Collection of tests in geometry on the topic "Body of revolution" (Grade 11). A collection of tests in geometry on the topic "Body of revolution" (Grade 11) Two mutually perpendicular sections are drawn in a ball

3.1. The radius of the base of the cone is R, the generatrix is inclined to the plane of the base at an angle  . In a cone through the top at an angle  a plane is drawn to its height. Find the area of the resulting section.

3.2. The areas of the bases of the truncated cone are 81  cm 2 and 225  cm 2, the generatrix relates to the height as 5: 4. Find the area of the axial section.

3.3. The diagonals of the axial section of a truncated cone are mutually perpendicular. The area of the axial section is 324 cm 2 . Find the area of the bases of the cone, knowing that the radius of one base is 2 cm greater than the other.

3.4. Dana trapezoid ABCD, in which AD= 15 cm, BC= 9 cm, AB = CD\u003d 5 cm. The trapezoid rotates around an axis passing through the top A and perpendicular AD. Find the surface area of the resulting body of revolution.

3.5. A straight line cuts off from the sides of a right-angled triangle, the angle between which is 60, segments whose lengths are a quarter of the length of the hypotenuse, counting from the vertex of this angle. Find the ratio of the area of the triangle to the surface area of the body obtained by rotating this triangle around a straight line.

3.6. The cone lies on a plane and rolls along it, rotating around its fixed vertex. The height of the cone is h, forming - b. Find the surface area described by the height of the cone.

3.7. Two cones have a common base. In the general axial section, the generatrix of one of the cones is perpendicular to the opposite generatrix of the other. The volume of one of them is half the volume of the other. Find the angle between the generatrix of the larger cone and the plane of the bases of the cones.

3.8. Triangle ABC, which one AB= 13 cm, Sun= 20 cm, AC\u003d 21 cm, rotates around an axis passing through the top A perpendicular AC. Find the volume of the resulting body of revolution.

3.9. The parallelogram rotates about an axis passing through the vertex of an acute angle perpendicular to the larger diagonal. Find the volume of the body of revolution if the sides of the parallelogram and its major diagonal are 15 cm, 37 cm and 44 cm, respectively.

3.10. The generatrix of a truncated cone, equal to l, is inclined to the plane of the base at an angle . The ratio of the areas of the bases of the cone is 4. Find the volume of the truncated cone.

12.6. Ball

Ball and sphere

sphere is the set of all points in space that are equidistant from a given point.

This point is called center spheres. A line segment connecting the center of a sphere with any point on it is called radius spheres. Chordoy A line segment that connects two points on a sphere is called. diameter called a chord passing through the center of the sphere (Fig. 12.40).

ball called a geometric body bounded by a sphere. The center, radius, chord and diameter of a sphere are named respectively center ,radius ,chord And diameter ball (Fig. 12.40).

A ball can be considered as a body obtained by rotating a semicircle about an axis containing the diameter of the semicircle.

A sphere is also called the surface of a sphere.

A plane that has a single point in common with a sphere is called a tangent. plane to the sphere (ball). The common point is called touch point spheres (ball) and planes.

Theorem . In order for a plane to be tangent to a sphere (ball), it is necessary and sufficient that this plane be perpendicular to the radius of the sphere (ball) drawn to the point of contact.

The correct formulas for a sphere are:

Where S is the surface area of the ball (area of the sphere); R is the radius of the ball; V is the volume of the sphere.

Ball segment and spherical segment

ball segment called the part of the ball cut off from it by a plane. The circle, which turned out in section, is called basis segment. The segment connecting the center of the base of the segment with a point on the surface of the ball, perpendicular to the base, is called tall ball segment (Fig. 12.41). The surface of the spherical part of a spherical segment is called spherical segment .

For a spherical segment, the following formulas are true:

Where S is the area of the spherical part of the spherical segment (the area of the spherical segment); R is the radius of the ball; h– segment height; S full is the total surface area of the spherical segment; r is the radius of the base of the spherical segment; V is the volume of the spherical segment.

Spherical layer and spherical belt

ball layer The part of a sphere enclosed between two parallel cutting planes is called. The circles obtained in the section are called grounds layer. The distance between cutting planes is called tall layer (Fig. 12.42). The surface of the spherical part of the spherical layer is called spherical belt .

A ball, a spherical segment and a spherical layer can be considered as geometric bodies of revolution. When rotating a semicircle around an axis containing the diameter of a semicircle, a ball is obtained, respectively, when rotating parts of a circle, parts of a ball are obtained: a spherical segment and a spherical layer.

For a spherical layer, the following formulas are true:

Where S 1 , S 2 - areas of bases; R 1 , R 2 - base radii; S is the area of the spherical part of the spherical layer (the area of the spherical belt); R is the radius of the ball; h- height; S full is the total surface area; V is the volume of the spherical layer.

Ball sector

Spherical sector called a geometric body obtained by rotating a circular sector (with an angle less than 90) around an axis containing one of the side radii. The addition of such a body to a ball is also called ball sector . Thus, a spherical sector consists of a spherical segment and a cone, or a spherical segment without a cone (Fig. 12.43 a, b).

For a spherical sector, the formulas are correct:

Where S is the surface area of the spherical sector; R is the radius of the ball; r is the segment base radius; h is the height of the spherical segment; V is the volume of the spherical sector.

Example 1 The radius of the sphere is divided into three equal parts. Two sections were drawn through the division points, perpendicular to the radius. Find the area of the spherical belt if the radius of the sphere is 15 cm.

Solution. Let's make a drawing (Fig. 12.44).

In order to calculate the area of a spherical belt, you need to know the radius of the ball and the height. The radius of the ball is known, and we find the height, knowing that the radius is divided into three equal parts:

Then the area

Example 2 The ball is crossed by two parallel planes passing perpendicular to the diameter and on opposite sides of the center of the ball. The areas of the spherical segments are 42  cm 2 and 70  cm 2. Find the radius of the sphere if the distance between the planes is 6 cm.

Solution. Consider two spherical segments with areas:

Where R- ball (sphere) radius, h, H – segment heights. We get the equations:
And
We have two equations with three unknowns. Let's make another equation. The ball diameter is
Let's solve the system:

From the first two equations of the system we express:

we substitute into the third equation of the system:
We solve the resulting equation:
we get

According to the condition of the problem, the value

Example 3 The section of a sphere by a plane perpendicular to its diameter divides the diameter in a ratio of 1: 2. How many times is the cross-sectional area less than the surface area of the ball?

Solution . Let's make a drawing (Fig. 12.45).

Consider the diametrical section of the ball: AD– diameter, O- center, OE= R is the radius of the ball, BE is the radius of the section perpendicular to the ball diameter,

Express BE through R:

From  OBE express BE through R:

Cross-sectional area
sphere surface area
We get the ratio

Hence, S 1 less S 2 4.5 times.

ball layer The part of a sphere enclosed between two parallel cutting planes is called. The circles obtained in the section are called grounds layer. The distance between cutting planes is called tall layer (Fig. 42). The surface of the spherical part of the spherical layer is called spherical belt .

A ball, a spherical segment and a spherical layer can be considered as geometric bodies of revolution. When a semicircle is rotated around an axis containing the diameter of the semicircle, a ball is obtained, respectively, when the parts of the circle are rotated, parts of the ball are obtained: a spherical segment and a spherical layer.

For a spherical layer, the following formulas are true:

Where R is the radius of the ball;

R1, R2 are the radii of the bases;

h- height;

S1, S2- base areas;

S is the area of the spherical part of the spherical layer (the area of the spherical belt);

S full is the total surface area;

V is the volume of the spherical layer.

Ball sector

Spherical sector called a geometric body obtained by rotating a circular sector (with an angle less than ) around an axis containing one of the side radii. The addition of such a body to a ball is also called ball sector . Thus, a spherical sector consists of a spherical segment and a cone, or a spherical segment without a cone (Fig. 43a, 43b).

Rice. 43a. Rice. 43b.

For a spherical sector, the formulas are correct:

Where R is the radius of the ball;

r is the segment base radius;

h- height of the spherical segment;

S is the surface area of the spherical sector;

V is the volume of the spherical sector.

Example 1 The radius of the sphere is divided into three equal parts. Two sections perpendicular to the radius were drawn through the division points. Find the area of the spherical belt if the radius of the sphere is 15cm.

Solution. Let's make a drawing (Fig. 44).

Then the area

Answer:

Example 2 The ball is crossed by two parallel planes passing perpendicular to the diameter and on opposite sides of the center of the ball. The areas of the spherical segments are 42p cm 2 and 70p cm 2 . Find the radius of the sphere if the distance between the planes is 6 cm.

Solution. Consider two spherical segments with areas: where R- ball (sphere) radius, h, H segment heights. We get the equations: and We have two equations with three unknowns. Let's make another equation. The diameter of the ball is Solving the system, we find the radius of the ball.

Û Þ Û

According to the condition of the problem, the value

Answer: 7 cm

Example 3 The cross section of a sphere by a plane perpendicular to its diameter divides the diameter in a ratio of 1:2. How many times is the cross-sectional area smaller than the surface area of the sphere?

Solution. Let's make a drawing (Fig. 45).

Consider the diametrical section of the ball: AD– diameter, O- center, OE=R is the radius of the ball, BE is the radius of the section perpendicular to the diameter of the ball,

Express BE through R:

From DOBE express BE through R:

Sectional area surface area of the ball We get the ratio . Means, S1 less S2 4.5 times.

Answer: 4.5 times.

Example 4 In a sphere with a radius of 13 cm, two mutually perpendicular sections are drawn at a distance of 4 cm and 12 cm from the center. Find the length of their common chord.

Solution. Let's make a drawing (Fig. 46).

The sections are perpendicular, because OO 2- distance and OO 1 - distance. Thus, and OC- the diagonal of the rectangle OO 2 CO 1 and equal to

a) Two parallel sections of the ball are drawn. Prove that the center of the ball lies on the line passing through the centers of these sections.
b) A section of radius r is drawn in a ball of radius R. What is the distance between it and the great circle parallel to it?
c) In a ball of radius 3, two sections with radii 1 and 2 are drawn, the planes of which are parallel. Calculate the distance between them,
d) Write problems inverse to problems b) and c).

a) Given two circles in one ball, the circles of which lie on the sphere and have a single common point. Prove that the line of intersection of the planes in which these circles lie has a single common point with the ball,
b) Two circles are drawn on the sphere, having a single common point. Prove that the center of the sphere, the centers of both circles and their common point lie in the same plane,
c) On a sphere of radius R, two sections of the same radius r are drawn, having one common point. Their planes form an angle cf. Establish a relationship between R, r, φ.

III. 3. In a ball of radius R, two sections of radius r intersect at an angle φ. Their intersection is a chord of length d. Establish a relationship between R, r, d, φ.

III. 4. This area includes:

a) a cylinder
b) a cone;
c) truncated cone.

Their sizes are known. How to find the distances from the center of the sphere to the bases and side surfaces of the cylinder, cone and truncated cone?

III. 5. Four equal balls of radius R are placed so that each touches the other three. Three of these balls lie on a horizontal plane, and the fourth ball lies above them. What is the height of this building? How to find the radius of the sphere described near this structure.

III. 6. Three cylinders are arranged so that every two have a single common point. This common point is inside the generatrix of each of the cylinders. The axes of the cylinders are mutually perpendicular, and one of them is vertical. The radius of each cylinder is R. Find the radius of the ball that, falling vertically, will pass through the gap formed by the cylinders.

III. 7. In a ball of radius R there is a cylinder with the largest axial section. What are the dimensions of this cylinder?

III. 8. Consider all possible cylinders with an axial section diagonal equal to d. Calculate the radius of the largest ball contained in such a cylinder and the radius of the smallest ball containing such a cylinder.

III. 9. In a cylinder whose height is equal to the diameter of the base and equal to d, two identical balls must be placed. What is their largest radius?

III. 10. Two equal cones have a common vertex. Their lateral surfaces intersect along two generators. Prove that the plane passing through these generators is perpendicular to the plane containing the axes of the cones.

III.11. Two equal cones have parallel axes. Do they have a common reference plane passing through the surfaces that form them?

III.12. Prove that the circle is a line of intersection (if such exists):

a) the lateral surfaces of the cone and the cylinder, the axes of which lie on the same straight line);
b) the side surfaces of two cones whose axes lie on the same straight line.

III.13. The center of the sphere lies at the apex of the cone. The radius of the sphere is less than the generatrix of the lateral surface of the cone. Prove that the sphere intersects the lateral surface of the cone in a circle.

a) A circle is drawn on a real sphere. How to calculate its radius?
b) How to calculate the radius of a real sphere (ball)?

We use a computer

III.15. Given a line p and a segment AB on a line parallel to p. Find a point X on the line p such that the angle AXB is the largest.

III.16. Among all isosceles triangles ABC circumscribed about a given circle tangent to the base of AC, find the triangle with the smallest area.

III.17. Is there a point on a given line from which two equal circles are visible at equal angles?

III.18. Inscribe a rectangle of the largest area in the given circle.

III.19. Given a circle with center O. A chord AB, different from the diameter, and a radius OS perpendicular to this chord are drawn in it. Let D be the intersection point of this radius and this chord. Point X moves along the larger arc of the circle. Two chords are drawn from it: XK, passing through point D, and XC. Let L be the intersection point of the chords XC and AB. Which of the segments is longer: KD or LC?

Results of Chapter III

In § 16-19 only three theorems are proved:

theorem 17 on the intersection of a ball with a plane (section 16.2),
Theorem 18 on the tangency of a sphere and a plane (Sec. 16.3) and
Theorem 19 on the section of a cone (Sec. 19.1).

In chapter III, a discussion of the important question of the symmetry of spatial figures began.

In § 20, more complex questions of the geometry of a circle are studied than in the course of the basic school.

1. Lines a and b are parallel, and lines a and c intersect. What is the relative position of b and c? (done)
2. A plane is drawn through three points lying on three skew edges of a cube. Find the sum of the interior angles of the polygon obtained in the section. (done)
3. All side edges of the pyramid are equal to 13. The radius of the circle inscribed at the base of the pyramid is 5, and the radius of the circle circumscribed near the base of the pyramid is 12. Find the height of the pyramid. instructed
4. All dihedral angles at the edges of the base of a quadrangular pyramid are 45. The radius of the circle inscribed in the base of the pyramid is 8, and the radius of the circle circumscribed near the base of the pyramid is 52. Find the height of the pyramid. (made)
5. The planes of the three side faces of the triangular pyramid form an angle of 60 with the plane of its base. The radius of the circle inscribed in the base of the pyramid is 8, and the radius of the circle circumscribed near the base of the pyramid is 52. Find the height of the pyramid. (done)
6. The distance between the centers of two spheres of radii 4 and 7 is equal to 2. Describe the set of common points of these spheres. (made)
7. Two generators of the cone are mutually perpendicular. Can the angle in the development of the cone be equal to 252. (done)
8. ABCD - axial section of the cylinder. B and C are the points of the upper base, and A and D are the points of the lower base. Point K divides the arc AD in relation to AK:KD=1:2. Find the value of the angle AKC. (made)
9. The section passing through the middle of the lateral edge of the pyramid and parallel to the base, split the pyramid into two bodies, the volume of one of which is 6 m ^ 3 less than the other. Find the volume of the pyramid. (made)
10. MABC is a tetrahedron. How many different planes are there from which all the vertices of this tetrahedron are at the same distance?
11. At what value of x is the length of the vector with coordinates (1-x; 4 + x; x) the smallest? (made)
12. What part of the volume of the parallelepiped ABCDA1B1C1D1 is occupied by the volume of the tetrahedron A1C1BD? (made)
13. Can two planes of non-adjacent side faces of a quadrangular pyramid be perpendicular to the base plane?
14. The distance from the ends of the diameter of the ball to the plane touching it is 3 and 7 cm. Find the radius of the ball. (made)

In the 8th, I was only able to draw a picture and remember that the angle ACB is equal to the angle BAC, as if lying crosswise. Then I don't know what to do.

In 13 they can by the 3-perpendicular theorem. Yes?

In the 10th, maybe 4. I guess because the tetrahedron has 4 faces, but I don’t see the logic.

In the 9th it turned out 8.

k.black you wrote like this:
I argued the same.
The volume of one such pyramid to be cut off is equal to 1/6 of the volume of the parallelepiped (1/3 * half of the base * the same height)
So, the volume of the cut-off part is 4/6 = 2/3
Then the volume of the pyramid A1C1BD is 1/3 of the volume of par-yes

I can’t understand why you first have volumes as 1/6, and then as 1/3

GBOU SPO Fri 13 named after P.A. Ovchinnikov

Tests on the topic "Body of revolution"

teacher of mathematics Makeeva Elena Sergeevna

T E S T 1

Option 1

A1 . The lateral surface area of a right circular cylinder is 12π, and the height of the cylinder is 3. Find the total surface area of the cylinder.

¤ 1) 24π ¤ 2) 16π ¤ 3) 22π ¤ 4) 20π

A2 . The area of the axial section of the cylinder is 10 cm 2 , base area is 5 cm 2

¤ 1) ¤ 2) ¤ 3) ¤ 4)
A3 . Two sections are drawn through the generatrix of the cylinder, of which one is axial with an area equal toS. The angle between the planes of sections is 30 O

¤ 1) ¤ 2) S ¤ 3) ¤ 4)

B 1. The ends of the segment AB lie on the circles of the bases of the cylinder. The radius of the base is 10 cm, the distance between the straight line AB and the axis of the cylinder is 8 cm, AB \u003d 13 cm. Determine the height of the cylinder.

Answer:

AT 2 . The height of the cylinder ish, base radius -r. A square is inscribed obliquely to the axis in this cylinder so that all its vertices are on the circles of the bases. Find the side of the square.

Answer :________________________________________________________________________

C1 . The diagonal of the development of the side surface of the cylinder makes an angle β with the side of the base of the development. Calculate the angle between the diagonal of the axial section of the cylinder and the base plane.

Answer:________________________________________________________________________

T E S T 1

Cylinder. The surface area of a cylinder.

Option 2

A1. The lateral surface area of a right circular cylinder is 20π and the height of the cylinder is 5. Find the total surface area of the cylinder.

¤ 1) 24π ¤ 2) 32π ¤ 3) 28π ¤ 4) 36π

A2 . The area of the axial section of the cylinder is 16 cm 2 , base area is 8 cm 2 . Calculate the height and area of the lateral surface of the cylinder.

¤ 1) ¤ 2) ¤ 3) ¤ 4) A3. Two sections are drawn through the generatrix of the cylinder, of which one is axial with an area equal toS. The angle between the planes of sections is 45 O . Find the area of the second section.

¤ 1) ¤ 2) ¤ 3) ¤ 4) S

B 1. The ends of the segment AB lie on the circles of the bases of the cylinder. The radius of the base is 5 cm, the height of the cylinder is 6 cm, AB = 10 cm. Determine the distance between the straight line AB and the axis of the cylinder.

Answer: ________________________________________________________________________

AT 2 . The radius of the base of the cylinder isr. In this cylinder, a square is inscribed obliquely with sideaso that all its vertices are on the circles of the bases. Find the height of the cylinder.

Answer: ________________________________________________________________________

C1 . The angle between the diagonal of the axial section of the cylinder and the plane of its base is equal to β. Calculate the angle between the diagonal of the development of its side surface and the side of the base of the development.

Answer: ________________________________________________________________________

T E S T 2

Straight circular cone

Option 1

A1 . Find the height of a right circular cone if its axial section is 6 cm 2 , and the base area is 8 cm 2 .

¤ 1) 3 2) 3 ¤ 3) 6 ¤ 4) 4

A2. Determine the angle at the vertex of the axial section of the cone if the development of its lateral surface is a sector with an arc equal to 90 o

¤ 1) 60 o ¤ 2) 2 arcsin ¤ 3) 2 arcsin ¤ 4) 30 o

A3. The circumference of the bases of the truncated cone is 4π and 10π. The height of the cone is 4. Find the surface area of the truncated cone.

¤ 1) 64 π ¤ 2) 68 π ¤ 3) 52 π ¤ 1) 74 π

B 1. The height of the cone is equal to the radiusRits foundations. A plane is drawn through the top of the cone, cutting off an arc of 60 o

Answer:

AT 2. The generatrix of the cone is 13 cm, the height is 12 cm. This cone is crossed by a straight line parallel to the base. Its distance from the base is 6 cm, and from the height - 2 cm. Find the length of the segment of this straight line enclosed inside the cone.

Answer: ________________________________________________________________________________

C1 . The generatrix of the truncated cone is equal toLand makes an angle α with the base plane. The diagonal of its axial section is perpendicular to the generatrix. Find the area of the lateral surface of the cone.

Answer: ________________________________________________________________________________

T E S T 2

Straight circular cone

Option 2

A1 . Find the height of a right circular cone if its axial section is 8 cm 2 , and the base area is 12 cm 2 .

1) 4 ¤ 2) 4 ¤ 3) 6 ¤ 4) 6

A2 . Determine the angle at the vertex of the axial section of the cone if the development of its lateral surface is a sector with an arc equal to 120 o

¤ 1) 90 o ¤ 2) 2 arcsin ¤ 3) 2 arcsin ¤ 4) 60 o

A3 . The circumference of the bases of the truncated cone is 4π and 28π. The height of the cone is 5. Find the surface area of the truncated cone.

¤ 1) 420 π ¤ 2) 412 π ¤ 3) 416 π ¤ 1) 408 π

B 1. The height of the cone is equal to the radiusRits foundations. A plane is drawn through the top of the cone, cutting off an arc of 90 o . Determine the sectional area.

Answer: ________________________________________________________________________________

AT 2. The generatrix of the cone is 17 cm, the height is 8 cm. This cone is crossed by a straight line parallel to the base. Its distance from the base is 4 cm, and from the height - 6 cm. Find the length of the segment of this straight line enclosed inside the cone.

Answer: ________________________________________________________________________________

C1 . The generatrix of a truncated cone makes an angle α with the plane of the lower base. The diagonal of its axial section is perpendicular to the generatrix of the cone. The sum of the circumferences is 2 πm. Find the area of the lateral surface of the cone.

Answer: ________________________________________________________________________________

T E S T 3

Option 1

A1 . Points A and B lie on a sphere of radiusR. Find the distance from the center of the sphere to the line AB if AB=m.

¤ 1) ¤ 2) ¤ 3) ¤ 4)

A2. Find the coordinates of the center C and the radiusRsphere given by the equation

¤ 1) C (-3; 2; 0), R= ¤ 2) C (3; -2;0), R=5 ¤ 3) C (-3; 2;0), R=5 ¤ 4) C (3; -2;0), R=

A3. Write the equation of a sphere centered at point C (4; -1; 3) passing through point A (-2; 3; 1)

¤ 1) ¤ 2)

¤ 3) ¤ 4)

B 1 . Vertices of a right triangle with legs 25 and 5lie on the sphere. Find the radius of the sphere if the distance from the center to the plane of the triangle is 8.

Answer: ________________________________________________________________________________

B 2 athe equation

defines the scope.

Answer: ________________________________________________________________________________

C1. Two mutually perpendicular sections of the ball have a common chord of length 12. It is known that the areas of these sections are 100π and 64π . Find the radius of the ball.

Answer: ________________________________________________________________________________

T E S T 3

Sphere and ball. Sphere equation.

Option 2

A1. Points A and B lie on a sphere of radiusR. The distance from the center of the sphere to the line AB isa. Find the length of segment AB.

¤ 1) ¤ 2) ¤ 3) ¤ 4)

A2 . Find the coordinates of the center C and the radiusRsphere given by the equation

¤ 1) C (-4; 0; 3), R= ¤ 2) C (4; 0;-3), R=7 ¤ 3) C (-4; 0;3), R=7 ¤ 4) C (4; 0;-3), R=

A3. Write the equation of a sphere centered at point C (-3; 1; -2) passing through point A (3; 4; -1)

¤ 1) ¤ 2)

¤ 3) ¤ 4)

B 1 . The vertices of a right triangle with legs 15 and lie on the sphere. Find the radius of the sphere if the distance from the center to the plane of the triangle is 5.

Answer: ________________________________________________________________________________

B 2 . Determine at what values of the parameterathe equation

defines the scope.

Answer: ________________________________________________________________________________

C1. Two mutually perpendicular sections of the ball have a common chord of length 12. It is known that the areas of these sections are 256π and 100π . Find the radius of the ball.

Answer: ________________________________________________________________________________

T E S T 4

Option 1

A1. The line of intersection of the sphere and the plane 8 away from the center has a length of 12 π. Find the surface area of the sphere.

¤ 1) 396 π ¤ 2) 400 π ¤ 3) 408 π ¤ 4) 362π

A2. Sphere RadiusRtouches the faces of a dihedral angle, the value of which is equal toα . Determine the distance from the center of the sphere to the edge of the dihedral angle.

¤ 1) ¤ 2) Rtg ¤ 3) ¤ 4) Rctg

A3. Find the chord length of the sphere , belonging to the x-axis.

¤ 1) 2 ¤ 2) 4 ¤ 3) 8 ¤ 4) 2

IN 1. The cross section of a ball by two parallel planes, between which lies the center of the ball, have areas of 144π and 25π . Calculate the surface area of a sphere if the distance between parallel planes is 17.

AT 2.

And

Answer

C1.

Answer:________________________________________________________________________________

T E S T 4

The mutual arrangement of a sphere and a plane, a sphere and a straight line.

Option 2

A1. Ball sectionplane 15 away from its center has an area of 64 π. Find the surface area of the sphere.

¤ 1) 1156 π ¤ 2) 1024 π ¤ 3) 1172 π ¤ 4) 1096π

A2. The sphere touches the faces of a dihedral angle, the value of which is equal toα . The distance from the center of the sphere to the edge of the dihedral angle isl. Determine the radius of the sphere.

¤ 1) l tg ¤ 2) l sin ¤ 3) l cos ¤ 4) lctg

A3. Find the chord length of the sphere , belonging to the y-axis..

¤ 1) 2 ¤ 2) 10 ¤ 3) 4 ¤ 4) 2

IN 1. The cross section of a ball by two parallel planes that lie on the same side of the center of the ball have areas of 576π and 100π . Calculate the surface area of a sphere if the distance between parallel planes is 14.

Answer:________________________________________________________________________________

AT 2. Write the equation of the plane containing the common points of the spheres given by the equations

And

Answer:________________________________________________________________________________

C1. Find the coordinates of the points of intersection of the line given by the equation and the sphere given by the equation

Answer:________________________________________________________________________________

T E S T 5

Combinations of figures of rotation.

Option 1

A1. A right triangle with legs 5 cm and 12 cm revolves around the hypotenuse. Calculate the surface area of the resulting body of revolution.

¤ 1) cm 2 ¤ 2) 82π cm 2 ¤ 3) cm 2 ¤ 4) 78π cm 2

A2. A sphere is inscribed in a cylinder. Find the ratio of the total surface area of the cylinder to the surface area of the sphere.

¤ 1) 3:2 ¤ 2) 2:1 ¤ 3) 4:3 ¤ 4) 5:2

A3. r, height -H

¤ 1) ¤ 2) ¤ 3) π( ¤ 4)

B 1 . A cylinder is inscribed in a cone, the height of which is equal to the radius of the base of the cone. Find the angle between the axis of the cone and its generatrix, if the area of the total surface of the cylinder is related to the area of the base of the cone as 3:2, and the axis of the cylinder coincides with the axis of the cone.

Answer: ________________________________________________________________________________

C1 . Three identical balls of radius lie on a planeRrelating to each other. A fourth ball of the same radius is placed on top of the hole formed by the balls. Find the distance from the top of the fourth ball to the plane.

Answer :________________________________________________________________________________

T E S T 5

Combinations of figures of rotation.

Option 2

A1. A right triangle with legs 8 cm and 15 cm revolves around the hypotenuse. Calculate the surface area of the resulting body of revolution.

¤ 1) 162π cm 2 ¤ 2) cm 2 ¤ 3) 164π cm 2 ¤ 4) cm 2

A2. A sphere is inscribed in a cylinder. Find the ratio of the lateral surface area of the cylinder to the surface area of the sphere.

¤ 1) 2:1 ¤ 2) 3:2 ¤ 3) 1:1 ¤ 4) 2:3

A3. A cone is inscribed in a sphere whose base radius isr, height -L. Determine the surface area of the sphere.

¤ 1) π ( ¤ 2) ¤ 3) pr ¤ 4) πL

B 1 . A cylinder is inscribed in a cone, the height of which is equal to the radius of the base of the cone. Find the angle between the axis of the cone and its generatrix, if the area of the total surface of the cylinder is related to the area of the base of the cone as 8:9, and the axis of the cylinder coincides with the axis of the cone.

Answer: ________________________________________________________________________________

C1 . Four identical balls of radius lie on a planeRso that each of the balls touches two adjacent ones. A fifth ball of the same radius is placed on top of the hole formed by the balls. Find the distance from the top of the fifth ball to the plane.

Answer :________________________________________________________________________________

T E S T 6

Option 1

A1. A cylinder is inscribed in a regular triangular prism. Find its surface area if the side of the base of the prism is 2, and the height is 3.

¤ 1) 6π ¤ 2) 8π ¤ 3) 10π ¤ 4) 5π

A2. A cone is described around a regular triangular pyramid. Calculate the area of the lateral surface of the cone if the side of the base of the pyramid isa, side ribs are inclined to the base at an angle of 30 o .

¤ 1) ¤ 2) ¤ 3) 4)

A3. A sphere is inscribed in a regular quadrangular prism. Find the ratio of the total surface area of the prism to the area of the sphere.

¤ 1) ¤ 2) ¤ 3) ¤ 4)

IN 1. aAndb. Find the area of the side surface of the pyramid.

Answer:________________________________________________________________________________

AT 2. Into a cube with an edge equal toa, a ball is inscribed. Calculate the radius of a ball that touches the given ball and three faces of a cube that share a common vertex.

Answer:________________________________________________________________________________

C1. The axial section of the cone is an equilateral triangle. A regular triangular pyramid is inscribed in this cone. Find the ratio of the areas of the lateral surfaces of the pyramid and the cone.

Answer:________________________________________________________________________________

T E S T 6

Combinations of polyhedra and bodies of revolution.

Option 2

A1. A cylinder is described around a regular triangular prism. Find its surface area if the height of the prism is 4 and the height of the base of the prism is 6.

¤ 1) 64π ¤ 2) 56π ¤ 3) 68π ¤ 4) 60π

A2. In a regular triangular pyramid, the side of the base is equal toa, the side faces are inclined to the plane of the base at an angle of 45 o . Calculate the lateral surface area of a cone inscribed in the pyramid.

¤ 1) ¤ 2) ¤ 3) 4)

A3. A sphere is described around the cube. Find the ratio of the area of the sphere to the total surface area of the cube.

¤ 1) ¤ 2) ¤ 3) ¤ 4)

IN 1. A regular triangular truncated pyramid is described near the ball, the sides of the bases of which are equalaAndb. Find the surface area of the sphere.

Answer:________________________________________________________________________________

AT 2. A ball is inscribed in a cube. The radius of a ball touching the given ball and three faces of a cube that have a common vertex is equal toR. Calculate the edge length of the cube.

Answer:________________________________________________________________________________

C1. The axial section of the cone is an equilateral triangle. A regular quadrangular pyramid is inscribed in this cone. Find the ratio of the areas of the lateral surfaces of the pyramid and the cone.

Answer:________________________________________________________________________________

T E S T 7

Option 1

A1. A rectangle with sides equal to 10 cm and 12 cm rotates around the larger side. Find the total surface area of the resulting body of revolution.

¤ 1) 460π cm 2 ¤ 2) 420π cm 2 ¤ 3) 440 π cm 2 ¤ 4) 400π cm 2

A2 a. Calculate the area of the section passing through two generators of the cone, the angle between which is 60 o .

¤ 1) A 2 ¤ 2) A 2 ¤ 3) A 2 ¤ 4) A 2

A3 . Determine the total surface area of a truncated cone, if the radii of its bases are 6 cm and 10 cm, the height is 3 cm.

¤ 1)212π cm 2 ¤ 2)224π cm 2 ¤ 3)220π cm 2 ¤ 4)216π cm 2

A4. + + +6 x-8 y+2 z-7=0

¤ 1) 132 π ¤ 2) 136 π ¤ 3) 140 π ¤ 4) 128p

A5. The sides of the triangle touch a sphere of radius 5 cm. Determine the distance from the center of the sphere to the plane of the triangle if its sides are 15 cm, 15 cm and 24 cm.

A6. In a cone with an angle rinscribed sphere of radiusR. Find the valuerif knownRAnd .

¤ 1) Rtg( - ¤ 2) Rtg( + ¤ 3) Rtg ¤ 4) Rctg

IN 1 . Two mutually perpendicular planes are drawn through the generatrix of the cylinder. The areas of the obtained sections are cm 2 And

Answer: _______________________________________________________________________________

AT 2. An isosceles triangle rotates around its axis of symmetry. Find the sides of this triangle if its perimeter is 30 cm and the total surface area of the body of revolution is 60

Answer: ________________________________________________________________________________

AT 3 . Sphere RadiusRtouches all edges of a regular triangular prism. Find the length of the side edge of the prism and the distance from the center of the sphere to the planes of the side faces.

Answer: ________________________________________________________________________________

C1 DD: D.B.=1:2:3. Determine the ratio of the radii of the sections (smaller to larger), if the straight line containing the given diameter forms an angle with the planes .

Answer: ________________________________________________________________________________

C2. The sphere touches all the edges of a regular quadrangular pyramid. Find the radius of such a sphere if all the edges of the pyramid are 18 cm.

Answer: ________________________________________________________________________________

T E S T 7

Generalization of the theme "Cylinder, cone, ball".

Option 2

A1. A rectangle with sides equal to 8 cm and 10 cm rotates around the smaller side. Find the total surface area of the resulting body of revolution.

¤ 1) 360π cm 2 ¤ 2) 354π cm 2 ¤ 3) 368 π cm 2 ¤ 4) 376π cm 2

A2 . The axial section of the cone is a right triangle with hypotenuse equal toa. Calculate the area of the section passing through two generators of the cone, the angle between which is 45 o .

¤ 1) A 2 ¤ 2) A 2 ¤ 3) A 2 ¤ 4) A 2

A3 . Determine the area of the total surface of a truncated cone, if the radii of its bases are 5 cm and 8 cm, the height is 4 cm.

¤ 1)150π cm 2 ¤ 2)154π cm 2 ¤ 3)158π cm 2 ¤ 4)146π cm 2

A4. Find the surface area of the sphere given by the equation + + -4 x+2 y+6 z-4=0

¤ 1) 68 π ¤ 2) 80 π ¤ 3) 76 π ¤ 4) 72 π

A5. The sides of the triangle touch a sphere of radius 5 cm. Determine the distance from the center of the sphere to the plane of the triangle if its sides are 10 cm, 10 cm and 12 cm.

¤ 1) 1 cm ¤ 2) 2 cm ¤ 3) 3 cm ¤ 4) 4 cm

A6. In a cone with an angle at the top of the axial section and the radius of the baserinscribed sphere of radiusR. Find the valueRif known

Answer: ________________________________________________________________________________

AT 3 . Sphere RadiusRtouches all edges of a regular triangular prism. Find the length of the edge of the base of the prism and the distance from the center of the sphere to the planes of the bases of the prism.

Answer: ________________________________________________________________________________

C1 . Two parallel planes intersect the diameter of the sphere AB at points C andDdividing it with respect to AC:CD: D.B.=1:3:4. Determine the ratio of the radii of the sections (smaller to larger), if the straight line containing the given diameter forms an angle with the planes .

Answer: ________________________________________________________________________________

C2. The sphere touches all the edges of a regular quadrangular pyramid. Find the radius of such a sphere if all the edges of the pyramid are 22 cm.

Answer: ________________________________________________________________________________

676π

4x-6y+2z+7=0

(-4 ;5;2), (; )

2704π

3x-4y+8z-12=0

(3;0;7), (1;2;3)

(2+ )R

2(2+ )R

12 cm, 9 cm, 9 cm

11 cm