14 A rectangular box is circumscribed around a sphere of radius 1. Find its volume. 54 The base of a right triangular prism is a right triangle with legs 3 and 5. The volume of the prism is 30. Find its side edge. 94 A ball is inscribed in a cube with edge 3. Find the volume of this ball divided by π. 134 The volume of a cube is 12. Find the volume of a triangular prism cut off from it by a plane passing through the midpoints of two edges emerging from one vertex and parallel to the third edge emerging from the same vertex. 174 Find the volume of a polyhedron whose vertices are points A, B, C, A 1 of a regular triangular prism ABCDA 1 B 1 C 1 D 1 whose base area is 2, and the side edge is 3. Aleksandrova Ekaterina (issue 2012)

14 (prototype B) A rectangular parallelepiped is circumscribed around a sphere of radius 1. Find its volume. ABCDA 1 B 1 C 1 D 1 - cube V = a 3 a = d = 2 R = 2 1 = 2 V = 2 3 = 8 Answer: 8

54 (prototype B) The base of a right triangular prism is a right triangle with legs 3 and 5. The volume of the prism is 30. Find its side edge. V \u003d S main h 30 \u003d 7.5 h Answer: 4

94 (prototype B) A ball is inscribed in a cube with edge 3. Find the volume of this ball divided by π. Answer: 4.5

134 (prototype B) The volume of a cube is 12. Find the volume of a triangular prism cut off from it by a plane passing through the midpoints of two edges emerging from one vertex and parallel to the third edge emerging from the same vertex. Answer: 1.5

174 (prototype B) Find the volume of a polyhedron whose vertices are points A, B, C, A 1 of a regular triangular prism ABCDA 1 B 1 C 1 D 1 whose base area is 2, and the side edge is 3. Answer: 2 B C1C1 A1A1 B1B1 C A Alexandrova Ekaterina 11 "A"

A rectangular parallelepiped into which a ball is inscribed will be a cube, the edge of which is equal to the diameter of the ball. V=a3 a=2 => 2?2?2=8. Answer: 8. Prototype of task B9 (No. 27043). A rectangular parallelepiped is circumscribed about a sphere of radius 1. Find its volume. Decision.

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USE in mathematics

“Unified State Examination Tasks in Mathematics” - Task B 5. Task B 13. Task B 3. We need to solve a couple more examples. After rain, the water level in the well may rise. Find the speed of the motorcyclist. Task B 12. Task B 6. Preparation for the exam. Independent work. How much should the water level rise after rain? Task B 1. Find the area.

"B3 in Mathematics" - Skills in CT. Logarithms with the same base. Find the root of the equation. Let's solve the linear equation. Tasks for independent decision. Note to the student. The equation. Degree. Job prototype. Logarithms. Properties of logarithms. Preparation for the exam in mathematics. The content of the task B3. Root of the equation.

"B8 in the exam in mathematics" - Find the value of the derivative of the function. Speed. The value of the derivative of the function. The derivative of the function is negative. Intervals of increasing function. The number of extremum points of the function. The value of the derivative at the point of contact. The line is tangent to the graph of the function. Time. Decreasing intervals of a function.

"B1 in math" - Shampoo bottle. speed on the speedometer. Income tax. Taxi driver. Decision. Income tax. A pack of butter. How many notebooks at a price of 6.6 rubles can be bought for 80 rubles. Sale day discount. Customer. Quarter. Advertising campaign. Ticket. Marmalade. Mobile phone. Tasks B1 USE in mathematics.

"The solution of tasks B11" - Find the smallest value of the function. Tasks. Examination. Find the smallest value of the function on the segment. Beginnings of mathematical analysis. Note to the student. Find the largest value of the function. Decision. CT skills. Formulas. Find the highest value. Tasks for independent decision. Job prototype B11.

"Math assignment B9" - Surface area. The surface area of ​​a cylinder. The volume of the ball. Decision. The surface area of ​​a cone. Volume. Tasks for independent decision. volume of the pyramid. The surface area of ​​a sphere. verifiable requirements. Cone volume. Job prototype. Note to the student. The volume of the cube. The volume of a rectangular parallelepiped.

Total in the topic 33 presentations

Alexandrova Natalia (issue of 2012) 15 A rectangular parallelepiped is circumscribed near a sphere of radius 8.5. Find its volume 55 At the base of a right prism lies a right triangle with legs 3 and 3. The lateral edges are equal to 5/p. Find the volume of the cylinder circumscribed by this prism. 95 The area of ​​the great circle of the ball is 3. Find the surface area of ​​the ball. 135 Find the surface area of ​​the polyhedron shown in the figure (all dihedral angles are right). 175 Find the volume of the polyhedron shown in the figure (all dihedral angles are right).

At the base of a right prism lies a right triangle with legs 3 and 3. The lateral edges are equal to 5/n. Find the volume of the cylinder circumscribed by this prism. Task B11 (4969) A right triangle with legs 3 and 3 lies at the base of a right prism. The side edges are equal to 5/p. Find the volume of the cylinder circumscribed by this prism. A C B H \u003d 5 / n From ABC (angle C - straight): Answer: 22.5

Answer: 30 Task B11 (25583) Find the surface area of ​​the polyhedron shown in the figure (all dihedral angles are right angles). 1) Left and right edges: 2 * (2 *3)=12 2) Front and back edges: 2 * (2 *3)=12 3) Top and bottom edges: 2 * (2 * 2– 1 *1) =2*3=6

No. 1. The side of the base of a regular quadrangular pyramid is 4 cm. The flat angle at the top of the pyramid is 60 degrees. find: a) the volume of the pyramid; b) the angle that the side face forms with the base plane.

SO \u003d H - the height of the pyramid, draw OM perpendicular to AB. Then SM is perpendicular to AB (by the theory of 3 perpendiculars).

By condition AB=4, angle ASB=60º, then angle ASM=30º.

In ASM 3: SM = AM ctg 30º = 2√3. In 3rd SOM: SO2 = SM2- OM2 =(2√3)2-22 = 12 - 4 =8. SO = √8 = 2√2

a) V = Sbase H/3 = 4 4 2√2/3 = 32√2 / 3.

b) angle1 = angleSMO. From 3 SOM: OM / SM = cos (SMO angle) = 2/(2√3) = 1/√3.

SMO angle = arccos(1/√3)

or SO / MO = tan angle SMO = 2√2 / 2 = √2 --> angle SMO = arctg √2.

No. 2. A rectangular parallelepiped is circumscribed about a cylinder whose base radius and height are equal to 1. Find the volume of the parallelepiped.

The base of the parallelepiped is a square. Its sides are equal to the diameter of the base of the cylinder, i.e. a=d=2.

V= Sprim H = a2 H = 22 1=4. Answer: 4.

No. 3. A rectangular parallelepiped is circumscribed around a sphere of radius 7.5. Find its volume.

If a cuboid is circumscribed near a sphere, then it is a cube. Its edges are equal to the diameter of the sphere, i.e. a \u003d 7.5 2 \u003d 15.

V= a3 = 153 = 3375.

No. 4. The cylinder and the cone have a common base and a common height. Calculate the volume of the cylinder if the volume of the cone is 27

Vcylinder \u003d Son H,

Vcone \u003d Son H / 3 \u003d 27.

We see that the volume of the cone is 3 times less than the volume of the cylinder, so Vcylinder = Vcone * 3 = 27 * 3 = 81.

No. 5. In a regular 4-sided pyramid, the angle between the height and the side edge is 45 degrees. Find the flat corner at the vertex.

Angle OSB and angle OBS are 45°, then BO=SO=x.

In rectangular 3-ke AOB: BO=OA=x. 3-to SOB = 3-ku AOB on two legs --> SB=BA and SB=SA.

3-way ABS - equilateral --> all angles in it are 60°.

Answer: AOB=60°

A cuboid is circumscribed about a cylinder whose base radius and height are equal to 1. Find the volume of the cuboid.

27042

A cuboid is circumscribed about a cylinder whose base radius is 4. The volume of the cuboid is 16. Find the height of the cylinder.

27043

A rectangular parallelepiped is circumscribed about a sphere of radius 1. Find its volume.

27044

Find the volume of the polyhedron shown in the figure (all dihedral angles of the polyhedron are right).

2000 cm 3 of water was poured into a cylindrical vessel. The liquid level turned out to be 12 cm. The part was completely immersed in water. At the same time, the liquid level in the vessel rose by 9 cm. What is the volume of the part? Express your answer in cm3.

27046

In a cylindrical vessel, the liquid level reaches 16 cm. At what height will the liquid level be if it is poured into a second cylindrical vessel, the diameter of which is 2 times greater than the diameter of the first? Express your answer in centimeters.

27047

2300 cm 3 of water was poured into a vessel having the shape of a regular triangular prism and the part was completely immersed in it. At the same time, the liquid level in the vessel rose from 25 cm to 27 cm. What is the volume of the part? Express your answer in cm3.

27048

Water is poured into a vessel shaped like a regular triangular prism. The water level reaches 80 cm. At what height will the water level be if it is poured into another similar vessel, whose base side is 4 times larger than the first one? Express your answer in cm.

27049

At the base of a straight prism lies a right triangle with legs 6 and 8. The side edges are equal. Find the volume of the cylinder circumscribed by this prism.

27050

The base of a straight prism is a square with a side of 2. The side edges are equal. Find the volume of the cylinder circumscribed by this prism.

27051

The cone and the cylinder have a common base and a common height (the cone is inscribed in the cylinder). Calculate the volume of the cylinder if the volume of the cone is 25.

27052

The volume of the cone is 16. Through the middle of the height, a section is drawn parallel to the base of the cone, which is the base of a smaller cone with the same vertex. Find the volume of the smaller cone.

27056

The volume of a cube is 8. Find its surface area.

27074

The volume of the parallelepiped ABCDA 1 B 1 C 1 D 1 is 9. Find the volume of the triangular pyramid ABC A 1 .

27076

The area of ​​a face of a cuboid is 12. The edge perpendicular to this face is 4. Find the volume of the cuboid.

27077

The volume of a cuboid is 24. One of its edges is 3. Find the area of ​​the face of the cuboid that is perpendicular to this edge.

27078

The volume of a cuboid is 60. The area of ​​one of its faces is 12. Find the edge of the cuboid that is perpendicular to this face.

27079

The two edges of the cuboid outgoing from the same vertex are 2 and 6. The volume of the cuboid is 48. Find the third edge of the cuboid outgoing from the same vertex.

27080

Three edges of a cuboid coming out of the same vertex are equal to 4, 6, 9. Find the edge of a cube of equal area.

27081

How many times will the volume of a cube increase if its edges are tripled?

27082

The base of a right triangular prism is a right triangle with legs 6 and 8, the side edge is 5. Find the volume of the prism.

27083

The base of a right triangular prism is a right triangle with legs 3 and 5. The volume of the prism is 30. Find its side edge.

27084

Find the volume of a regular hexagonal prism with base sides equal to 1 and side edges equal to .

27085

How many times will the volume of a regular tetrahedron increase if all its edges are doubled?

27086

The base of the pyramid is a rectangle with sides 3 and 4. Its volume is 16. Find the height of this pyramid.

27087

Find the volume of a regular triangular pyramid whose base sides are 1 and whose height is .

27088

Find the height of a regular triangular pyramid whose base sides are 2 and whose volume is .

27089

How many times will the volume of the pyramid increase if its height is quadrupled?

27091

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? Express your answer in liters.

27093

Find the volume V of a cone whose generatrix is ​​equal to 2 and is inclined to the plane of the base at an angle of 30 0 . Please indicate in your answer.

27094

By how many times will the volume of a cone decrease if its height is tripled?

27095

How many times will the volume of a cone increase if its base radius is increased by 1.5 times?

27096

The cone and the cylinder have a common base and a common height (the cone is inscribed in the cylinder). Calculate the volume of the cone if the volume of the cylinder is 150.

27097

How many times will the volume of a sphere increase if its radius is tripled?

27098

The diagonal of a cube is . Find its volume.

27099

The volume of a cube is 24. Find its diagonal.

27100

Two edges of a cuboid coming out of the same vertex are 2, 4. The diagonal of the cuboid is 6. Find the volume of the cuboid.

27101

Two edges of a cuboid coming out of the same vertex are equal to 2, 3. The volume of the cuboid is 36. Find its diagonal.

27102

If each edge of the cube is increased by 1, then its volume will increase by 19. Find the edge of the cube.

27103

The diagonal of a rectangular parallelepiped is equal to and forms angles 30 0 , 30 0 and 45 0 with the planes of the faces of the parallelepiped. Find the volume of the parallelepiped.

27104

The face of the parallelepiped is a rhombus with a side of 1 and an acute angle of 60 0 . One of the edges of the parallelepiped makes an angle of 60 0 with this face and is equal to 2. Find the volume of the parallelepiped.

27105

The volume of a cuboid circumscribed about a sphere is 216. Find the radius of the sphere.

27106

Through the midline of the base of a triangular prism, the volume of which is 32, a plane is drawn parallel to the side edge. Find the volume of the cut off triangular prism.

27107

A plane parallel to the lateral edge is drawn through the midline of the base of the triangular prism. The volume of the cut off triangular prism is 5. Find the volume of the original prism.

27108

Find the volume of a prism whose bases are regular hexagons with sides 2, and side edges equal to 2 and inclined to the plane of the base at an angle of 30 0 .

27109

In a regular quadrangular pyramid, the height is 6, the side edge is 10. Find its volume.

27110

The base of the pyramid is a rectangle, one side face is perpendicular to the base plane, and the other three side faces are inclined to the base plane at an angle of 60 0 . The height of the pyramid is 6. Find the volume of the pyramid.

27111

The side edges of a triangular pyramid are mutually perpendicular, each of them is equal to 3. Find the volume of the pyramid.

27112

From a triangular prism, the volume of which is equal to 6, a triangular pyramid is cut off by a plane passing through the side of one base and the opposite vertex of the other base. Find the volume of the rest.

27113

The volume of the triangular pyramid SABC, which is part of the regular hexagonal pyramid SABCDEF, is equal to 1. Find the volume of the hexagonal pyramid.

27114

The volume of a regular quadrangular pyramid SABCD is 12. Point E is the midpoint of edge SB. Find the volume of the triangular pyramid EABC.

27115

From a triangular pyramid, the volume of which is equal to 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.

27116

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.

27117

Find the volume of the spatial cross shown in the figure and composed of unit cubes.

27118

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.

27120

The height of the cone is 6, the generatrix is ​​10. Find its volume divided by

27121

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

27122

A cone is obtained by rotating an isosceles right triangle ABC around a leg equal to 6. Find its volume divided by.

27123

The cone is described near a regular quadrangular pyramid with a base side of 4 and a height of 6. Find its volume divided by

27124

How many times greater is the volume of a cone circumscribed near a regular quadrangular pyramid than the volume of a cone inscribed in this pyramid?

27125

The radii of the three balls are 6, 8 and 10. Find the radius of the ball whose volume is equal to the sum of their volumes.

27126

A sphere is inscribed in a cube with edge 3. Find the volume of this sphere divided by

27127

A sphere is described near a cube with an edge. Find the volume of this sphere divided by

27141

The surface area of ​​a cube is 24. Find its volume.

27146

Two edges of a cuboid coming out of the same vertex are 1, 2. The volume of the cuboid is 6. Find its surface area.

27162

The volume of one ball is 27 times the volume of the second. How many times greater is the surface area of ​​the first sphere than the surface area of ​​the second?

27168

The volume of one cube is 8 times the volume of the other cube. How many times greater is the surface area of ​​the first cube than the surface area of ​​the second cube?

27174

The volume of the sphere is 288 . Find its surface area divided by .

27176

Find the volume of a pyramid whose height is 6 and whose base is a rectangle with sides 3 and 4.

27178

In a regular quadrangular pyramid, the height is 12, the volume is 200. Find the side edge of this pyramid

27179

The side of the base of a regular hexagonal pyramid is 2, the side edge is 4. Find the volume of the pyramid.

27180

The volume of a regular hexagonal pyramid is 6. The side of the base is 1. Find the side edge.

27181

The side of the base of a regular hexagonal pyramid is 4, and the angle between the side face and the base is 45 0 . Find the volume of the pyramid.

27182

The volume of the parallelepiped ABCDA 1 B 1 C 1 D 1 is 12. Find the volume of the triangular pyramid B 1 ABC

27183

The volume of a cube is 12. Find the volume of a triangular prism cut off from it by a plane passing through the midpoints of two edges emerging from one vertex and parallel to the third edge emerging from the same vertex.

27184

The volume of a cube is 12. Find the volume of a quadrangular pyramid whose base is the face of the cube and whose apex is the center of the cube.

27187

27188 27189

27190 27191

27196 27197

Find the volume V of the part of the cylinder shown in the figure. Please indicate in your answer.

Find the volume V of the part of the cylinder shown in the figure. Please indicate in your answer

27198 27199

27200 27201

27202 27203

Find the volume V of the part of the cone shown in the figure. Please indicate in your answer

27209

The volume of the parallelepiped ABCDA 1 B 1 C 1 D 1 is 4.5. Find the volume of the triangular pyramid AD 1 CB 1

27210 27211

Find the volume of the polyhedron shown in the figure (all dihedral angles are right).

27212 27213

27214

The volume of a tetrahedron is 1.9. Find the volume of a polyhedron whose vertices are the midpoints of the edges of the given tetrahedron.

27216

Find the volume of the polyhedron shown in the figure (all dihedral angles are right).

77154

Find the volume of the parallelepiped ABCDA 1 B 1 C 1 D 1 if the volume of the triangular pyramid ABDA 1 is 3.

245335

Find the volume of a polyhedron whose vertices are points A, D, A 1 , B, C, B 1 of a cuboid ABCDA 1 B 1 C 1 D 1 with AB=3, AD=4, AA 1 =5

245336

Find the volume of a polyhedron whose vertices are points A, B, C, D 1 , B, B 1 of the cuboid ABCDA 1 B 1 C 1 D 1 with AB=4, AD=3, AA 1 =4

245337

Find the volume of a polyhedron whose vertices are points A 1 , B, C, C 1 , B 1 of a cuboid ABCDA 1 B 1 C 1 D 1 with AB=4, AD=3, AA 1 =4

245338

Find the volume of a polyhedron whose vertices are points A, B, C, B 1 of a cuboid ABCDA 1 B 1 C 1 D 1 with AB=3, AD=3, AA 1 =4

245339

Find the volume of a polyhedron whose vertices are points A, B, B 1 , C 1 of a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1 with AB=5, AD=3, AA 1 =4

245340

Find the volume of a polyhedron whose vertices are points A, B, C, A 1 of a regular triangular prism ABCA 1 B 1 C 1 whose base area is 2 and whose side edge is 3.

245341

Find the volume of a polyhedron whose vertices are points A, B, C, A 1 , C 1 of a regular triangular prism ABCA 1 B 1 C 1 whose base area is 3 and whose side edge is 2.

245342



The cylinder is described beside the ball. The volume of the cylinder is 33. Find the volume of the sphere.
245349

The cylinder is described beside the ball. The volume of the sphere is 24. Find the volume of the cylinder.

245350

The cone and the cylinder have a common base and a common height (the cone is inscribed in the cylinder). Calculate the volume of the cylinder if the volume of the cone is 5.

245351

A cone is inscribed in a sphere. The radius of the base of the cone is equal to the radius of the ball. The volume of the sphere is 28. Find the volume of the cone.

245352

A cone is inscribed in a sphere. The radius of the base of the cone is equal to the radius of the ball. The volume of the cone is 6. Find the volume of the sphere.

245353

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.

245355

The cube is inscribed in a sphere of radius . Find the volume of the cube.

245357

Find the volume of a regular hexagonal prism all edges of which are equal

318145

In a vessel shaped like a cone, the liquid level reaches a height. The volume of liquid is 70 ml. How many milliliters of liquid must be added to completely fill the vessel?

318146