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The date of the designation will set the cone. Cone і th element. Ways to remove end surfaces on a turning bench

Be it clear, be it a line l (a curve for a laman), but lie in a deyak area (Fig. 386, a, b), and a point M, but not lie at the same area. All the straight lines, so that the point M is connected with the points of the line to set the surface a; such a surface is called an end surface, a point is a top, a line is a direct, straight line. In fig. 386 mi is not surrounded by the surface but by the top, ale rosum is not intertwined in the offense of the sides towards the summit.

If the end surface is rosy, be it a flat area, parallel to a straight line, then in a period of a straight line (a curve for a laman, in a fallowness because of the fact that the top is crooked or a laman line of homogeneity), a homosexual Effectively, the position of any kind of general appearance will be permanent:

Otzhe, overflowing conic surface areas, parallel to areas directly, similar and similarly rosette, centered on the top of the conical surface; it is good for any parallel areas, but do not pass through the top of the surface.

Nekhai is now straight - the line opucleus is closed (curve in Fig. 387, a, lamana in Fig. 387, b). Tilo, surrounded from the sides by a flat surface, taken by a top and a directing area, and a flat edge in a straight area, is called a cone (which is a curved line) or a crooked line.

Let us classify ourselves for a number of sides of the bagatokutnik, but lie in its foundation. Seems about tricot, chotiric and zagal-coal pyramids. Savoringly, scho-angular pyramid of maє between: bichny faces and eddies. At the tops of the pyramid, there is a faceted cut with flat and dihedral cut.

Stinks are called flat kuts at the tops and dihedral kutas at the side edges. At the tops of the mamo trigranny kutіv; These flat kuti, set by the edges and sides, are called flat kuts when presented, dihedral kuti and by the side edges and the area of the base - by dihedral kutas when presented.

Trikutna pіramіda inakshe is called a tetrahedron (i.e., Chetirekhgrannіkami). Be-yak from її facets can be taken as a basis.

Piracy is called correct when two minds are victorious:

2) the height, lowered from the top of the pedestrian to the subway, rewounding to the center of the bagatokutnik (іnakshe kazhuchi, the top of the pіramidi project to the center of the submission).

It’s great, that the correct pirama is not є, vzagali, apparently, a regular polyhedron!

Significantly, the actions of the power of the correct-angle piracy. Draw through the top of such a pyramid to the height SO (Fig. 388).

We will turn the whole period as a whole around a circle of height on the cube. With such a turn, the bagatokutnik let go in itself: the skin from the top of the loan is suspended. The summit of the feast and the twilight of the face (wrap around!)

The stars are: all the common edges are equal to oneself, all the edges are the same are the two-sided cutouts, all the two-sided cutouts are at the base of the cutouts, all the cutouts are at the base of the cutouts, all the cutouts are at the base of the cutouts.

Three cones in the course of elementary geometry are straight circular cone, i.e. such a cone, the base of such a spike, and the top is projected to the center of the spike.

A straight circular cone of readings in Fig. 389. If it is drawn through the top of the cone at the height of SO and if the cone is turned around the center of the height to the full cut, then the circumference of the cone will be on its own; The height and the top fall on the ground, so when you turn to be a kut cone, it will succeed by itself. It can be seen that the spring is growing, but everything is going to make the cone equal to itself and, however, to the area of the base. The overflowing of the cone with areas, which passes through its height, will be equal-sided tricytes, equal to oneself. The entire cone goes through the wrapping of the SOA rectangular tricycle near the th leg (like the length of the cone). To that, a straight circular cone є wraparound і is also called a wraparound cone. It is not meant for the sake of it, but for the stiffness of the nadal, we simply say "a cone", a cone wrap around.

Peretinu a cone with areas parallel to the area of your sleep, the essence of a stake (I wish that the stench is homothetical).

Zavdannya. Two-sided kuti when presented with the correct three-sided pyramid. Know dihedral kuti with bichny ribs.

Decision. Apparently, the time is the side of the base of the pіramіdi through а. It is carried out overriding the area to avenge the SO and the median presented by the AM (Fig. 390).

( tops cone) і pass through the flat surface. In some cases, a cone is called a part of such a body, so that it is possible to encircle the obsyag and reject all of them, so that the top and point of a flat surface can be found (I will call before cone, and call the cone spiral on the given pіdstavu). Yaksho base of the cone is a bogatokutnik, such a cone is pyramid.

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✪ Yak cut a cone with a paper.

subtitle

knitted

Vidrizok, where the summit and the cordon are sent, are called I will make a cone.
Create a cone to be called I'll say(abo bichny) on top of the cone... I will make the surface of the cone є a tapering surface.
As a result, descents perpendicularly from the top to the base area (as well as such a drop), are called with a cone.
Kut to the length of the cone- kut mіzh two houses opposed to fit (kut at the top of the cone, in the middle of the cone).
Also, the base of the cone has a center of symmetry (for example, є by a pole or an ellipse) and an orthogonal projection of the apex of the cone onto the area, and be set with the center of the cone, then the cone is called straight... When there is a straight line, it is vissyu cone.
oblique (abduction) A cone is a cone, at which the orthogonal projection of the vertex on the side does not go beyond the center of the symmetry.
circular cone- a cone, the base of the yak є stake.
Straight circular cone(Often called yogo just a cone) you can trim the wraps of a rectangular tricycle to a straight line, so that the leg can be avenged (a straight line є the length of the cone).
A cone that spirals onto an ellips, a parabola or a hyperbola, is called eliptic, parabolicі hyperbolic cone(Remain two may not endless volume).
The part of the cone, which lies between the ground and the area, parallel to the front and located between the top and the bottom, is called increase with a cone, abo end ball.

power

If the area of the base is kintsev, then the cone is also kintseviy and the third additional space is on the area of the base.

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

de S- area for presentation, H- visota. In such a rank, all the cones, which spiral on the ground at the end (the end of the area) and the top, which is located on the same area, parallel to the top, may be large and small, some of the height.

The center of gravity of any cone with a kintsevym volume to lie on the four-point height from the front.
Tіlesny kut at the apex of a straight circular cone dorіvnyu

2 π (1 - cos ⁡ α 2), (\ displaystyle 2 \ pi \ left (1 \ cos (\ alpha \ over 2) \ right),) de α - cut to the length of the cone.

The area of the bichesky surface of such a cone is

S = π R l, (\ displaystyle S = \ pi Rl,)

but on the surface of the surface (tobto the sum of the area of the common surface and the front)

S = π R (l + R), (\ displaystyle S = \ pi R (l + R),) de R- radius submission, l = R 2 + H 2 (\ displaystyle l = (\ sqrt (R ^ (2) + H ^ (2))))- Dovzhina tvіrnoї.

Obsyag circular (not obov'yazkovo straight) cone dorіvnyu

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 following should be done:

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

de S 1 and S 2 - areas along the upper (closest to the top) and lower parts, hі H- from the area from the top and bottom to the top.

Peretin the area with a straight circular cone є one of the end peretsiniv (in non-vigorous fall - ellips, a parabola or a hyperbola, in fallowness from the position of the other area).

cone flush

Rivnyannya, to set the bichnu surface of a straight circular cone with a cutout 2Θ, with the top on the cob of coordinates and vissyu, to get out of vissyu Oz :

In spherical coordinate systems 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) abo r = z ⋅ tg ⁡ Θ. (\ Displaystyle r = z \ cdot \ operatorname (tg) \ theta.)

Cartesian coordinate systems with coordinates (x, y, z) :

z = ± x 2 + y 2 ⋅ ctg ⁡ Θ. (\ Displaystyle z = \ pm (\ sqrt (x ^ (2) + y ^ (2))) \ cdot \ operatorname (ctg) \ theta.) Tse rivnyannya in the canonical viglyadi enroll yak

de constant a, s in proportion to c / a = cos ⁡ Θ / sin ⁡ Θ. (\ Displaystyle c / a = \ cos \ theta / \ sin \ theta.) It can be seen that the lateral surface of a straight circular cone is a surface of a different order (I will call it end surface). In the zagalny viglyadі, the surface of a different order spirals onto the elips; in all types of Cartesian coordinate systems (axes Ohі OU parallel to the axes of the ellipse, the apex of the cone is set on the cob of coordinates, the center of the ellipse is to lie on the axis Oz) Її рівняння maє viglyad

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,)

whereby a / cі b / c to the semi-axes of the ellipse. At the most extreme, if the cone spirals onto a fairly flat surface, it is possible to show how the equal surface of the cone (from the top to the cob of coordinates) is set to equal f (x, y, z) = 0, (\ displaystyle f (x, y, z) = 0,) de function f (x, y, z) (\ displaystyle f (x, y, z))є one-sided, to be happy with the mind f (α x, α y, α z) = α nf (x, y, z) (\ displaystyle f (\ alpha x, \ alpha y, \ alpha z) = \ alpha ^ (n) f (x, y , z)) for any useful number α.

branch

Straight circular cone yak tilo wrapping statements h- the height of the cone from the center to the top to the top - є the leg of the rectangular tricycle, near which one is wrapped. The other leg of the rectangular tricycle r- radius at the base of the cone. Hypotenuse of rectangular tricycle є l- I will fix the cone.

At the stem of the cone, there may be vicoristovuvat less than two sizes rі l... radius pіdstavi r form in the rozgorttsi colo before the cone, and the sector of the bichniy surface of the cone is l, Scho є the radius of the sector of the bichniy surface. sector cut φ (\ displaystyle \ varphi) in the rozgorttsi bichniy surface of the cone, start for the formula:

φ = 360 ° ( r/l) .

Lesson topic: Cone і th element

Meta lesson:to introduce an understanding of the cone, scho asserted, with a head and a present; to introduce an understanding of the area of the bichesky surface of the cone yak of the area of the rosette; formulate the solution of tasks for the knowledge of the elements of the cone.

Lesson type:combinations.

ustatkuvannya:PC, multimedia projector, interactive board, cone models.

Go to lesson:

Perevirka homework at the nursery.
Self-operation of the robot (Dodatok 1.)
Explanation of the new material.

The understanding of the cone, its elements (top, hanging, setting, pidstava, lateral surface). Image of the cone.

cone(More precisely, a circular cone) is called just, as it folds out of a circle - the base of the cone, the point that does not lie in the area of the stake, - the top of the cone and all the edges, so that the top of the cone is drawn from the base points (Fig. 1).

Vidrizki, where to draw the top of the cone with the base stake points, are called pretend cone. The top of the cone is folded from the base and the biconny surface.

cone be called straight, It is straight, but the apex of the cone with the center of the front is perpendicular to the area of the front. Nadal we will look at only a straight cone, which is simply called a cone for the stiffness. A pointedly straight circular cone can be seen to be tilted, but not trimmed when a rectangular tricycle is wrapped near the th leg yak axis (Fig. 2).

hanging a cone is called a perpendicular, the lowering of the third apex to the base area. At a straight cone, the base of the hanging is located at the center of the front. The face of a straight circular cone is called straight, yak to take revenge on its height.

^ Peretin of the cone with small areas.
The peretin of the cone with an area that passes through its top is a tricycle, at which side of the cone there is a cone (Fig. 3). Zokrema, tricot tricycle є axial overrun of the cone. The chain will pass through the line of the cone (Fig. 4).

Theorem. The area, parallel to the area of the base of the cone, overflows the cone along the stake, and the bichu surface - along the stake with the center on the axis of the cone.

Delivered. hey - the area is parallel to the area of the base of the cone and the overflowing of the cone (Fig. 5). The re-creation of the homothety of the top of the cone, which gives the area

Back forward

Uwaga! The frontal re-glance of the slides is victorious, in particular, for the purpose of understanding and perhaps not giving an indication of all the possibilities of the presentation. If a robot is given to you, be a weasel, add a new version.

Meta lesson:

lighting: Introduce an understanding of the cone, yogo elements; look out for a straight cone; view the sign on the surface of the cone; formvati vmіnnya razv'yazuvati tasks on the knowledge of the elements of the cone.
expanding: Development of mathematics literate, logical misleading.
Vikhovna: Vihovuvati pіznavalnu activity, culture spіlkuvannya, culture dialogue.

Lesson form: a lesson in the formation of new knowledge and intelligence.

Form of initial activity: the collective form of the robot.

Methods, like vikoristoyutsya at urotsi: explanatory and illustrative, productive.

Didactic material: zoshit, handler, pen, olivets, line, board, kreida and kolorovu kreidu, projector and presentation “Cone. Basic understanding. The area of the surface of the cone. "

Lesson plan:

Organizational moment (1 min).
Preparatory stage (motivation) (5 min).
Vivchennya new material (15 hv).
Solution of problems on the knowledge of elements of the cone (15 min).
Leading the lesson (2 mins).
Zavdannya to the booth (2 minutes).

XID LESSON

1. Organizational moment

Meta: pidgotuvati to mastering new material.

2. Preparatory stage

Shape: Robot Sleep.

Meta: knowledge of the new style of wrapping.

The cone in the crossbar from the walnut "konos" means "pine cone".

It is created in the form of a cone. You can see it in small objects, fixing it from the bitter frost and ending it with technology, so it is in childish igrashkas (pyramidka, popping і іn.), In nature (ice hockey, mountains, volcanoes, tornadoes).

(Wicked Slide 1-7)

teacher's strength	scholarship
3. Explanation of the new material Meta: introduce new understanding and power of the cone.
1. The cone can be trimmed to the wraps of the rectangular tricycle near one of the th legs. (Slide 8) Now it is clear, like a cone. The image is a circle with the center O і straight line OP, perpendicular to the area of the whole circle. The skin point of the stake is drawn from the point P (the driver will be the cone in stages). The surface, adopted by the cimi as a form, is called end surface, And most of them - fit end surfaces.	Zoshites will have a cone.
(Dictate of value) (Slide 9) Tilo, surrounded by an end surface and around with a cordon L, called cone.	Write down the value.
The final surface is called the side surface of the cone, And colo - before the cone... Straight OP, to pass through the center of the entrance and the top, be called vissyu cone... The axis of the cone is perpendicular to the surface area. Відрізок OP be called with a cone... Point P is called the top of the cone, And set the end surfaces - make a cone.	On the chair, write the elements of the cone.
Name two twin cones and name them?	PA і PB, stench rіvnі.
What should be done?	Projections of stolen rivni yak radiusi cola, which means they themselves set up rivni.
Write in zoshiti: the power of the cone:	(Slide 10)
1. Try to fit the cone to the door. Name kuti nahila finish to the core? Check it out. What, to bring the price?	Kuti: PSO, PDO. Stink rіvnі. So yak trikutnik PAB - equestrian.
2. Kuti nahilu set up to the base of the rіvnі. Name kuti mіzh vіssu and approve? How can you tell us about tsі kutah?	SRO and DPO Stink rіvnі.
3. Kuti mіzh vіssu and approve rіvnі. Name kuti mіzh vіssu and pіdstava? Why Rivni Tsi Kuti?	POC and POD. 90 pro
4. Kuti mіzh vіssu and іnta straight. We will only look at the straight cone.
2. It is clear that the cone overflows with small areas. What is the usual area, how to pass through the hanging of the cone?	Trikutnik.
Yaky tse tricutnik?	Win rivnobedreniy.
To what?	Two yogo side є make up, and the stench rіvnі.
Who is the witness of the given trikutnik?	Diameter of the base of the cone.
Such a peretin is called axial. (Slide 11) Nahrelit in zoshits and write a chain of pereretin. What is the normal plane perpendicular to the axis OP of the cone?	Colo.
De-stitching the center of the whole stake?	On the axis of the cone.
Tse peretin is called a circular pererez. (Silent 12) Seat in zoshita and write the price of peretin. Look and see the cone overhanging, not axial or parallel to the base of the cone. Visible on the stocks. (Slide 13)	Seating in zoshites.
3. Now we have introduced the formula for the surface of the cone. (Slide 14) For the whole bichnu surface of the cone, like the bichnu surface of the cylinder, it is possible to flutter up to the area, spreading out one by one.
What rozgortkoy bichnoy surface of the cone? (Sit on doshtsі)	Circular sector.
What is the radius of the sector?	I will make a cone.
And what about the arc of the sector?	Dovzhina circle.
For the area of the bichesky surface of the cone, there is the area of the rosette. (Slide 15)	, De - degree arc world.
Why do you need a circular sector?
So, why is the large area of the cone's bicheskoy surface? Visibly through that. (Slide 16) Why do you need to go to the arc?
From the side of the tsya zh arc is the half of the stake of the base of the cone. Why won’t you get home?
It is possible to put into the formula the bichesky surfaces of the cone are The flat surface of the cone is called the sum of the surface of the cone. . Write down the formulas.	Write :, .h	(Slide 21) L = 5

6. Homework. P.55, 56, No. 548 (b), 549 (b). (Slide 22)

Viznachennya. apex of a cone- the whole point (K), from which there are promenades.

Viznachennya. at the bottom of the cone- the whole area, established as a result of the overflow of the flat surface and all changes, which go from the top of the cone. The cone can have such bases, such as colo, elips, hyperbola and parabola.

Viznachennya. I will make a cone(L) be called a be-like v_drizok, which is the one at the top of the cone with a cordon in front of the cone. Establish є from the exchange, to go from the top of the cone.

Formula. I'll make it up(L) of a straight circular cone through radius R and height H (through Pyfagor's theorem):

Viznachennya. directing the cone is a straight curve, which describes the contour of the cone.

Viznachennya. bichna surface cone - the sukupn_st of all constituting cones. Tobto, on the surface, I will pretend to collapse along the direction of the cone.

Viznachennya. surface the cone is stored on the side of the cone and at the front of the cone.

Viznachennya. Visota cone (H) - a chain of edges, which go from the apex of the cone і perpendicular to its base.

Viznachennya. axis cone (a) - a straight line that passes through the apex of the cone and the center of the base of the cone.

Viznachennya. Taper (C) cone - the difference between the diameter of the cone to the thigh height. At a single truncated cone, the difference between the diameters of the transverse transitions D and d of the truncated cone to the appearance between them is the same: de R is the radius of the base, and H is the height of the cone.