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Spheres described around polyhedra.
Definition. A polyhedron is said to be inscribed in a sphere (and a sphere described about 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 given points, passing through its middle (the plane of the perpendicular bisectors to this segment). AB ┴ α AO=OB α A B O
Theorem 2. Set of points equidistant from n given points, lying on the same circle, there is a straight line perpendicular to the plane of these points, passing through the center of the circle circumscribed about them. C E A B D O a . . . . . . C E A B D . . . . .
A 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 . X 1. .A .B .C .D E. X. a a 1 . O. O 1
Consequences. 1)A sphere can be described around a straight triangular prism, because You can always describe a circle around a triangle. 2) A sphere can be described around any regular prism, because a regular prism is straight and a circle can always be described around a regular polyhedron. O. O. .
Task No. 1. The ball is circumscribed around a 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: Rw = ? 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 sh C 1 B 1 A 1 A C B
Task No. 3. The dimensions 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: Rw = ? Solution: 1) AC 2 =a 2 +b 2 +c 2. 2) A 1 C 2 =25+9+4=38 (Property of the diagonals of a rectangular parallelepiped) 3) A 1 C=√38; R w = O w C = √38 /2 Answer: √38 /2 D 1 C 1 B 1 A 1 A B C D 5 2 3 . . . O sh
Task No. 3. The side of the base of a regular triangular prism is equal to a, and the side edge is equal to 2 a. Find the radius of the circumscribed sphere. Given: AB=BC=AC=a, AA 1 ┴ABC ; AA 1= 2a. Find: Rw = ? Solution: 1)AB=AO √3; AO=a/√3. 2)R w =√ a 2 + a 2 /3=2a/ √ 3 Answer: 2a/ √ 3 C 1 B A 1 C B 1 A O w R w. O O 1
Consequences. 1) You can always describe a sphere around a triangular pyramid, since you can always describe a circle around a triangle. 2)About regular pyramid you can always describe the sphere. 3) If the lateral edges of the pyramid are equal (equally inclined to the base), then a sphere can always be described around such a pyramid. *In the last two cases, the center of the sphere lies on the straight line containing the height of the pyramid. O. O.
Problems (sphere described near the pyramid). Near the PABC pyramid, the base of which is regular triangle ABC with side 4√3, a sphere is described. 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: Rw = ? Solution: 1) OO SF ┴(ABC); O – center of a circle circumscribed about ∆ABC; K O SF ┴ PA; KP=AK (KO SF One of the midperpendiculars to the side edge PA); O SF is the center of the circumscribed sphere. 2) OO SF ┴(ABC); OO SF belongs to (AKO); PA ┴(ABC); AK belongs to (AKO) ; means KA|| OO SF; . O SF. O K. P. A. B. C
Problems (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 = R w =5 Answer: 5
Problems (sphere described near the pyramid). In a regular quadrangular pyramid, the side 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 – regular pyramid; (AP^(ABC))=45 ˚; PO=h. Find: Rw = ? Solution: 1) AO=OP=h; AP=h √ 2; 2) ∆PAP 1 – rectangular; PP 1 – ball diameter; PP 1 = 2 R w; AP 2 = PP 1 *OP; (h √ 2) 2 =2 R w *h; R w = 2h 2 /2h=h. Answer: h. C. B A. .D .P .P 1 . O
Tasks (sphere described near the pyramid). On one's own. The radius of a sphere circumscribed about a regular tetrahedron is equal to R. Find the total surface area of the tetrahedron.
Problems (sphere described near the pyramid). On one's own. Given: DABC – 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, which means the angle DAD 1 is an inscribed angle based on the diameter, DD 1; angle DAD 1 =90 ˚; 4) AO – height ∆ ADD 1 drawn from the vertex 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 = 8R 2 /3; .D 1 .D .O .B .C A. a a
Problems (sphere described near the pyramid). On one's own. 6) S full tet. = 8R 2 √ 3/3 Answer: 8R 2 √ 3/3
The topic “Different problems on polyhedra, cylinder, cone and 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 and others on this topic (p. 138) one can only find definitions of a polyhedron described around a sphere, a polyhedron inscribed in a sphere, a sphere inscribed in a polyhedron, and a sphere described around a polyhedron. IN methodological recommendations this textbook (see the book “Studying geometry in grades 10–11” by S.M. Saakyan and V.F. Butuzov, p. 159) says what combinations of bodies are considered when solving problems No. 629–646, and addresses attention to the fact that “when solving a particular problem, first of all, it is necessary to ensure that students have a good understanding of the relative positions of the bodies indicated in the condition.” The following is the solution to problems No. 638(a) and No. 640.
Considering all of the above, and the fact that the most difficult problems for students are the combination of a ball with other bodies, it is necessary to systematize the relevant theoretical principles and communicate them to students.
Definitions.
1. A ball is called inscribed in a polyhedron, and a polyhedron described around a ball if the surface of the ball touches all faces of the polyhedron.
2. A ball is called circumscribed about a polyhedron, and a polyhedron inscribed in a ball, if the surface of the ball passes through all the vertices of the polyhedron.
3. A ball is said to be inscribed in a cylinder, truncated cone (cone), and a cylinder, truncated cone (cone) is said to be inscribed around the ball if the surface of the ball touches the bases (base) and all the generatrices of the cylinder, truncated cone (cone).
(From this definition it follows that a circle can be inscribed in any axial section of these bodies great circle ball).
4. A ball is said to be circumscribed about a cylinder, a truncated cone (cone), if the circles of the bases (base circle and apex) belong to the surface of the ball.
(From this definition it follows that around any axial section of these bodies the circle of a larger circle of the ball can be described).
General notes on the position of the center of the ball.
1. The center of a ball inscribed in a polyhedron lies at the point of intersection of the bisector planes of all dihedral angles of the polyhedron. It is located only inside the polyhedron.
2. The center of a ball circumscribed about a polyhedron lies at the intersection point of planes perpendicular to all edges of the polyhedron and passing through their midpoints. It can be located inside, on the surface or outside the polyhedron.
Combination of a sphere and a prism.
1. A ball inscribed in a straight prism.
Theorem 1. A sphere can be inscribed into a straight prism if and only if a circle can be inscribed at the base of the prism, and the height of the prism is equal to the diameter of this circle.
Corollary 1. The center of a sphere inscribed in a right prism lies at the midpoint of the altitude of the prism passing through the center of the circle inscribed in the base.
Corollary 2. A ball, in particular, can be inscribed in straight lines: triangular, regular, quadrangular (in which the sums of the 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 circumscribed about a prism.
Theorem 2. A sphere can be described around a prism if and only if the prism is straight and a circle can be described around its base.
Corollary 1. The center of a sphere circumscribed about a straight prism lies at the midpoint of the height of the prism drawn through the center of a circle circumscribed about the base.
Corollary 2. A ball, 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 equal to 180 degrees.
From the textbook by L.S. Atanasyan, problems No. 632, 633, 634, 637(a), 639(a,b) can be suggested for the combination of a ball and a prism.
Combination of a ball with a pyramid.
1. A ball described near a pyramid.
Theorem 3. A ball can be described around a pyramid if and only if a circle can be described around its base.
Corollary 1. The center of a sphere circumscribed about a pyramid lies at the point of intersection of a straight line perpendicular to the base of the pyramid passing through the center of a circle circumscribed about this base and a plane perpendicular to any lateral edge drawn through the middle of this edge.
Corollary 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 around 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 extension) with the axis of symmetry of the side edge lying in the plane lateral edge and height.
Corollary 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 ball can be inscribed in such a pyramid.
Corollary 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.
Corollary 2. You can fit a ball into a regular pyramid.
From the textbook by L.S. Atanasyan, problems No. 635, 637(b), 638, 639(c), 640, 641 can be suggested for the combination of a ball with a pyramid.
Combination of a ball with a truncated pyramid.
1. A ball circumscribed about a regular truncated pyramid.
Theorem 5. A sphere can be described around any regular truncated pyramid. (This condition is sufficient, but not necessary)
2. A ball inscribed in a regular truncated pyramid.
Theorem 6. A ball can be inscribed into 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 the combination of a ball with a truncated pyramid in L.S. Atanasyan’s textbook (No. 636).
Combination of ball with round bodies.
Theorem 7. A sphere can be described around a cylinder, a truncated cone (straight circular), or a cone.
Theorem 8. A ball can be inscribed into a (straight circular) cylinder if and only if the cylinder is equilateral.
Theorem 9. You can fit a ball into any cone (straight circular).
Theorem 10. A ball can be inscribed into a truncated cone (straight circular) if and only if its generator 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 suggested for the combination of a ball with round bodies.
To more successfully study the material on this topic, it is necessary to include oral tasks in the lessons:
1. The edge of the cube is equal to a. Find the radii of the balls: inscribed in the cube and circumscribed around it. (r = a/2, R = a3).
2. Is it possible to describe a sphere (ball) around: a) a cube; b) rectangular parallelepiped; c) an inclined parallelepiped with a rectangle at its base; d) straight parallelepiped; e) an inclined parallelepiped? (a) yes; b) yes; c) no; d) no; d) no)
3. Is it true that a sphere can be described around 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 should be a polygon around which a circle can be described)
6. A pyramid is inscribed in a sphere, the side 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 loci of points in space. The first is a perpendicular drawn to the plane of the base of the pyramid, through the center of a circle circumscribed around it. The second is a plane perpendicular to a given side edge and drawn through its middle)
7. Under what conditions can you describe a sphere around 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 for a sphere to be described around it? (The prism must be straight, and its base must be a polygon around which a circle can be described)
9. A sphere is described around a triangular prism, the center of which lies outside the prism. Which triangle is the base of the prism? (Obtuse triangle)
10. Is it possible to describe a sphere around 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 lateral 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 a 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 described around 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 around a rectangular parallelepiped whose edges are 1 dm, 2 dm and 2 dm. Calculate the radius of the sphere. (1.5 dm)
16. What truncated cone can a sphere fit into? (In a truncated cone, into 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, the sum of the radii of the bases of the cone must be equal to the generator)
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 for a sphere to be inscribed into 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 that cannot fit a sphere? (For example, a quadrangular pyramid with a rectangle or parallelogram at its base)
20. At the base of a straight prism is a rhombus. Is it possible to fit a sphere into this prism? (No, you can’t, because near the rhombus in general case can't describe a circle)
21. Under what condition can a sphere be inscribed into 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 into a regular quadrangular truncated pyramid? (If the cross-section of a given pyramid is a plane passing through the middle of the side of the base perpendicular to it, it is an isosceles trapezoid into which a circle can be inscribed)
23. A sphere is inscribed in a triangular truncated pyramid. Which 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 bisectral planes of angles formed by the lateral 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 around a cone, a truncated cone (straight circular)? (Yes, you can, in both cases)
26. Can a sphere be inscribed into any cylinder? What properties must a cylinder have in order to fit a sphere into it? (No, not every time: the axial section of the cylinder must be square)
27. Can a sphere be inscribed into any cone? How to determine the position of the center of a sphere inscribed in a cone? (Yes, absolutely. The center of the inscribed sphere is at the intersection of the altitude 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 planning lessons on the topic “Different problems on polyhedra, cylinder, cone and ball”, it is advisable to devote two lessons to solving problems on combining a ball with other bodies. It is not recommended to prove the theorems given above due to insufficient time in class. You can invite students who have sufficient skills for this to prove them by indicating (at the teacher’s discretion) the course or plan of the proof.
A ball can be described around a pyramid if and only if a circle can be described around its base.
To construct the center O of this ball, you need:
1. Find the center O of the circle circumscribed about the base.
2. Through point O, draw a straight line perpendicular to the plane of the base.
3. Draw a plane through the middle of any lateral edge of the pyramid perpendicular to this edge.
4. Find the point O of the intersection of the constructed line and plane.
Special case: the lateral 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 described around a prism if and only if the prism is straight and a circle can be described around 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; - radius of the circle described near the base; H is the height of the prism.
5.3. Ball and cylinder
A ball can always be described around a cylinder. The center of the ball is the center of symmetry of the axial section of the cylinder.
5.4. Ball and cone
A ball can always be described around 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 around a sphere. Calculate the ratio of the total surface area of the prism and the volume of the sphere
A prism is called regular if its bases are regular polygons and its side edges are perpendicular to the base. A regular quadrilateral is a square. The point of intersection of the diagonals of a square is its center, as well as the center of the circle inscribed in it. Let's prove this fact. although this proof is unlikely to be asked and can be omitted
As a special type of parallelogram, rectangle and rhombus, the square has their properties: the diagonals are equal and bisected by the point of intersection, and are bisectors of the corners of the square. Through point E we draw a straight line TK parallel to AB. AB is perpendicular to BC, which means TC is also perpendicular to BC (if one of two parallel lines is perpendicular to any third line, then the second parallel line is perpendicular to this (third) line). In the same way we will carry out direct MR. Right triangles BET and AEK are equal in hypotenuse and acute angle (BE=AE - half of the diagonals, ∠ EBT=∠ EAK - half of the right angle), which means ET=EK. In the same way we prove that EM=EP. And from the equality of triangles CEP and CET (the same sign) we see that ET = EP, i.e. ET=EP=EK=EM or simply say that 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 AVTC (this quadrilateral is a rectangle, since all the angles in it are right angles by construction). In a rectangle, opposite sides are equal - AB = CT (it should be noted that CT 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 planes through parallel (two straight lines perpendicular to the same plane are parallel) AA 1, CC 1 and BB 1 and DD 1, respectively (parallel straight lines define only one plane). 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 point H (intersection of diagonals) in 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, draw a perpendicular to another plane, then this perpendicular lies completely in the first plane - we find that this perpendicular must lie both in the plane AA 1 C 1 C and in the plane BB 1 D 1 D. This is only possible if this the 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 (since it is NOT equidistant from the planes of the lateral faces, and this in turn follows from the equidistance of points E and H from the vertices of the corresponding bases (according to what has been 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 into a regular prism if and only if its height is equal to the diameter of the circle inscribed in the base. Well, it is already inscribed in our prism ball, which means 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 is h, then using this theorem we conclude A=h and then the volume of the prism is found like this: 
Next, using the fact that the height is equal to the diameter of the inscribed ball and the side of the base of the prism, we find the radius of the ball and then its volume: 
It must be said that the lateral edges are equal to the height (segments of parallel straight lines enclosed between parallel planes 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 relate it to the volume of the ball: 
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 lateral 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 lateral edge of this prism.
Answer: 10.
3. Find the lateral 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 circumscribed about 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 circumscribed about a cylinder whose base radius is 2. The lateral surface area of the prism is 48. Find the height of the cylinder.
Right 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 whose side edges are equal to .
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, all edges of which are equal to √3.
Answer: 13.5
4. Find the volume of the polyhedron whose vertices are points A, B, C, D, A1, B1, C1, D1 of a regular hexagonal prism ABCDEFA1B1C1D1E1F1, the base area of which is 6, and the lateral edge is 2.
Straight prism (arbitrary n-coal)
A prism whose side edges are perpendicular to the bases, and the 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. The section parallel to the base of the pyramid is a polygon similar to the base. The cross-sectional areas and base areas 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 its apex is projected into the center of the base.
3. All lateral edges of a regular pyramid are equal, and the lateral 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 equal to 10, the side edges are equal to 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, the adjacent sides of which are perpendicular, and one of the side edges is perpendicular to the plane of the base and equal to 3.
Answer: 27.
6. Find the volume of a regular triangular pyramid whose base sides are equal to 1 and whose height is equal to .
Answer: 0.25.
7. The lateral 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 lateral edge is 5. Find the volume of the pyramid.
Answer: 32.
9. In a regular quadrangular pyramid, the height is 12 and 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 equal to 6, the side edges are equal to 5. Find the surface area of the pyramid.
Answer: 84.
11. The volume of a regular hexagonal pyramid is 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 the 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 volume 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 equal to 2 and whose volume is equal to .
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, is cut off triangular pyramid plane passing through the top of the pyramid and the midline of the base. Find the volume of the cut-off triangular pyramid.
Answer: 3.
Cylinder
A cylinder is a body bounded by a cylindrical surface and two circles with boundaries.
| H |
| R |
| Body volume | Lateral surface area | Base area | Total surface area |
 |  |  |  |
 |  |  |
1. Generators of a cylinder - segments of generatrixes enclosed between the bases.
2. The height of the cylinder is the length of the generatrix.
3. The axial section is a rectangle, two sides of which are generatrices, and the other two are the diameters of the bases of the cylinder.
4. Circular section - a section whose cutting plane is perpendicular to the axis of the cylinder.
5. Development of the side surface of the cylinder - a rectangle representing two edges of the cut of the side 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 total surface area of a cylinder is called the sum of the areas of the lateral surface and the two bases.
8. You can always describe a sphere around a cylinder. Its center lies at 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. You can fit a ball into 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, its generatrix is 6 and is inclined to the plane of the base at an angle of 30°.
Answer: 3.
3. The cylinder and 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, located 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 axial cross-sectional area of the cylinder is equal to . Find the lateral surface area of the cylinder.
Answer: 2.
6. A regular quadrangular prism is circumscribed about a cylinder whose base radius and height are equal to 2. Find the lateral surface area 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 tall 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 whose diameter is 3 times more than the first?
Answer: 2.
Cone
A cone is a body bounded by a conical surface and a circle.
  |
| cone axis |
| R |
| vertex |
| forming |
| 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 sweep angle and the apex angle 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, its generatrix is 6 and is inclined to the plane of the base at an angle of 30°.
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 about 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 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 apex of the axial section is 90°. Calculate the volume of the cone divided by .
8. A cone is described around a regular quadrangular pyramid with a base side of 4 and a height of 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 ball 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 ball by a plane is a circle.
3. A tangent plane to a sphere is a plane that has only one common point with the sphere.
4. The radius of the sphere, drawn to the point of contact of 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 circumscribed about a sphere if the sphere touches all its faces.
7. Segments of tangents to a sphere drawn from one point are equal and amount to 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 parallelepiped is described 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 parallelepiped is described around a sphere of radius 2. Find its volume.
8. The volume of a rectangular parallelepiped circumscribed about a sphere is 216. Find the radius of the sphere.
Answer: 3.
9. The surface area of a rectangular parallelepiped circumscribed around a sphere is 96. Find the radius of the sphere.
Answer: 2.
10. A cylinder is described around the ball, the lateral surface area of which is 9. Find the surface area of the ball.
Answer: 9.
11. How many times the surface area of a sphere circumscribed around a cube? more area surface 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 angles. 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 angles. Give your answer in degrees.
Answer: 60.
3. Find the surface area of the polyhedron shown in the figure (all dihedral angles are right angles).
Answer: 18.
4. Find the surface area of the polyhedron shown in the figure (all dihedral angles are right angles).
Answer: 132
5. Find the surface area of the spatial cross shown in the figure and composed 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 angles. Find the tangent of angle ABB3.
Answer: 2
10. The figure shows a polyhedron; all dihedral angles of the polyhedron are right angles. Find the tangent of angle C3D3B3.
Answer: 3
11. Through the middle line of the base of the triangular prism, a plane is drawn parallel to the side edge. Find the lateral surface area of the prism if the lateral surface area of the trimmed triangular prism is 37.
Answer: 74.
12. The figure shows a polyhedron; all dihedral angles of the polyhedron are right angles. Find the square of the distance between vertices B2 and D3.
Answer: 11.
