle (base), a point that does not lie in the plane of the base (top) and all segments connecting the top with points of the base.
Definition 3 ... The height of the cone is the perpendicular dropped from the top to the plane of the base of the cone. You can select a cone, the height of which falls to the center of the flat shape of the base.
Definition 4 ... A straight cone is a cone whose base height contains the center of the base of the cone.
If we connect these two definitions, we get a cone, the base of which is a circle, and the height falls to the center of this circle.
Definition 5 ... A straight circular cone is called a cone, the base of which is a circle, and its height connects the top and center of the base of this cone. Such a cone is obtained by rotating a right-angled triangle around one of the legs. Therefore, a straight circular cone is a body of revolution and is also called a cone of revolution. Unless otherwise stated, for brevity, in what follows we will simply say a cone.
So here are some of the properties of the cone:
Theorem 1. All generators of the cone are equal. Proof. The MO height is perpendicular to all straight lines of the base by definition, perpendicular to the plane. Therefore, the triangles MOA, MOB and MOS are rectangular and equal in two legs (MO - common, OA = OB = OS - radii of the base. Therefore, the hypotenuses, ie 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 axis of the cone, so 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 isosceles triangle.
Theorem 1.1. The axial section of the cone is an isosceles triangle. So triangle AMB is isosceles, because its two sides MV and MA are generators. The AMB angle is the apex angle of the axial section.

In the section of a conical surface by a plane, curves of the second order are obtained - a circle, an ellipse, a parabola and a hyperbola. In a frequent case, with a certain location of the secant plane and when it passes through the apex of the cone (S∈γ), the circle and the ellipse degenerate into a point or one or two generators of the cone fall into the section.

Gives - a circle when the cutting plane is perpendicular to its axis and intersects all generating surfaces.

Gives - an ellipse when the cutting plane is not perpendicular to its axis and intersects all generating surfaces.

Let's construct an elliptical ω plane α occupying a general position.

Solving the problem on section of a straight circular cone plane is greatly simplified if the clipping plane is in the projection position.

By changing the projection planes, we translate the plane α from general position in the private - front-projection. On the frontal plane of projections V 1 build a trace of the plane α and the projection of the cone surface ω plane gives an ellipse, since the cutting plane intersects all generators of the cone. The ellipse is projected on the projection plane as a second-order curve.
On the trail of the plane α V take an arbitrary point 3" measure its distance from the projection plane H and put it off along the communication line already on the plane V 1 getting point 3" 1 ... The trail will pass through it αV 1... Cone section line ω - points A "1, E "1 coincides here with the trace of the plane. Next, we construct an auxiliary secant plane γ3, drawing on the frontal plane of projections V 1 her trace γ 3V 1... A construction plane intersecting with a tapered surface ω will give a circle, and intersecting with a plane α will give the horizontal line h3. In turn, the straight line intersecting with the circle gives the required points C` and K` intersection plane α with conical surface ω ... Frontal projections of the required points C "and K" plot as points belonging to the cutting plane α .

To find a point E (E`, E ") section lines, draw a horizontal projection plane through the top of the cone γ 2 H that will cross the plane α in a straight line 1-2(1`-2`, 1"-2") ... Crossing 1"-2" with a connection line gives a point E "- the highest point of the section line.

To find a point indicating the boundary of visibility of the frontal projection of the section line, draw a horizontal projection plane through the apex of the cone γ 5 H and find the horizontal projection F` the desired point. Also, the plane γ 5 H will cross the plane α frontally f (f`, f ")... Crossing f " with a connection line gives a point F "... We connect the points of a smooth curve obtained on a horizontal projection, marking on it the extreme left point G - one of the characteristic points of the intersection line.
Then, we build the projections G on the frontal planes of the projections V1 and V. We connect all the constructed points of the section line on the frontal plane of the projections V with a smooth line.

Gives - a parabola when the cutting plane is parallel to one generatrix of the cone.

When constructing projections of curves - conic sections, it is necessary to remember the theorem: the orthogonal projection of a plane section of a cone of revolution onto a plane perpendicular to its axis is a second-order curve and has one of its focuses an orthogonal projection onto this plane of the top of the cone.

Consider the construction of section projections when the cutting plane α parallel to one generatrix of the cone (SD).

In the section, you get a parabola with apex at the point A (A`, A ")... According to the theorem, the vertex of the cone S projected into focus S`... According to the well-known = R S` we determine the position of the directrix of the parabola. Subsequently, the points of the curve are plotted according to the equation p = R.

Creation of section projections when the cutting plane α parallel to one generatrix of the cone, the following can be performed:

Using auxiliary horizontal projection planes passing through the top of the cone γ 1 H and γ 2 H.

First, the frontal projections of the points are determined F ", G"- at the intersection of generators S "1", S "2" and the trail of the cutting plane α V... At the intersection of communication lines with γ 1 H and γ 2 H be defined F`, G`.

Other points of the section line can be defined similarly, for example D ", E" and D`, E`.

With the help of the auxiliary front-projection planes ⊥ of the cone axis γ 3 V and γ 4 V.

Projections of the section of construction planes and the cone on the plane H, there will be circles. Lines of intersection of construction planes with a cutting plane α there will be front-projection straight lines.

Gives - a hyperbola when the cutting plane is parallel to the two generatrices of the cone.

Definition of the cone. Cone as a geometric figure

Properties

Generalizations

see also

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

Properties

Generalizations

see also

See what "Straight circular cone" is in other dictionaries: