A sphere inscribed in a regular triangular prism. Polyhedra circumscribed about a sphere A polyhedron is said to be circumscribed about a sphere if the planes of all its faces touch the sphere

Polyhedra circumscribed about a sphere A polyhedron is said to be circumscribed about a sphere if the planes of all its faces touch the sphere. The sphere itself is said to be inscribed in a polyhedron. Theorem. A sphere can be inscribed in a prism if and only if a circle can be inscribed in its base, and the height of the prism is equal to the diameter of this circle. Theorem. Any triangular pyramid can be inscribed with a sphere, and moreover, only one.

Exercise 1 Erase the square and draw two parallelograms representing the top and bottom faces of the cube. Connect their vertices with segments. Get an image of a sphere inscribed in a cube. Draw a sphere inscribed in a cube, as in the previous slide. To do this, draw an ellipse inscribed in a parallelogram obtained by compressing a circle and a square by 4 times. Mark the poles of the sphere and the tangent points of the ellipse and parallelogram.

Exercise 1 A sphere is inscribed in a right quadrangular prism, at the base of which is a rhombus with a side of 1 and an acute angle of 60 o. Find the radius of the sphere and the height of the prism. Solution. The radius of the sphere is half the height DG of the base, i.e. The height of the prism is equal to the diameter of the sphere, i.e.

Exercise 4 The sphere is inscribed in a right quadrangular prism, at the base of which is a quadrilateral, perimeter 4 and area 2. Find the radius r of the inscribed sphere. Solution. Note that the radius of the sphere is equal to the radius of the circle inscribed in the base of the prism. Let us use the fact that the radius of a circle inscribed in a polygon is equal to the area of ​​this polygon divided by its half-perimeter. We get

Exercise 3 Find the radius of a sphere inscribed in a regular triangular pyramid, the base side of which is 2, and the dihedral angles at the base are 60 o. Solution. Let us use the fact that the center of the inscribed sphere is the point of intersection of the bisectoral planes of the dihedral angles at the base of the pyramid. The sphere radius OE satisfies the equality Therefore,

Exercise 4 Find the radius of a sphere inscribed in a regular triangular pyramid, the side edges of which are equal to 1, and the flat angles at the top are 90 o. Answer: Decision. In the tetrahedron SABC we have: SD = DE = SE = From the similarity of triangles SOF and SDE we obtain an equation, solving which, we find

Exercise 1 Find the radius of a sphere inscribed in a regular quadrangular pyramid, all edges of which are equal to 1. Let's use the fact that for the radius r of a circle inscribed in a triangle, the formula takes place: r = S / p, where S is the area, p is the semiperimeter of the triangle . In our case S = p = Solution. The radius of the sphere is equal to the radius of the circle inscribed in the triangle SEF, in which SE = SF = EF=1, SG = Therefore,

Exercise 2 Find the radius of a sphere inscribed in a regular quadrangular pyramid, the base side of which is equal to 1, and the side edge is 2. Let's use the fact that for the radius r of a circle inscribed in a triangle, the formula takes place: r = S / p, where S - area, p is the half-perimeter of the triangle. In our case S = p = Solution. The radius of the sphere is equal to the radius of the circle inscribed in the triangle SEF, in which SE = SF = EF=1, SG = Therefore,

Exercise 3 Find the radius of a sphere inscribed in a regular quadrangular pyramid, the base side of which is 2, and the dihedral angles at the base are 60 o. Solution. Let us use the fact that the center of the inscribed sphere is the point of intersection of the bisectoral planes of the dihedral angles at the base of the pyramid. The sphere radius OG satisfies the equality Therefore,

Exercise 4 The unit sphere is inscribed in a regular quadrangular pyramid, the base side of which is 4. Find the height of the pyramid. Let's take advantage of the fact that for the radius r of a circle inscribed in a triangle, the formula takes place: r = S/p, where S is the area, p is the half-perimeter of the triangle. In our case S = 2h, p = Solution. Let's denote the height SG of the pyramid as h. The radius of the sphere is equal to the radius of the circle inscribed in the triangle SEF, in which SE = SF = EF=4. Therefore, we have an equality from which we find

Exercise 1 Find the radius of a sphere inscribed in a regular hexagonal pyramid, in which the base edges are 1 and the side edges are 2. Let's use the fact that for the radius r of a circle inscribed in a triangle, the formula takes place: r \u003d S / p, where S is the area, p is the half-perimeter of the triangle. In our case, S = p = Therefore, Solution. The radius of the sphere is equal to the radius of the circle inscribed in the triangle SPQ, in which SP = SQ = PQ= SH =

Exercise 2 Find the radius of a sphere inscribed in a regular hexagonal pyramid with base edges equal to 1 and dihedral angles at the base equal to 60 o. Solution. Let us use the fact that the center of the inscribed sphere is the point of intersection of the bisectoral planes of the dihedral angles at the base of the pyramid. The sphere radius OH satisfies the equality Therefore,
Exercise Find the radius of a sphere inscribed in a unit octahedron. Answer: Decision. The radius of the sphere is equal to the radius of the circle inscribed in the rhombus SESF, in which SE = SF = EF=1, SO = Then the height of the rhombus, lowered from the vertex E, will be equal to The desired radius is equal to half the height, and is equal to O

Exercise Find the radius of a sphere inscribed in a unit icosahedron. Solution. Let us use the fact that the radius OA of the circumscribed sphere is equal to and the radius AQ of the circle circumscribed about an equilateral triangle with side 1 is equal to By the Pythagorean theorem applied to right triangle OAQ, we get Exercise Find the radius of a sphere inscribed in a unit dodecahedron. Solution. We use the fact that the radius OF of the circumscribed sphere is equal to and the radius FQ of a circle circumscribed about an equilateral pentagon with side 1 is Equal to the Pythagorean theorem, applied to a right triangle OFQ, we obtain

Exercise 1 Can a sphere be inscribed in a truncated tetrahedron? Solution. Note that the center O of a sphere inscribed in a truncated tetrahedron must coincide with the center of a sphere inscribed in a tetrahedron, which coincides with the center of a sphere semi-inscribed in a truncated tetrahedron. Distances d 1, d 2 from the point O to the hexagonal and triangular faces are calculated using the Pythagorean theorem: where R is the radius of the semi-inscribed sphere, r 1, r 2 are the radii of the circles inscribed in the hexagon and triangle, respectively. Since r 1 > r 2, then d 1 r 2, then d 1

"Sphere of politics" - The relationship of social actors about state power. Scientific and theoretical. The process of interaction between politics and the economy. Together with the state. The regulation of social relations is determined by social interests. The process of interaction between politics and morality. The power of the state, persuasion, stimulation.

"Prism geometry" - A straight quadrangular prism ABCDA1B1C1D1 is given. Euclid probably thought practical guides by geometry. Straight prism - a prism in which the lateral edge is perpendicular to the base. Prism in geometry. By property 2 of volumes, V=V1+V2, that is, V=SABD h+SBDC h=(SABD+SBDC) h. So triangles A1B1C1 and ABC are equal in three sides.

"Volume of a prism" - How to find the volume of a straight prism? The volume of the original prism is equal to the product S · h. Basic steps in proving the direct prism theorem? The area S of the base of the original prism. Draw the altitude of triangle ABC. A task. direct prism. Lesson goals. The concept of a prism. The volume of a straight prism. The solution of the problem. The prism can be divided into straight triangular prisms with height h.

"Surface of the sphere" - Mars. Is the ball a ball? Ball and sphere. Earth. Encyclopedia. We support our high school baseball team. Venus. Uranus. Is it a ball in the picture? A bit of history. Atmosphere. I decided to do a little research……. Saturn. Are you ready to answer questions?

The topic “Different problems on polyhedra, a cylinder, a cone and a ball” is one of the most difficult in the 11th grade geometry course. Before solving geometric problems, they usually study the relevant sections of the theory that are referred to when solving problems. In the textbook by S. Atanasyan et al. on this topic (p. 138) one can find only the definitions of a polyhedron circumscribed about a sphere, a polyhedron inscribed in a sphere, a sphere inscribed in a polyhedron, and a sphere circumscribed near a polyhedron. IN guidelines this textbook (see the book “Studying geometry in grades 10–11” by S.M. Saakyan and V.F. Butuzov, p. 159) says which combinations of bodies are considered when solving problems No. 629–646, and attention to the fact that "when solving a particular problem, first of all, it is necessary to ensure that students have a good idea of ​​the relative position of the bodies indicated in the condition." The following is the solution of problems No. 638 (a) and No. 640.

Considering all of the above, and the fact that the most difficult tasks for students are the tasks of combining a ball with other bodies, it is necessary to systematize the relevant theoretical positions and communicate them to students.

Definitions.

1. A ball is called inscribed in a polyhedron, and a polyhedron is said to be circumscribed near the ball, if the surface of the ball touches all the faces of the polyhedron.

2. A ball is called circumscribed near a polyhedron, and a polyhedron is called inscribed in a ball if the surface of the ball passes through all the vertices of the polyhedron.

3. A ball is called inscribed in a cylinder, a truncated cone (cone), and a cylinder, a truncated cone (cone) is called circumscribed near the ball, if the surface of the ball touches the bases (base) and all generators of the cylinder, truncated cone (cone).

(It follows from this definition that the circumference of the great circle of the ball can be inscribed in any axial section of these bodies).

4. A ball is called circumscribed near a cylinder, a truncated cone (cone) if the circles of the bases (the circle of the base and the top) belong to the surface of the ball.

(From this definition it follows that about any axial section of these bodies, the circumference of the larger circle of the ball can be described).

General remarks about the position of the center of the ball.

1. The center of a ball inscribed in a polyhedron lies at the intersection point of the bisector planes of all dihedral angles of the polyhedron. It is located only inside the polyhedron.

2. The center of a sphere circumscribed about a polyhedron lies at the point of intersection of planes perpendicular to all edges of the polyhedron and passing through their midpoints. It can be located inside, on the surface and outside of the polyhedron.

A combination of a sphere and a prism.

1. A sphere inscribed in a straight prism.

Theorem 1. A sphere can be inscribed in a right prism if and only if a circle can be inscribed in the base of the prism, and the height of the prism is equal to the diameter of this circle.

Consequence 1. The center of a sphere inscribed in a right prism lies at the middle of the height of the prism passing through the center of a circle inscribed in the base.

Consequence 2. The ball, in particular, can be inscribed in straight lines: triangular, regular, quadrangular (in which the sums of opposite sides of the base are equal to each other) under the condition H = 2r, where H is the height of the prism, r is the radius of the circle inscribed in the base.

2. A sphere described near a prism.

Theorem 2. A sphere can be circumscribed about a prism if and only if the prism is straight and a circle can be circumscribed near its base.

Corollary 1. The center of a sphere circumscribed near a straight prism lies at the middle of the height of the prism drawn through the center of a circle circumscribed near the base.

Consequence 2. A 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 180 degrees.

From the textbook by L.S. Atanasyan, problems No. 632, 633, 634, 637 (a), 639 (a, b) can be proposed for the combination of a ball with a prism.

Combination of a sphere with a pyramid.

1. The ball described near the pyramid.

Theorem 3. A sphere can be circumscribed near a pyramid if and only if a circle can be circumscribed near its base.

Consequence 1. The center of a sphere circumscribed about a pyramid lies at the point of intersection of a line perpendicular to the base of the pyramid, passing through the center of a circle circumscribed near this base, and a plane perpendicular to any side edge drawn through the middle of this edge.

Consequence 2. If the side edges of the pyramid are equal to each other (or equally inclined to the plane of the base), then a ball can be described near such a pyramid. The center of this ball in this case lies at the point of intersection of the height of the pyramid (or its continuation) with the axis of symmetry of the side edge lying in the plane lateral edge and height.

Consequence 3. A ball, in particular, can be described: near a triangular pyramid, near a regular pyramid, near a quadrangular pyramid, in which the sum of opposite angles is 180 degrees.

2. A ball inscribed in a pyramid.

Theorem 4. If the side faces of the pyramid are equally inclined to the base, then a sphere can be inscribed in such a pyramid.

Consequence 1. The center of a ball inscribed in a pyramid, whose side faces are equally inclined to the base, lies at the point of intersection of the height of the pyramid with the bisector of the linear angle of any dihedral angle at the base of the pyramid, the side of which is the height of the side face drawn from the top of the pyramid.

Consequence 2. A ball can be inscribed in a regular pyramid.

From the textbook by L.S. Atanasyan, problems No. 635, 637 (b), 638, 639 (c), 640, 641 can be proposed for the combination of a ball with a pyramid.

Combination of a sphere with a truncated pyramid.

1. A ball circumscribed near a regular truncated pyramid.

Theorem 5. Near any regular truncated pyramid, a sphere can be described. (This condition is sufficient but not necessary)

2. A ball inscribed in a regular truncated pyramid.

Theorem 6. A ball can be inscribed in a regular truncated pyramid if and only if the apothem of the pyramid is equal to the sum of the apothems of the bases.

There is only one problem for combining a ball with a truncated pyramid in L.S. Atanasyan's textbook (No. 636).

A combination of a ball with round bodies.

Theorem 7. Near a cylinder, a truncated cone (right circular), a cone, a sphere can be described.

Theorem 8. A sphere can be inscribed in a cylinder (right circular) if and only if the cylinder is equilateral.

Theorem 9. A sphere can be inscribed in any cone (right circular).

Theorem 10. A ball can be inscribed in a truncated cone (right circular) if and only if its generatrix is ​​equal to the sum of the radii of the bases.

From the textbook by L.S. Atanasyan, problems No. 642, 643, 644, 645, 646 can be proposed for the combination of a ball with round bodies.

For a more successful study of the material of this topic, it is necessary to include oral tasks in the course of the lessons:

1. The edge of the cube is equal to a. Find the radii of the balls: inscribed in a cube and circumscribed near it. (r = a/2, R = a3).

2. Is it possible to describe a sphere (ball) around: a) a cube; b) a rectangular parallelepiped; c) an inclined parallelepiped, at the base of which lies a rectangle; d) a straight parallelepiped; e) an inclined parallelepiped? (a) yes; b) yes; c) no; d) no; e) no)

3. Is it true that a sphere can be described near any triangular pyramid? (Yes)

4. Is it possible to describe a sphere around any quadrangular pyramid? (No, not near any quadrangular pyramid)

5. What properties must a pyramid have in order to describe a sphere around it? (At its base there must be a polygon, around which a circle can be described)

6. A pyramid is inscribed in the sphere, the lateral edge of which is perpendicular to the base. How to find the center of a sphere? (The center of the sphere is the point of intersection of two geometric places of points in space. The first is a perpendicular drawn to the plane of the base of the pyramid, through the center of the circle described around it. The second is a plane perpendicular to this side edge and drawn through its middle)

7. Under what conditions can a sphere be described near a prism, at the base of which is a trapezoid? (Firstly, the prism must be straight, and, secondly, the trapezoid must be isosceles so that a circle can be described around it)

8. What conditions must a prism satisfy in order to describe a sphere around it? (The prism must be straight and its base must be a polygon around which a circle can be circumscribed)

9. A sphere is described near a triangular prism, the center of which lies outside the prism. What triangle is the base of the prism? (obtuse triangle)

10. Is it possible to describe a sphere near an inclined prism? (No you can not)

11. Under what condition will the center of a sphere circumscribed about a right triangular prism be located on one of the side faces of the prism? (The base is a right triangle)

12. The base of the pyramid is an isosceles trapezoid. The orthogonal projection of the top of the pyramid onto the plane of the base is a point located outside the trapezoid. Is it possible to describe a sphere around such a trapezoid? (Yes, you can. The fact that the orthogonal projection of the top of the pyramid is located outside its base does not matter. It is important that at the base of the pyramid lies an isosceles trapezoid - a polygon around which a circle can be described)

13. A sphere is described near the regular pyramid. How is its center located relative to the elements of the pyramid? (The center of the sphere is on a perpendicular drawn to the plane of the base through its center)

14. Under what condition does the center of a sphere circumscribed about a right triangular prism lie: a) inside the prism; b) outside the prism? (At the base of the prism: a) an acute triangle; b) obtuse triangle)

15. A sphere is described near a rectangular parallelepiped whose edges are 1 dm, 2 dm and 2 dm. Calculate the radius of the sphere. (1.5 dm)

16. In which truncated cone can a sphere be inscribed? (In a truncated cone, in the axial section of which a circle can be inscribed. The axial section of the cone is an isosceles trapezoid, the sum of its bases must be equal to the sum of its lateral sides. In other words, for a cone, the sum of the radii of the bases must be equal to the generatrix)

17. A sphere is inscribed in a truncated cone. At what angle is the generatrix of the cone visible from the center of the sphere? (90 degrees)

18. What property must a straight prism have in order to be able to inscribe a sphere in it? (Firstly, at the base of a straight prism there must be a polygon in which a circle can be inscribed, and, secondly, the height of the prism must be equal to the diameter of the circle inscribed in the base)

19. Give an example of a pyramid in which a sphere cannot be inscribed? (For example, a quadrangular pyramid, at the base of which lies a rectangle or parallelogram)

20. A rhombus lies at the base of a straight prism. Can a sphere be inscribed in this prism? (No, you can’t, since in the general case it is impossible to describe a circle near a rhombus)

21. Under what condition can a sphere be inscribed in a right triangular prism? (If the height of the prism is twice the radius of the circle inscribed in the base)

22. Under what condition can a sphere be inscribed in a regular quadrangular truncated pyramid? (If the section of this pyramid by a plane passing through the middle of the side of the base perpendicular to it is an isosceles trapezoid into which a circle can be inscribed)

23. A sphere is inscribed in a triangular truncated pyramid. What point of the pyramid is the center of the sphere? (The center of the sphere inscribed in this pyramid is at the intersection of three bisectoral planes of angles formed by the side faces of the pyramid with the base)

24. Is it possible to describe a sphere around a cylinder (right circular)? (Yes you can)

25. Is it possible to describe a sphere near a cone, a truncated cone (right circular ones)? (Yes, you can, in both cases)

26. Can a sphere be inscribed in any cylinder? What properties must a cylinder have in order for a sphere to be inscribed in it? (No, not in everyone: the axial section of the cylinder must be a square)

27. Can a sphere be inscribed in any cone? How to determine the position of the center of a sphere inscribed in a cone? (Yes, in any. The center of the inscribed sphere is at the intersection of the height of the cone and the bisector of the angle of inclination of the generatrix to the plane of the base)

The author believes that out of the three lessons that are given for planning on the topic “Different problems for polyhedra, a cylinder, a cone and a ball”, it is advisable to take two lessons for solving problems for combining a ball with other bodies. It is not recommended to prove the theorems given above due to the insufficient amount of time in the lessons. You can offer students who have sufficient skills to prove them by indicating (at the discretion of the teacher) the course or plan of the proof.