A sphere is described near a regular quadrangular prism. Direct prism (quadrangular right). Pyramid and ball
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Spheres described near polyhedra.
Definition. A polyhedron is said to be inscribed in a sphere (and a sphere circumscribed near a polyhedron) if all the vertices of the polyhedron belong to this sphere. Consequence. The center of the circumscribed sphere is a point equidistant from all vertices of the polyhedron. O o o . . .
Theorem 1. The set of points equidistant from two given points is a plane perpendicular to a segment with ends at these points, passing through its middle (the plane of perpendicular bisectors to this segment). AB ┴ α AO=OB α A B O
Theorem 2. The set of points equidistant from n given points lying on the same circle is a straight line perpendicular to the plane of these points, passing through the center of the circumscribed circle. C E A B D O a . . . . . . C E A B D . . . . .
Prism inscribed in a sphere. OA=OB=…=OX=R sf. O 1 . O. O sf a 1 a .A 1 .B 1 .C 1 .D 1 E 1 . X1. .A .B .C .D E. X. a a 1 . O. O 1
Consequences. 1) Near a right triangular prism, a sphere can be described, because a circle can always be circumscribed around a triangle. 2) A sphere can be described near any regular prism, because a regular prism is a straight line and a circle can always be circumscribed near a regular polyhedron. O. O. .
Task number 1. The ball is described near the prism, at the base of which lies a right triangle with legs 6 and 8. The lateral edge of the prism is 24. Find the Radius of the ball. Given: ∆ ABC – rectangular; AC=6, BC=8, AA 1=24. Find: R w = ? Solution: 1)OO 1 ┴AB 1 ; OO 1 =AA 1 =24. 2) ABC: AB=10. 3) O w OB: R w = O w B=√OO w 2 + OB 2 = = √144+25=13 Answer: 13. O 1 O. . . R w O w C 1 B 1 A 1 A C B
Task number 3. The measurements of a cuboid are 2,3 and 5. Find the radius of the circumscribed sphere. Given: AB=a=2; BC=b=3; CC 1=c=5. Find: R w = ? Solution: 1) AC 2 =a 2 +b 2 +c 2 . 2) A 1 C 2 =25+9+4=38 (Property of diagonals of a rectangular parallelepiped) 3) A 1 C=√38; R w \u003d O w C \u003d √38 / 2 Answer: √38 / 2 D 1 C 1 B 1 A 1 A B C D 5 2 3. . . O sh
Task number 3. The side of the base of a regular triangular prism is a, and the side edge is 2 a. Find the radius of the circumscribed sphere. Given: AB=BC=AC=a, AA 1 ┴ABC ; AA 1= 2a. Find: R w = ? Solution: 1)AB=AO √3; AO=a/√3. 2) R w \u003d √ a 2 + a 2 / 3 \u003d 2a / √ 3 Answer: 2a / √ 3 C 1 B A 1 C B 1 A O w R w. O O 1
Consequences. 1) A sphere can always be described near a triangular pyramid, since a circle can always be described near a triangle. 2) Near a regular pyramid, you can always describe a sphere. 3) If the lateral edges of the pyramid are equal (similarly inclined to the base), then a sphere can always be described near such a pyramid. *In the last two cases, the center of the sphere lies on the line containing the height of the pyramid. O. O.
Tasks (the sphere described near the pyramid). A ball is described near the pyramid PABC, whose base is a regular triangle ABC with side 4√3. The lateral edge PA is perpendicular to the plane of the base of the pyramid and is equal to 6. Find the radius of the ball. Given: AB=BC=AC=4 √3 ; PA┴(ABC); PA=6. Find: R w = ? Solution: 1) OO SF ┴(ABC); O is the center of the circumscribed about ∆ABC circle; K O SF ┴ PA; KP=AK (KO SF One of the perpendicular bisectors to the side edge PA); O SF is the center of the circumscribed sphere. 2) OO SF ┴(ABC); OO SF belongs (AKO) ; PA┴(ABC); AK belongs (AKO) ; means KA|| OO SF; . O SF. O K.P.A.B.C
Tasks (the sphere described near the pyramid). 3) KO c f ┴AP; KO c f belongs to (AOK); AO ┴AP; AO belongs to (AOK) ; means KO c f || AO; 4) From (2) and (3) : AOO c f K-rectangle, AK=PA/2=3; 5) AO=AB/ √3 =4; 6) ∆ AO O c f: AO c f \u003d R w \u003d 5 Answer: 5
Tasks (the sphere described near the pyramid). In a regular quadrangular pyramid, the lateral edge is inclined to the base at an angle of 45 ˚. The height of the pyramid is h. Find the radius of the circumscribed sphere. Given: PABCD is a regular pyramid; (AP^(ABC))=45 ˚; PO=h. Find: R w = ? Solution: 1) AO=OP=h; AP=h √ 2; 2) ∆PAP 1 – rectangular; PP 1 - ball diameter; PP 1 \u003d 2 R w; AP 2 = PP 1 *OP; (h √ 2) 2 =2 Rw *h; R w \u003d 2h 2 / 2h \u003d h. Answer: h. C. B A. .D .P .P 1 . O
Tasks (the sphere described near the pyramid). On one's own. The radius of a sphere circumscribed about a regular tetrahedron is R. Find the total surface area of the tetrahedron.
Tasks (the sphere described near the pyramid). On one's own. Given: DABC is a regular tetrahedron; R is the radius of the sphere. Find: S full tetra. =? Solution: 1) Since the tetrahedron is regular, the center of the circumscribed sphere belongs to the straight line containing the height of the pyramid; 2) S full tetra. = a 2 √ 3/4*4= a 2 √ 3; 3) Points D, A, D 1 belong to the same circle - the section of the sphere by the plane DAD 1, so the angle DAD 1 is an inscribed angle based on the diameter, DD 1; angle DAD 1 =90 ˚; 4) AO is the height ∆ ADD 1 drawn from the vertex of the right angle. AD 2 = DO*DD 1 ; 5) AO=a/ √ 3; DO= √ a 2 -a 2 / 3=a √ 2 / √ 3; a 2 = a √ 2 / √ 3*2R; a= √ 2 / √ 3*2R; a 2 \u003d 8R 2 / 3; .D 1 .D .O .B .C A. a a
Tasks (the sphere described near the pyramid). On one's own. 6) S full tetra. = 8R 2 √ 3/3 Answer: 8R 2 √ 3/3
2. Base side 
Tasks
1. Find the surface area of a straight prism, at the base of which lies a rhombus with diagonals equal to 3 and 4, and a side edge equal to 5.
Answer: 62.
2. At the base of a straight prism lies a rhombus with diagonals equal to 6 and 8. Its surface area is 248. Find the side edge of this prism.
Answer: 10.
3. Find the side edge of a regular quadrangular prism if the sides of its base are 3 and the surface area is 66.
Answer: 4.
4. A regular quadrangular prism is described near a cylinder whose base radius and height are equal to 2. Find the lateral surface area of the prism.
Answer: 32.
5. A regular quadrangular prism is described near a cylinder whose base radius is 2. The lateral surface area of the prism is 48. Find the height of the cylinder.
Straight prism (hexagonal regular)
A prism in which the side edges are perpendicular to the bases, and the bases are equal squares.
1. Side faces - equal rectangles
2. Base side 
Tasks
1. Find the volume of a regular hexagonal prism whose base sides are equal to 1 and the side edges are equal.
Answer: 4.5.
2. Find the lateral surface area of a regular hexagonal prism whose base sides are 3 and whose height is 6.
Answer: 108.
3. Find the volume of a regular hexagonal prism with all edges equal to √3.
Answer: 13.5
4. Find the volume of a polyhedron whose vertices are points A, B, C, D, A1, B1, C1, D1 of a regular hexagonal prism ABCDEFA1B1C1D1E1F1, whose base area is 6, and whose side edge is 2.
Direct prism (arbitrary n-coal)
A prism whose side edges are perpendicular to the bases and whose bases are equal n-gons.
1. If the base is a regular polygon, then the side faces are equal rectangles.
2. Base side 
.
Pyramid
A pyramid is a polyhedron composed of an n-gon A1A2...AnA1 and n triangles (A1A2P, A1A3P, etc.).
1. A section parallel to the base of the pyramid is a polygon similar to the base. The areas of the section and the base are related as the squares of their distances to the top of the pyramid.
2. A pyramid is called regular if its base is a regular polygon, and the vertex is projected into the center of the base.
3. All side edges of a regular pyramid are equal, and the side faces are equal isosceles triangles.
4. The height of the side face of a regular pyramid is called apothem.
5. The area of the lateral surface of a regular pyramid is equal to half the product of the perimeter of the base and the apothem.
Tasks
1. How many times will the volume of a regular tetrahedron increase if all its edges are doubled?
Answer: 8.
2. The sides of the base of a regular hexagonal pyramid are 10, the side edges are 13. Find the area of the lateral surface of the pyramid.
Answer: 360.
5. Find the volume of the pyramid shown in the figure. Its base is a polygon whose adjacent sides are perpendicular, and one of the side edges is perpendicular to the plane of the base and is equal to 3.
Answer: 27.
6. Find the volume of a regular triangular pyramid whose base sides are 1 and whose height is .
Answer: 0.25.
7. The side edges of a triangular pyramid are mutually perpendicular, each of them is equal to 3. Find the volume of the pyramid.
Answer: 4.5.
8. The diagonal of the base of a regular quadrangular pyramid is 8. The side edge is 5. Find the volume of the pyramid.
Answer: 32.
9. In a regular quadrangular pyramid, the height is 12, the volume is 200. Find the side edge of the pyramid.
Answer: 13.
10. The sides of the base of a regular quadrangular pyramid are 6, the side edges are 5. Find the surface area of the pyramid.
Answer: 84.
11. The volume of a regular hexagonal pyramid 6. The side of the base is 1. Find the side edge.
12. How many times will the surface area of a regular tetrahedron increase if all its edges are doubled?
Answer: 4.
13. The volume of a regular quadrangular pyramid is 12. Find the volume of the pyramid cut off from it by a plane passing through the diagonal of the base and the middle of the opposite side edge.
Answer: 3.
14. How many times will the volume of an octahedron decrease if all its edges are halved?
Answer: 8.
15. The volume of a triangular pyramid is 15. The plane passes through the side of the base of this pyramid and intersects the opposite side edge at a point dividing it in a ratio of 1: 2, counting from the top of the pyramid. Find the largest of the volumes of the pyramids into which the plane divides the original pyramid.
Answer: 10.
16. Find the height of a regular triangular pyramid whose base sides are 2 and whose volume is .
Answer: 3.
17. In a regular quadrangular pyramid, the height is 6, the side edge is 10. Find its volume.
Answer: 256.
18. From a triangular pyramid, the volume of which is 12, a triangular pyramid is cut off by a plane passing through the top of the pyramid and the middle line of the base. Find the volume of the cut off triangular pyramid.
Answer: 3.
Cylinder
Cylinder - a body bounded by a cylindrical surface and two circles with borders.
| H |
| R |
| body volume | Lateral surface area | Base area | Total surface area |
 |  |  |  |
 |  |  |
1. Generators of a cylinder - segments of generators enclosed between the bases.
2. The height of the cylinder is the length of the generatrix.
3. Axial section - a rectangle, two sides of which are generators, and the other two are the diameters of the bases of the cylinder.
4. Circular section - a section, the secant plane of which is perpendicular to the axis of the cylinder.
5. Development of the lateral surface of the cylinder - a rectangle representing two edges of the cut of the lateral surface of the cylinder along the generatrix.
6. The area of the lateral surface of the cylinder is the area of its development.
7. The area of the full surface of the cylinder is called the sum of the areas of the lateral surface and the two bases.
8. It is always possible to describe a sphere near a cylinder. Its center lies in the middle of the height. 
, where R is the radius of the ball, r is the radius of the base of the cylinder, H is the height of the cylinder.
9. A ball can be inscribed in a cylinder if the diameter of the base of the cylinder is equal to its height, 
.
Tasks
1. A part is lowered into a cylindrical vessel containing 6 liters of water. At the same time, the liquid level in the vessel rose 1.5 times. What is the volume of the part?
Answer: 3.
2. Find the volume of a cylinder whose base area is 1, and the generatrix is 6 and is inclined to the base plane at an angle of 30o.
Answer: 3.
3. The cylinder and the cone have a common base and height. Find the volume of the cylinder if the volume of the cone is 50.
Answer: 150.
4. Water, which was in a cylindrical vessel at a level of 12 cm, was poured into a cylindrical vessel, twice as large in diameter. At what height will the water level be in the second vessel?
5. The area of the axial section of the cylinder is . Find the lateral surface area of the cylinder.
Answer: 2.
6. A regular quadrangular prism is described near a cylinder whose base radius and height are equal to 2. Find the area of the lateral surface of the prism.
Answer: 32.
7. The circumference of the base of the cylinder is 3. The lateral surface area is 6. Find the height of the cylinder.
8. One cylindrical mug is twice as high as the second, but the second is one and a half times wider. Find the ratio of the volume of the second mug to the volume of the first.
Answer: 1.125.
9. In a cylindrical vessel, the liquid level reaches 18 cm. At what height will the liquid level be if it is poured into a second vessel, the diameter of which is 3 times greater than the first?
Answer: 2.
Cone
A cone is a body bounded by a conical surface and a circle.
  |
| cone axis |
| R |
| vertex |
| generators |
| side surface |
| r |
| body volume | Lateral surface area | Base area | Total surface area |
 |  | |  |
 |  |
1. The area of the lateral surface of the cone is the area of its development.
2. Relationship between the development angle and the angle at the apex of the axial section 
.
1. A cylinder and a cone have a common base and height. Find the volume of the cylinder if the volume of the cone is 50.
Answer: 150.
2. Find the volume of a cone whose base area is 2, and the generatrix is 6 and is inclined to the base plane at an angle of 30o.
Answer: 2.
3. The volume of the cone is 12. A section is drawn parallel to the base of the cone, dividing the height in half. Find the volume of the cut off cone.
Answer: 1.5.
4. How many times is the volume of a cone circumscribed near a regular quadrangular pyramid greater than the volume of a cone inscribed in this pyramid?
Answer: 2.
5. The height of the cone is 6, the generatrix is 10. Find its volume divided by.
Answer: 128.
6. The cylinder and the cone have a common base and height. Find the volume of the cone if the volume of the cylinder is 48.
Answer: 16.
7. The diameter of the base of the cone is 6, and the angle at the top of the axial section is 90°. Calculate the volume of the cone divided by .
8. The cone is described near a regular quadrangular pyramid with base side 4 and height 6. Find its volume divided by .
9. A cone is obtained by rotating an isosceles right triangle around a leg equal to 6. Find its volume divided by.
Sphere and ball
A sphere is a surface consisting of all points in space located at a given distance from a given point. A sphere is a body bounded by a sphere.
1. A section of a sphere by a plane is a circle if the distance from the center of the sphere to the plane is less than the radius of the sphere.
2. The section of a sphere by a plane is a circle.
3. The tangent plane to the sphere is a plane that has only one common point with the sphere.
4. The radius of the sphere, drawn at the point of contact between the sphere and the plane, is perpendicular to the tangent plane.
5. If the radius of a sphere is perpendicular to the plane passing through its end lying on the sphere, then this plane is tangent to the sphere.
6. A polyhedron is said to be inscribed near a sphere if the sphere touches all its faces.
7. Segments of tangents to the sphere, drawn from one point, are equal and make equal angles with a straight line passing through this point and the center of the sphere.
8. A sphere is inscribed in a cylindrical surface if it touches all its generators.
9. A sphere is inscribed in a conical surface if it touches all its generators.
Tasks
1. The radii of two balls are 6 and 8. Find the radius of a ball whose surface area is equal to the sum of their surface areas.
Answer: 10.
2. The area of the great circle of the ball is 1. Find the surface area of the ball.
3. How many times will the surface area of the ball increase if its radius is doubled?
4. The radii of three balls are 3, 4 and 5. Find the radius of a ball whose volume is equal to the sum of their volumes.
Answer: 6.
5. A rectangular box is circumscribed around a sphere of radius 2. Find its surface area.
Answer: 96.
6. A cube is inscribed in a ball of radius . Find the surface area of the cube.
Answer: 24.
7. A rectangular box is circumscribed around a sphere of radius 2. Find its volume.
8. The volume of the cuboid circumscribed around the sphere is 216. Find the radius of the sphere.
Answer: 3.
9. The surface area of a cuboid circumscribed about a sphere is 96. Find the radius of the sphere.
Answer: 2.
10. A cylinder is described near the sphere, the lateral surface area of which is 9. Find the surface area of the sphere.
Answer: 9.
11. How many times is the surface area of a sphere circumscribed about a cube greater than the surface area of a sphere inscribed in the same cube?
Answer: 3.
12. A cube is inscribed in a ball of radius . Find the volume of the cube.
Answer: 8.
Composite polyhedra
Tasks
1. The figure shows a polyhedron, all dihedral angles of the polyhedron are right. Find the distance between vertices A and C2 .
Answer: 3.
2. Find the angle CAD2 of the polyhedron shown in the figure. All dihedral angles of a polyhedron are right. Give your answer in degrees.
Answer: 60.
3. Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).
Answer: 18.
4. Find the surface area of the polyhedron shown in the figure (all dihedral angles are right).
Answer: 132
5. Find the surface area of the spatial cross shown in the figure and made up of unit cubes.
Answer: 30
6. Find the volume of the polyhedron shown in the figure (all dihedral angles are right).
Answer: 8
7.Find the volume of the polyhedron shown in the figure (all dihedral angles are right).
Answer: 78
8. The figure shows a polyhedron, all dihedral angles of the polyhedron are right. Find the tangent of angle ABB3.
Answer: 2
10. The figure shows a polyhedron, all dihedral angles of the polyhedron are right. Find the tangent of angle C3D3B3.
Answer: 3
11. Through the midline of the base of a triangular prism, a plane is drawn parallel to the side edge. Find the area of the lateral surface of the prism if the area of the lateral surface of the cut off triangular prism is 37.
Answer: 74.
12. The figure shows a polyhedron, all dihedral angles of the polyhedron are right. Find the squared distance between vertices B2 and D3 .
Answer: 11.
A ball can be circumscribed near a pyramid if and only if a circle can be circumscribed near its base.
To build the center O of this ball, you need:
1. Find the center O, the circle circumscribed near the base.
2. Through point O, draw a straight line perpendicular to the plane of the base.
3. Through the middle of any side edge of the pyramid, draw a plane perpendicular to this edge.
4. Find the point O of the intersection of the constructed line and plane.
Special case: the side edges of the pyramid are equal. Then:
the ball can be described;
the center O of the ball lies at the height of the pyramid;
Where is the radius of the circumscribed sphere; - side rib; H is the height of the pyramid.
5.2. ball and prism
A sphere can be circumscribed near a prism if and only if the prism is straight and a circle can be circumscribed near its base.
The center of the ball is the middle of the segment connecting the centers of the circles described near the bases.
where is the radius of the circumscribed sphere; is the radius of the circumscribed circle near the base; H is the height of the prism.
5.3. ball and cylinder
A sphere can always be described near a cylinder. The center of the sphere is the center of symmetry of the axial section of the cylinder.
5.4. ball and cone
A sphere can always be described near a cone. the center of the ball; serves as the center of a circle circumscribed about the axial section of the cone.
A regular quadrangular prism, the volume of which is 65 dm 3, is described near the ball. Calculate the ratio of the area of the total surface of the prism and the volume of the sphere
A prism is called regular if its bases are regular polygons and its lateral edges are perpendicular to the base. A regular quadrilateral is a square. The intersection point of the diagonals of a square is its center, as well as the center of a circle inscribed in it. Let's prove this fact. although this proof is unlikely to be asked and can be omitted
How private view a parallelogram, a rectangle and a rhombus square has their properties: the diagonals are equal and bisect the intersection point, and are the bisectors of the corners of the square. Draw a line through point E parallel to AB. AB is perpendicular to BC, which means that TC is also perpendicular to BC (if one of the two parallel lines is perpendicular to any third of the line, then the second parallel line is also perpendicular to this (third) line). In the same way, we draw a straight line MR. Rectangular triangles BET and AEK are equal in hypotenuse and acute angle (BE=AE - half of the diagonals, ∠ EBT=∠ EAK - half of the right angle), so ET=EK. In the same way we prove that EM=EP. And from the equality of triangles CEP and SET (the same sign) we will see that ET = EP, i.e. ET = EP = EK = EM or simply say that the point M is equidistant from the sides of the square, and this is necessary condition in order to recognize it as the center of a circle inscribed in this square.
Consider the rectangle ABTK (this quadrilateral is a rectangle, since all angles in it are right by construction). In a rectangle, opposite sides are equal - AB \u003d KT (it should be noted that KT is the diameter of the base) - this means that the side of the base is equal to the diameter of the inscribed circle.
Let's draw a plane through parallel (two lines perpendicular to the same plane are parallel) AA 1, CC 1 and BB 1 and DD 1, respectively (parallel lines define a plane, moreover, only one). Planes AA 1 C 1 C and BB 1 D 1 D are perpendicular to the base ABCD, because pass through straight lines (lateral ribs) perpendicular to it.
From the point H (the intersection of the diagonals) in the plane AA 1 C 1 C perpendicular to the base ABCD. Then we will do the same in the plane BB 1 D 1 D. From the theorem: if from a point belonging to one of the two perpendicular planes we draw a perpendicular to the other plane, then this perpendicular lies completely in the first plane, we get that this perpendicular must lie and in the plane AA 1 C 1 C and in the plane BB 1 D 1 D. This is possible only if this perpendicular coincides with the line of intersection of these planes - NOT. Those. the segment is NOT a straight line on which the center of the inscribed circle lies (because it is NOT equidistant from the planes of the side faces, and this in turn follows from the equidistance of points E and H from the vertices of the corresponding bases (according to the proven: the point of intersection of the diagonals is equidistant from the sides of the square ), and from the fact that NOT is perpendicular to the bases, we can conclude that NOT is the diameter of the ball. Theorem: A ball can be inscribed in a regular prism if and only if its height is equal to the diameter of the circle inscribed in the base. ball, so its height is equal to the diameter of the circle inscribed in the base.If we designate the side of the base as A, and the height of the prism for h, then using this theorem, we conclude A=h and then the volume of the prism can be found as follows: 
Further, using the fact that the height is equal to the diameter of the inscribed sphere and the side of the base of the prism, we find the radius of the sphere and then its volume: 
It must be said that the side edges are equal to the height (segments of parallel lines enclosed between parallel planes are equal), and since the height is equal to the side of the base, then in general all the edges of the prism are equal to each other, and all the faces are essentially squares with an area A 2. In fact, such a figure is called a cube - a special case of a parallelepiped. It remains to find the total surface of the cube and correlate it with the volume of the ball: 
The topic “Different problems on polyhedra, a cylinder, a cone and a ball” is one of the most difficult in the 11th grade geometry course. Before solving geometric problems, they usually study the relevant sections of the theory that are referred to when solving problems. In the textbook by S. Atanasyan et al. on this topic (p. 138) one can find only definitions of a polyhedron circumscribed about a sphere, a polyhedron inscribed in a sphere, a sphere inscribed in a polyhedron, and a sphere circumscribed near a polyhedron. IN guidelines this textbook (see the book “Studying geometry in grades 10–11” by S.M. Saakyan and V.F. Butuzov, p. 159) says which combinations of bodies are considered when solving problems No. 629–646, and attention to the fact that "when solving a particular problem, first of all, it is necessary to ensure that students have a good idea of the relative position of the bodies indicated in the condition." The following is the solution of problems No. 638 (a) and No. 640.
Considering all of the above, and the fact that the most difficult tasks for students are the tasks of combining a ball with other bodies, it is necessary to systematize the relevant theoretical provisions and communicate them to students.
Definitions.
1. A ball is called inscribed in a polyhedron, and a polyhedron is said to be circumscribed near the ball, if the surface of the ball touches all the faces of the polyhedron.
2. A ball is called circumscribed near a polyhedron, and a polyhedron is called inscribed in a ball if the surface of the ball passes through all the vertices of the polyhedron.
3. A ball is called inscribed in a cylinder, a truncated cone (cone), and a cylinder, a truncated cone (cone) is called circumscribed near the ball, if the surface of the ball touches the bases (base) and all generators of the cylinder, truncated cone (cone).
(It follows from this definition that the circumference of the great circle of the ball can be inscribed in any axial section of these bodies).
4. A ball is called circumscribed near a cylinder, a truncated cone (cone) if the circles of the bases (the circle of the base and the top) belong to the surface of the ball.
(From this definition it follows that about any axial section of these bodies, the circumference of the larger circle of the ball can be described).
General remarks about the position of the center of the ball.
1. The center of a ball inscribed in a polyhedron lies at the intersection point of the bisector planes of all dihedral angles of the polyhedron. It is located only inside the polyhedron.
2. The center of a sphere circumscribed about a polyhedron lies at the point of intersection of planes perpendicular to all edges of the polyhedron and passing through their midpoints. It can be located inside, on the surface and outside of the polyhedron.
A combination of a sphere and a prism.
1. A sphere inscribed in a straight prism.
Theorem 1. A sphere can be inscribed in a right prism if and only if a circle can be inscribed in the base of the prism, and the height of the prism is equal to the diameter of this circle.
Consequence 1. The center of a sphere inscribed in a right prism lies at the middle of the height of the prism passing through the center of a circle inscribed in the base.
Consequence 2. A ball, in particular, can be inscribed in straight lines: triangular, regular, quadrangular (in which the sums of opposite sides of the base are equal to each other) under the condition H = 2r, where H is the height of the prism, r is the radius of the circle inscribed in the base.
2. A sphere described near a prism.
Theorem 2. A sphere can be circumscribed about a prism if and only if the prism is straight and a circle can be circumscribed near its base.
Corollary 1. The center of a sphere circumscribed near a straight prism lies at the middle of the height of the prism drawn through the center of a circle circumscribed near the base.
Consequence 2. A sphere, in particular, can be described: near a right triangular prism, near a regular prism, near a rectangular parallelepiped, near a right quadrangular prism, in which the sum of the opposite angles of the base is 180 degrees.
From the textbook by L.S. Atanasyan, problems No. 632, 633, 634, 637 (a), 639 (a, b) can be proposed for the combination of a ball with a prism.
Combination of a sphere with a pyramid.
1. The ball described near the pyramid.
Theorem 3. A sphere can be circumscribed near a pyramid if and only if a circle can be circumscribed near its base.
Consequence 1. The center of a sphere circumscribed near a pyramid lies at the point of intersection of a line perpendicular to the base of the pyramid, passing through the center of a circle circumscribed near this base, and a plane perpendicular to any side edge drawn through the middle of this edge.
Consequence 2. If the side edges of the pyramid are equal to each other (or equally inclined to the plane of the base), then a ball can be described near such a pyramid. The center of this ball in this case lies at the point of intersection of the height of the pyramid (or its continuation) with the axis of symmetry of the side edge lying in the plane lateral edge and height.
Consequence 3. A ball, in particular, can be described: near a triangular pyramid, near a regular pyramid, near a quadrangular pyramid, in which the sum of opposite angles is 180 degrees.
2. A ball inscribed in a pyramid.
Theorem 4. If the side faces of the pyramid are equally inclined to the base, then a sphere can be inscribed in such a pyramid.
Consequence 1. The center of a ball inscribed in a pyramid, whose side faces are equally inclined to the base, lies at the point of intersection of the height of the pyramid with the bisector of the linear angle of any dihedral angle at the base of the pyramid, the side of which is the height of the side face drawn from the top of the pyramid.
Consequence 2. A ball can be inscribed in a regular pyramid.
From the textbook by L.S. Atanasyan, problems No. 635, 637 (b), 638, 639 (c), 640, 641 can be proposed for the combination of a ball with a pyramid.
Combination of a sphere with a truncated pyramid.
1. A ball circumscribed near a regular truncated pyramid.
Theorem 5. Near any regular truncated pyramid, a sphere can be described. (This condition is sufficient but not necessary)
2. A ball inscribed in a regular truncated pyramid.
Theorem 6. A ball can be inscribed in a regular truncated pyramid if and only if the apothem of the pyramid is equal to the sum of the apothems of the bases.
There is only one problem for combining a ball with a truncated pyramid in L.S. Atanasyan's textbook (No. 636).
A combination of a ball with round bodies.
Theorem 7. Near a cylinder, a truncated cone (right circular), a cone, a sphere can be described.
Theorem 8. A sphere can be inscribed in a cylinder (right circular) if and only if the cylinder is equilateral.
Theorem 9. A sphere can be inscribed in any cone (right circular).
Theorem 10. A ball can be inscribed in a truncated cone (right circular) if and only if its generatrix is equal to the sum of the radii of the bases.
From the textbook by L.S. Atanasyan, problems No. 642, 643, 644, 645, 646 can be proposed for the combination of a ball with round bodies.
For a more successful study of the material of this topic, it is necessary to include oral tasks in the course of the lessons:
1. The edge of the cube is equal to a. Find the radii of the balls: inscribed in a cube and circumscribed near it. (r = a/2, R = a3).
2. Is it possible to describe a sphere (ball) around: a) a cube; b) a rectangular parallelepiped; c) an inclined parallelepiped, at the base of which lies a rectangle; d) a straight parallelepiped; e) an inclined parallelepiped? (a) yes; b) yes; c) no; d) no; e) no)
3. Is it true that a sphere can be described near any triangular pyramid? (Yes)
4. Is it possible to describe a sphere around any quadrangular pyramid? (No, not near any quadrangular pyramid)
5. What properties must a pyramid have in order to describe a sphere around it? (At its base there must be a polygon, around which a circle can be described)
6. A pyramid is inscribed in the sphere, the lateral edge of which is perpendicular to the base. How to find the center of a sphere? (The center of the sphere is the intersection point of two geometric places points in space. The first is a perpendicular drawn to the plane of the base of the pyramid, through the center of the circle described around it. The second is a plane perpendicular to the given lateral edge and drawn through its middle)
7. Under what conditions can a sphere be described near a prism, at the base of which is a trapezoid? (Firstly, the prism must be straight, and, secondly, the trapezoid must be isosceles so that a circle can be described around it)
8. What conditions must a prism satisfy in order to describe a sphere around it? (The prism must be straight and its base must be a polygon around which a circle can be circumscribed)
9. A sphere is described near a triangular prism, the center of which lies outside the prism. What triangle is the base of the prism? (obtuse triangle)
10. Is it possible to describe a sphere near an inclined prism? (No you can not)
11. Under what condition will the center of a sphere circumscribed about a right triangular prism be located on one of the side faces of the prism? (The base is a right triangle)
12. The base of the pyramid is an isosceles trapezoid. The orthogonal projection of the top of the pyramid onto the plane of the base is a point located outside the trapezoid. Is it possible to describe a sphere around such a trapezoid? (Yes, you can. The fact that the orthogonal projection of the top of the pyramid is located outside its base does not matter. It is important that at the base of the pyramid lies an isosceles trapezoid - a polygon around which a circle can be described)
13. A sphere is described near the regular pyramid. How is its center located relative to the elements of the pyramid? (The center of the sphere is on a perpendicular drawn to the plane of the base through its center)
14. Under what condition does the center of a sphere circumscribed about a right triangular prism lie: a) inside the prism; b) outside the prism? (At the base of the prism: a) an acute triangle; b) obtuse triangle)
15. A sphere is described near a rectangular parallelepiped whose edges are 1 dm, 2 dm and 2 dm. Calculate the radius of the sphere. (1.5 dm)
16. In which truncated cone can a sphere be inscribed? (In a truncated cone, in the axial section of which a circle can be inscribed. The axial section of the cone is an isosceles trapezoid, the sum of its bases must be equal to the sum of its lateral sides. In other words, for a cone, the sum of the radii of the bases must be equal to the generatrix)
17. A sphere is inscribed in a truncated cone. At what angle is the generatrix of the cone visible from the center of the sphere? (90 degrees)
18. What property must a straight prism have in order to be able to inscribe a sphere in it? (Firstly, at the base of a straight prism there must be a polygon into which a circle can be inscribed, and, secondly, the height of the prism must be equal to the diameter of the circle inscribed in the base)
19. Give an example of a pyramid in which a sphere cannot be inscribed? (For example, a quadrangular pyramid, at the base of which lies a rectangle or parallelogram)
20. A rhombus lies at the base of a straight prism. Can a sphere be inscribed in this prism? (No, you can’t, since in the general case it is impossible to describe a circle near a rhombus)
21. Under what condition can a sphere be inscribed in a right triangular prism? (If the height of the prism is twice the radius of the circle inscribed in the base)
22. Under what condition can a sphere be inscribed in a regular quadrangular truncated pyramid? (If the section of this pyramid by a plane passing through the middle of the side of the base perpendicular to it is an isosceles trapezoid into which a circle can be inscribed)
23. A sphere is inscribed in a triangular truncated pyramid. What point of the pyramid is the center of the sphere? (The center of the sphere inscribed in this pyramid is at the intersection of three bisectoral planes of angles formed by the side faces of the pyramid with the base)
24. Is it possible to describe a sphere around a cylinder (right circular)? (Yes you can)
25. Is it possible to describe a sphere near a cone, a truncated cone (right circular ones)? (Yes, you can, in both cases)
26. Can a sphere be inscribed in any cylinder? What properties must a cylinder have in order for a sphere to be inscribed in it? (No, not in everyone: the axial section of the cylinder must be a square)
27. Can a sphere be inscribed in any cone? How to determine the position of the center of a sphere inscribed in a cone? (Yes, in any. The center of the inscribed sphere is at the intersection of the height of the cone and the bisector of the angle of inclination of the generatrix to the plane of the base)
The author believes that out of the three lessons that are given for planning on the topic “Different problems for polyhedra, a cylinder, a cone and a ball”, it is advisable to take two lessons for solving problems for combining a ball with other bodies. It is not recommended to prove the theorems given above due to the insufficient amount of time in the lessons. You can offer students who have sufficient skills to prove them by indicating (at the discretion of the teacher) the course or plan of the proof.
