Kulya is inscribed in the prism of yakosti.

Repair Golovna beauty and health. Or a sphere. Any cut that connects the center of the culi with a point on the culi surface is called. radiusThe cut that connects two points of the core surface and passes through the center of the core is called diameter The ends of any diameter are called the diametrically protruding points of the culi. Be-yake Peretin Kuliflat colo. The center of this stake is the base of a perpendicular, lowered from the center on the square to sit. The area that passes through the center of the coulee is called The center of this stake is the base of a perpendicular, lowered from the center on the square to sit.. diametrical area . The cross section of the culi with a diametrical plane is called great stake. , and the intersection of the sphere - Whatever the diametrical area of ​​the butt is symmetrical area .. Center of culturecenter of symmetry The area that passes through a point on the culo surface and is perpendicular to the radius drawn to that point is called. sub-flat. This point is calleddot torcannya The Stosovna Ploshchina has only one hidden point behind the culley - the torkannya point.A straight line that passes through a given point of the culo surface perpendicular to the radius drawn to that point is called doticThrough any point of the cul surface there are no people passing through, and all the smells lie near the cul surface.Kulova segment is called the part of the coulee that meets the surface. With a ballcalled a piece of a culi, it is spread between two parallel planes to cross the culi.Kulova sectoremerge from the cul-de-segment and cone.Since the culo segment is smaller than the culi, the culo segment is supplemented with a cone, the apex of which is in the center of the culi, and the base is the base of the segment. If the segment is larger than the cone, then the cone is visible from it.; Basic formulasKulya (R = OV - radius):S b = 4πR 2; V = 4πR 3/3.Since the culo segment is smaller than the culi, the culo segment is supplemented with a cone, the apex of which is in the center of the culi, and the base is the base of the segment. Kulova segment (R = OB - radius of the culi, h = SC - segment height, r = KV - radius of the base of the segment):. V segm = πh 2 (R - h/3)or else; V segm = πh(h 2 + 3r 2) / 6S segm = 2πRh.The volume of the cooler is 288π cm 3 .Find the diameter of the cooler.DecisionV = πd 3/6288π = πd 3/6πd 3 = 1728πd3 = 1728d = 12 div.Verdict: 12.butt 2.Find the diameter of the cooler. Three equal spheres of radius r are juxtaposed to each other and have the same area. Calculate the radius of the fourth sphere, which equals three given areas. Let O1, O2, O3 be the centers of these spheres and O is the center of the fourth sphere, which consists of three data and a given area. Let A, U, Z, T be the points of the spheres with the same plane. The spots of the two spheres lie on the line of centers of the two spheres, so O 1 O 2 = O 2 O 3 = O 3 O 1 = 2r . Points of equal distance from plane ABCAVO 2 O 1, AVO 2 O 3, AVO 3 O 1 - equal-sided zi side 2r.Let's gox is the searched radius of the fourth sphere.Todi VID = x.Ozhe, similarFind the diameter of the cooler.Otzhe, T is the center of the equilateral tricutule. Tom zvidsiSubmission: r/3.Sphere inscribed near the pyramidA sphere can be inscribed on a regular pyramid of skin.The center of the sphere lies at the height of the pyramid at the point of the crossbar with the bisector of the linear corner at the edge of the base of the pyramid.Respect.If a pyramid, which is not necessarily regular, can be inscribed into a sphere, then the radius r of the sphere can be calculated using the formula r = 3V / S pp, where V is the area of ​​the pyramid, S pp is the area of ​​the surface.butt 3. The final funnel, the radius of the base is R, and the height H is filled with water.The virvu's bag is lowered. What is the radius of the coulee so that the volume of water squeezing out of the shower with the enclosed part of the coulee is maximum? Let's draw a crossbar through the center of the cone.This transection is made by the isosfemoral tricumus.Find the diameter of the cooler.
O 1 O 2 = O 2 O 3 = O 3 O 1 = 2r RABC is the correct pyramid and the point H-center is the basis of ABC.< 90° . Центр первого шара, касающегося всех граней пирамиды, лежит на отрезке РН в точке его пересечения с биссектрисой . O 1 O 2 = O 2 O 3 = O 3 O 1 = 2r Let M be the middle of the edge ND. O 1 O 2 = O 2 O 3 = O 3 O 1 = 2r Todi is a linear cut of a dihedral cut, which is behind the mind α, and αPN 1 - the diameter of the first culi and the flatness that passes through the point H 1 perpendicular to the straight line RN, crosses the side ribs RA, PB, RS in parallel at points A 1, B 1, C 1.Then N 1 will be the center of the correct ∆A 1 B 1 C 1, and the pyramid RA 1 B 1 C 1 will be similar to the pyramid RABC with a similarity coefficient k = PH 1 / PH. Please note that another ball, centered at point O 1, is inscribed in the pyramid RA 1 B 1 C 1 and therefore the relationship of the radii of inscriptions is similar to the coefficient of similarity: VIN / VIN 1 = RN / RN 1.Equally tgα = 24/7 we know: AB = x. Todi Zvidsi Shukan setting VIN/O1N1=16/9.Update: 16/9. Todi Sphere inscribed in a prismDiameterD of a sphere inscribed in a prism, the same height H of the prism: D = 2R = H.Radius R of the sphere inscribed in the prism is equal to the radius of the stake inscribed in the perpendicular section of the prism.If a sphere is inscribed into a direct prism, then a circle can be inscribed into the base of this prism.R of the sphere inscribed in a straight prism is equal to the radius of the stake inscribed in the base of the prism.Radius
Let ABC...A 1 B 1 Z 1 ... - stolen prism and F - center of the stake with radius FK, inscribed in its perpendicular crossbar.The fragments of the perpendicular cut of the prism are perpendicular to the skin surface of the butt side, then the radius of the stake inscribed in the perpendicular cut is drawn to the sides of this cut, and perpendicular to the lateral edges of the prism.Also, point F is equally distant from all side faces.Let us draw through point F a straight line OO 1 perpendicular to the plane of the prism’s base, which crosses the base at points O and O 1.Todi GO 1 – prism height.Find the diameter of the cooler. The fragments behind the sink GO 1 = 2FK, then F is the middle of the fragment GO 1:FK = OO 1 / 2 = FO = FO 1, then.point F is equally distant from the surfaces of all the faces of the prism.This means that a sphere can be inscribed in this prism, the center of which coincides with point F - the center of the stake, inscribed in that perpendicular section of the prism, which divides the height of the prism, which passes through point F, completely.The theorem has been proven.Find the diameter of the cooler.
Butt 5.A rectangular parallelepiped has a core of radius 1. Find the volume of the parallelepiped. O 1 O 2 = O 2 O 3 = O 3 O 1 = 2r Paint the look of the beast.Or on the side.

The center of the inscribed kuli is the point of cross-section of the bisectoral planes created for all the dihedral kuli that are in the pyramid;

Since the bisectoral plane does not have a corner point, it is not possible to enter a core.

Partial fall: the side faces of the pyramid are evenly inclined to the base plane.

Todi:

you can enter the bag;

The center of the center lies at the height of the pyramid, more specifically, the point of the crossbar of height with the bisector between the apothem and the projection of the apothem onto the plane of the base.

6.2.

Kulya is a straight prism

You can add a hole and a side to a straight prism if:

a colo can be inscribed into the base of the prism,

The diameter of this stake is equal to the height of the prism.

The center of the core is the middle of the cut, which connects the centers of the inscriptions at the base of the core.

de is the radius of the inscribed coulee;

- radius of the stake inscribed in the base;

H – height of the prism.

6.3.

Kula and cylinder

The cylinder can have a cylinder of one or the other, if the axial cross-section of the cylinder is a square (such a cylinder is called equilateral).

The center of the core is the center of symmetry of the axial cross-section of the cylinder.

6.4.

2. The cule is called described by the culi, and the culi is inscribed at the cule, since the surface of the culi passes through the apex of the culi.

3. The core is called inscribed in the cylinder, truncated cone (cone), and the cylinder, truncated cone (cone) - we describe the core, as the surface of the core is covered with a base (base) and all the parts of the cylinder, truncated cone ( cone).

(From this significance it follows that in any axial section of these bodies the stake of a great stake could be inscribed).

4. The core is called a cylinder, a truncated cone (cone), because the core (base and apex) lies on top of the core.

(From this it follows that any axial cut of these bodies can be described around a larger stake).

Zagalny respect for the location of the center of the cool.

1. The center of the coulee, inscribed in the richahedron, lies at the point of the crossbar of the bisectoral planes of all dihedral corners of the richahedron.

There are no differences in the middle of the rich side.

2. The center of the core, described as a richhedron, lies at the point of the crossbar of the planes perpendicular to all the edges of the richhedron and passes through their midpoints.

There may be a ripple in the middle, on the surface and a rich-hedron position.

Combination of coolants and prisms. 1. Kulya, inscribed in a straight prism.

Theorem 1. The core can be inscribed into a straight prism in the same way and only in the same way that a stake can be inserted into the base of the prism, and the height of the prism is equal to the diameter of the stake.

Nasledok 1. The center of the stake, inscribed in a straight prism, lies at the middle of the height of the prism, which passes through the center of the stake, inscribed in the base.

Naslidok 2.

Kulya, zokrema, can be inscribed at the straight line: trikutna, regular, qotirikutna (in which sum of the protilegal sides the bases are equal to each other) behind the head H = 2r, where H is the height of the prism, r is the radius of the stake inscribed in the base. 2. Kulya, described in white prism.

Theorem 2. A coulee can be described as a prism in this and only in that way, since the prism is straight and its base can be described in a circle.

Nasledok 1. Nasledok 1

.

The center of the stake, described as a straight prism, lies at the middle of the height of the prism, drawn through the center of the stake, described as a base.

The cul-de-sac can be described as: a straight tricutaneous prism, a regular prism, a recticutaneous parallelepiped, a straight tricutaneous prism, in which the base of the protidal cutata is 180 degrees.

From assistant L.S. Atanasyan on the combination of coolants and prisms, you can refer to works No. 632, 633, 634, 637(a), 639(a,b). The pyramid can be used to describe the core in the same way and only in that way as the base can be described as a cole.

Theorem 1. The center of the beam, described by the pyramid, lies at the point of the straight crossbar, perpendicular to the base of the pyramid, which passes through the center of the stake, described by this base, and the plane, perpendicular to any side rib drawn through the middle of it ribs

Nasledok 1. Since the side ribs of the pyramid are equal to each other (or one layered to the plane of the base), then the core can be described using such a pyramid.

side rib and height. Nasledok 3.

The culi, in short, can be described as: white tricutaneous pyramid, white regular pyramid, white tricutaneous pyramid, in which the sum of the proximal cutas is more than 180 degrees.

2. Kulya, inscribed in the pyramid. Theorem 4.

Theorem 1. If the side edges of the pyramid are however extended to the base, then such a pyramid can be inscribed into a core.

Nasledok 1. The center of the coulee, inscribed in the pyramid, whose side faces are however extended to the base, lies at the point of the crossbar of the height of the pyramid with the bisector of the linear edge of any dihedral edge on the base of the pyramid, which serves as a side the height of the side face drawn from the top of the pyramid.

A ball can be inscribed in a regular pyramid.

From the assistant L.S. Atanasyan, for the combination of a coulisse with a pyramid, you can assign works No. 635, 637(b), 638, 639(c), 640, 641.

Combination of a coulee with a truncated pyramid.

1. Kule, described with a correctly cut pyramid. Theorem 5.

You can describe the butt using any correct cut pyramid.

(This intelligence is sufficient, but not necessary) 2. Kulya, inscribed in a regular truncated pyramid.

Theorem 6.

With a correctly cut pyramid, you can enter a core in this and only that way, since the apothem of the pyramid is the same as the apothem of the bases.

There is only one task for the combination of a coulisse with a truncated pyramid in L.S. Atanasyan’s handbook (No. 636). Combination of cores with round bodies.

Theorem 7. Whether a cylinder, a cut cone (straight circular), a cone can be described as a butt.

Theorem 8. For a cylinder (straight circular), you can enter a core in the same and only in that case as a straight cylinder.

Theorem 9. If you have a cone (straight circular) you can fit a sack.

Theorem 10.

When cutting a cone (straight circular), you can enter a core in this and only that way, as it creates the same amount of radii of the bases.

1. The edge of the cube is ancient a.

Find the radius of the cube: inscribed in the cube and described in the cube. (r = a/2, R = a3).

2. You can describe a sphere (cule) close to: a) a cube; b) rectangular parallelepiped;

c) a slender parallelepiped, the basis of which is an orthocutaneous plant; d) straight parallelepiped;

e) a frail parallelepiped? (a) so;

b) yes; c) no;

d) no; e) nі)

3. Is it fair to say that the sphere can be described by any kind of three-piece pyramid? (So)

4. How can you describe the sphere of the white pyramid? (No, not the cutaneous pyramid)

5. What kind of power does the pyramid have, so that the sphere can be described by it? (It is based on a rich bush, which can be described in a different way)

6. A pyramid is inscribed in the sphere, the side edge of which is perpendicular to the base. How to know the center of the sphere?

(The center of the sphere is the point of the cross between two geometrical points in space. Perche is a perpendicular drawn to the plane of the base of the pyramid, through the center of the stake described by the white. The other is a plane perpendicular to this side edge and drawn through its middle) 7. For what kind of minds can one describe the sphere of a prism, on the basis of what - a trapezoid?

(First of all, the prism is straight, and, in another way, the trapezoid is equifemoral, so that it can be described as a circle) 8. What minds would be satisfied with a prism so that it could describe a sphere?

(The prism is straight, and its basis is to blame for the presence of a rich-cow, which can be described in a circle) (Based on the prism: a) gostrokutny trikutnik;

b) blunt-cut tricutaneous)

15. For a rectangular parallelepiped, the edges of which extend 1 inch, 2 inch, and 2 inch, a sphere is described. Calculate the radius of the sphere.

(1.5 dm) 16. In what kind of cut cone can a sphere be inscribed?

(At the truncations there is a cone, in the axial section of which a column can be inscribed. The axial beam of the cone is the isosfemoral trapezium, the sum of which is responsible for aligning with the sum of its lateral sides. Otherwise, it seems that the cone has radii the base is responsible for strengthening the solid) 17. At the truncated cone the sphere is inscribed.

Under which part of the closing cone is visible from the center of the sphere? (90 degrees)

18. How is power responsible for the mother’s direct prism, so that she can inscribe a sphere? (First of all, at the base of the straight prism there is a rich body into which a stake can be inserted, and, in another way, the height of the prism is equal to the diameter of the stake inscribed in the base)

19. Point the butt of the pyramid where it is not possible to enter the sphere? (For example, there is a pyramid that is based on an orthocutaneous or parallelogram)

20. The basis of a straight prism is a rhombus. How can you inscribe a sphere up to this prism?

(No, it’s not possible, because it’s impossible to describe the color of the white rhombus in the zagal form) 21. Behind what kind of straight tricutaneous prism can a sphere be inscribed?

(Since the height of the prism is twice as large as the radius of the stake inscribed in the base) 22. Behind what mind can a sphere be inscribed in a regular truncated pyramid?

(As the crossbar of this pyramid has a square that passes through the middle of the side of the base perpendicular to it, there is an isosceles trapezoid in which a column can be entered) 23. A sphere is inscribed on the three-piece truncated pyramid.

What point of the pyramid is the center of the sphere? (The center of the sphere inscribed in this pyramid is located on the web of three cross-sectional planes of the corners, formed by the side faces of the pyramid from the base)

24. How can you describe the sphere of a right circular cylinder? (That's possible)

The author appreciates that from the three lessons that follow the planning on the topic “Surveys on polyhedrons, cylinders, cones and cores”, two lessons should be focused on the most important tasks on the combination of cores with other bodies .

According to the theory, it is not recommended to bring lessons through the lack of hours and hours.

You can teach the students who have sufficient skills for him, bring them up by indicating (for the humility of the teacher) a plan and proof.

The connection between problems on a cone, inscriptions on a sphere (cone, inscriptions on a sphere) comes down to the consideration of one or several tricutaneous systems.

The cone of inscriptions at the core, because its apex and base lie on the surface of the core, like on a sphere.

The center of the coolant lies on the axis of the cone.

With a strong focus on the cone, inscriptions at the core, manually look through the combination of bodies with a plane that passes through the entire cone and the center of the core.

The retin is a large bell (the same as the colo, the radius of which is the same as the radius of the coule) with an isosceles tricutaneum inscribed in it - the axial retina of the cone.

The sides of this tricube form a cone, the base being the diameter of the cone.

Whereas between those that create gostria, the center of the described stake lies in the middle of the tricubitule (apparently, the center of the described stake of the cole cone is in the middle of the cone).

As between the straight lines, the center of the stake lies in the middle of the base of the tricubitule (the center of the stake is close to the center of the base of the cone).

As the center of the stake lies in the position of a trikulet (the center of the described joint is a cone pose).

Although the problem does not say where the center of the described circle lies, it is important to look at how you can decide on different options for its development.