Slide 6

Axial section

If the secant plane passes through the axis of the cone, then the section is an isosceles triangle, the base of which is the diameter of the base of the cone, and the sides are the generatrices of the cone. This section is called axial.

Slide 7

If the secant plane is perpendicular to the axis OP of the cone, then the section of the cone is a circle with center O and located on the axis of the cone. The radius r1 of this circle is (OP / PO1) * r, where r is the radius of the base of the cone.

Slide 8

Cone surface area

Slide 9

The area of \u200b\u200bthe lateral surface of the cone is equal to the product of half the circumference of the base by the generator.

The total surface area of \u200b\u200ba cone is the sum of the lateral and base areas. To calculate the area SCON of the total surface of the cone, the formula is obtained

Slide 10

Sside \u003d πr (l + r)

Slide 11

Frustum

Slide 12

A plane perpendicular to the axis of the cone cuts off the smaller cone from it. The remainder is called a frusto-cone. A truncated cone can also be obtained as a body of revolution. A truncated cone is a body of revolution formed by the rotation of a rectangular trapezoid about a side perpendicular to the bases.

Slide 13

Slide 14

Truncated cone features

Base

Generating

Base

Side surface

Slide 15

Cones around us

Slide 16

Cones around us

L Consider a circle L with c with center O and a straight line OP perpendicular to the plane ß of this circle. O ß P Draw a straight line through point P and each point of the circle. The surface formed by these straight lines is called the conical surface, and the straight lines are called the generatrices of the conical surface.

L O ß P The circle is called the base of the cone. The circle is called the base of the cone. The top of the conical surface is the top of the cone. The top of the conical surface is the top of the cone. The segments of the generatrices, enclosed between the top and the base, are the generatrices of the cone, and the part of the conical surface formed by them is the lateral surface of the cone. The segments of the generatrices, enclosed between the top and the base, are the generatrices of the cone, and the part of the conical surface formed by them is the lateral surface of the cone.

L O ß P The axis of the conical surface is called the axis of the cone, and its segment. enclosed between the top and the base, - the height of the cone. The axis of the conical surface is called the axis of the cone, and its segment. enclosed between the top and the base, - the height of the cone.

If the section of the cone passes through the axis of the cone, then the section is an isosceles triangle, the base of which is the diameter of the base of the cone, and the sides are the generatrices of the cone. This section is called axial. If the section of the cone passes through the axis of the cone, then the section is an isosceles triangle, the base of which is the diameter of the base of the cone, and the sides are the generatrices of the cone. This section is called axial.

If the cutting plane is perpendicular to the axis of the cone, then the section of the cone is a circle with the center located on the axis of the cone. If the cutting plane is perpendicular to the axis of the cone, then the section of the cone is a circle with the center located on the axis of the cone. α r΄r΄r΄r΄ r О΄О΄О΄О΄ О Р The radius r΄ of this circle is equal to PO΄ / PO r, where r is the radius of the base of the cone. The radius r΄ of this circle is PO΄ / PO r, where r is the radius of the base of the cone.

The lateral surface of the cone, like the lateral surface of the cylinder, can be turned into a plane by cutting it along one of the generatrices. The lateral surface of the cone, like the lateral surface of the cylinder, can be turned onto a plane by cutting it along one of the generatrices. The sweep of the lateral surface of the cone is a circular sector, the radius of which is equal to the generatrix of the cone (PA \u003d r), and the length of the arc of the sector is equal to the circumference of the base of the cone. The sweep of the lateral surface of the cone is a circular sector, the radius of which is equal to the generatrix of the cone (PA \u003d r), and the length of the arc of the sector is equal to the circumference of the base of the cone. The area of \u200b\u200bits sweep is taken as the area of \u200b\u200bthe lateral surface of the cone, which is equal to the product of half the circumference of the base by the generator. The area of \u200b\u200bits sweep is taken as the area of \u200b\u200bthe lateral surface of the cone, which is equal to the product of half the circumference of the base by the generator. S \u003d πrl P A B R A B А΄А΄А΄А΄

The total surface area of \u200b\u200ba cone is the sum of the lateral and base areas. To calculate S of the total surface of the cone, the formula is obtained.The area of \u200b\u200bthe total surface of the cone is the sum of the areas of the lateral surface and the base. To calculate S of the total surface of the cone, we obtain the formula S \u003d πr (l + r) S \u003d πr (l + r)

Take an arbitrary cone and draw a cutting plane perpendicular to its axis. This plane intersects the cone in a circle and splits the cone into two parts. Take an arbitrary cone and draw a cutting plane perpendicular to its axis. This plane intersects the cone in a circle and splits the cone into two parts. О΄О΄О΄О΄ О Р One of the parts is a cone, and the other is called a truncated cone. The base of the original cone and the circle obtained in the section of this cone by a plane are called the bases of the truncated cone, and the segment connecting their centers is called the height truncated cone. One of the parts is a cone, and the other is called a truncated cone. The base of the original cone and the circle obtained in the section of this cone by a plane are called the bases of the truncated cone, and the segment connecting their centers is the height of the truncated cone.

The part of the conical surface that bounds the truncated cone is called its lateral surface, and the segments of the generatrices of the conical surface enclosed between the bases are called the generators of the truncated cone. All generators are equal to each other The part of the conical surface that bounds the truncated cone is called its lateral surface, and the segments of the generatrices of the conical surface enclosed between the bases are called generators of the truncated cone. All generators are equal to each other О΄О΄О΄О΄ О Р В А

A truncated cone can be obtained by rotating a rectangular trapezoid around its lateral side perpendicular to the bases A truncated cone can be obtained by rotating a rectangular trapezoid around its lateral side perpendicular to the bases B D А С

The area of \u200b\u200bthe lateral surface of the truncated cone is equal to the product of the half-sum of the lengths of the circles of the bases by the generator, i.e. The area of \u200b\u200bthe lateral surface of the truncated cone is equal to the product of the half-sum of the lengths of the circles of the bases by the generator, i.e. S \u003d π (r + r΄) l, where r and r΄ are the radii of the bases, l is the generatrix of the truncated cone. S \u003d π (r + r΄) l, where r and r΄ are the radii of the bases, l is the generatrix of the truncated cone. B D А С r r΄r΄r΄r΄

There are many interesting facts about the cone. In many religions and teachings, the cone has a cult significance. There are many rituals in which the magical properties of the cone are affected, for example, witches and witches have a ritual - the "cone of power". There are many interesting facts about the cone. In many religions and teachings, the cone has a cult significance. There are many rituals in which the magical properties of the cone are affected, for example, witches and witches have a ritual - the "cone of power".

And one more very interesting fact, no one wondered why ladies in the Middle Ages wore a long cone-cap on their heads? If you say that such was the fashion, then you are wrong. The answer is simple, they believed that energy was collected under the hood, which in turn would make them stronger and smarter. And one more very interesting fact, no one wondered why ladies in the Middle Ages wore a long cone-cap on their heads? If you say that the fashion was like that, then you are wrong. The answer is simple, they believed that energy was collected under the hood, which in turn would make them stronger and smarter.

Cone

Belobrova Tatyan A. Valerievna

Mathematics teacher of the highest category

MCOU Secondary School No. 1, Sim

Chelyabinsk region

Cone a body is called, which consists 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 connecting the top of the cone with the base points

• The cone is called straight if its height falls to the center of the base

• If the height of the cone does not fall to the center of the base, then the cone is called oblique

The elements cone

All generators of the cone are equal each other and make one corner with the base

Cone can be obtained by rotating a right-angled triangle around one of the legs.

In this case, the axis of rotation will be a straight line containing the height of the cone.

This line is called the axis of the cone.

SECTIONS OF THE CONE

Section of the cone by a plane passing through the apex and chord of the base

Axial section

Section of a cone with a plane parallel to the base

Section of a cone with a plane not parallel to the base

l \u003d R

L =2 π r

Flattening the lateral surface of the cone - a sector of a circle whose radius is equal to the length of the generatrix of the cone, and its arc length is equal to the circumference of the base of the cone, i.e. 2 π R

AREA OF THE SIDE SURFACE OF THE CONE

The area of \u200b\u200bits sweep is taken as the area of \u200b\u200bthe lateral surface of the cone

l \u003d R

S SIDE . = π rl

L =2 π r

AREA OF THE FULL SURFACE OF THE CONE

Full surface area

cone is called the sum

lateral surface areas

and grounds

l \u003d R

L =2 π r

S SIDE + S cr . = π rl + π r 2

S end = π r ( l + r )

Truncated cone

called the part of a full cone enclosed between the base and the secant plane parallel to the base

The lateral surface area of \u200b\u200bthe truncated cone

Municipal educational institution

secondary school No. 4

Lyudinovo, Kaluga region.

CONE

Geometry lesson in grade 11

Mathematic teacher

first qualification category:

Molotkova Svetlana Sergeevna

Cone

Geometry lesson in grade 11.

Goal:

introduce students to a geometric body - a cone.

Tasks:

Formation of concepts of conical surface, cone.

Ability to determine the types of sections.

Derivation of formulas for the lateral surface area and the total surface area of \u200b\u200bthe cone.

Ability to work with a drawing and read it.

Application of knowledge in solving problems.

Equipment in the lesson:

Media projector. The teacher's presentation, which is provided with hyperlinks, can be used in subsequent lessons to test knowledge. Each student has a paper model of a cone, scissors. Geometry: Textbook. For 10-11 cl. Wednesday School / L.S. Atanasyan, V.F.Butuzov, S.B. Kadomtsev and others.

1. Organizational moment.

Message of the topic of the lesson (slide 1), the objectives of the lesson. Checking readiness for the lesson.

We will build the lesson according to plan (slide 2).

2. Explanation of the new topic.

Consider a spatial figure - a "round" geometric body-cone. Consider the layout.

Consider the picture - slide 3. Using your knowledge of the topic "Cylinder", try to name the elements of the cone.

Let's write down new concepts in a notebook and build a drawing of a cone.

Let's consider what elements the cone consists of (slide 4).

3. Working off the acquired knowledge.

Reading the chart (slide 5).

Draw two generators on the layout. What can you say about them?

Cone-body of revolution .

With the help of rotation, what is the geometric shape of the cone? (slide 6)

4. Sections of the cone.

The teacher gives the definition of the section. The students take notes. On leading questions, students should say what this section is and its main properties. (slide 7-10).

5. Historical background.

Slide 11-12.

WORK IN GROUPS

6. Development of the cone.

Cut the paper layout along one of the cone generatrices.

What a figure it turned out. Sketch in a notebook. (slide 13).

SELF-CONSCIOUSNESS

7. Derivation of formulas for the lateral surface area and the total surface of the cone.

Using the unfolding of the cone, derive the formula for the lateral surface of the cone

Derive the formula for the full surface of a cone

PROMOTING RESULTS

(slide 14), (slide 15).

A representative from each group advertises the results.

8. Formation of skills and abilities.

Problem number 547 - orally (slide 16).

Problem number 549 (a) with a note in notebooks (slide 17).

9. Summing up. Reflection. Homework setting.

Slide 18.

If time permits, you can conduct a verification dictation with a self-test (slide 19).

Mathematics teacher Molotkova Svetlana Sergeevna