Lesson type: Lesson on introducing new material.
Lesson objectives:
Introduce the concept of a sphere inscribed in a polyhedron; sphere circumscribed about the polyhedron.
Compare the circumcircle and the circumscribed sphere, the inscribed circle and the inscribed sphere.
Analyze the conditions for the existence of an inscribed sphere and a circumscribed sphere.
Develop problem solving skills on the topic.
Developing students' independent work skills.
Development of logical thinking, algorithmic culture, spatial imagination, development of mathematical thinking and intuition, creative abilities at the level necessary for continuing education and for independent activity in the field of mathematics and its applications in future professional activities.
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Circumscribed circle.
Definition: If all the vertices of a polygon lie on a circle, then the circle is calleddescribed about a polygon, and the polygon isinscribed in a circle.
Theorem. Around any triangle you can describe a circle, and only one.
Unlike a triangle, it is not always possible to describe a circle around a quadrilateral. For example: rhombus.
Theorem. In any cyclic quadrilateral, the sum of the opposite angles is 180 0 .
If the sum of the opposite angles of a quadrilateral is 180 0 , then a circle can be described around it.
In order for the quadrilateral ABCD to be inscribed, it is necessary and sufficient that any of the following conditions be met:
- ABCD is a convex quadrilateral and ∟ABD=∟ACD;
- The sum of two opposite angles of a quadrilateral is 180 0 .
The center of the circle is equidistant from each of its vertices and therefore coincides with the point of intersection of the perpendicular bisectors to the sides of the polygon, and the radius is equal to the distance from the center to the vertices.
For a triangle:For a regular polygon:
Inscribed circle.
Definition: If all sides of a polygon touch a circle, then the circle is calledinscribed in a polygon,and the polygon is described around this circle.
Theorem. You can inscribe a circle into any triangle, and only one.
Not every quadrilateral can fit a circle. For example: a rectangle that is not a square.
Theorem. In any circumscribed quadrilateral, the sums of the lengths of opposite sides are equal.
If the sums of the lengths of opposite sides of a convex quadrilateral are equal, then a circle can be inscribed in it.
In order for a convex quadrilateral ABCD to be described, it is necessary and sufficient that the condition AB+DC=BC+AD be satisfied (the sums of the lengths of opposite sides are equal).
The center of the circle is equidistant from the sides of the polygon, which means it coincides with the point of intersection of the bisectors of the angles of the polygon (angle bisector property). The radius is equal to the distance from the center of the circle to the sides of the polygon.
For a triangle:For the right
Polygon:
Preview:
Inscribed sphere.
Definition: The sphere is called inscribed into a polyhedron if it touches all faces of the polyhedron. The polyhedron in this case is called described about the sphere.
The center of the inscribed sphere is the point of intersection of the bisector planes of all dihedral angles.
A sphere is said to be inscribed in a dihedral angle if it touches its faces. The center of a sphere inscribed in a dihedral angle lies on the bisector plane of this dihedral angle. A sphere is said to be inscribed in a polyhedral angle if it touches all the faces of the polyhedral angle.
Not every polyhedron can accommodate a sphere. For example: a sphere cannot be inscribed in a rectangular parallelepiped that is not a cube.
Theorem. You can fit a sphere into any triangular pyramid, and only one.
Proof. Consider the triangular pyramid CABD. Let us draw bisector planes of its dihedral angles with edges AC and BC. They intersect along a straight line that intersects the bisector plane of the dihedral angle with the edge AB. Thus, the bisector planes of dihedral angles with edges AB, AC and BC have a single common point. Let's denote it Q. Point Q is equidistant from all faces of the pyramid. Consequently, a sphere of the appropriate radius with center at point Q is inscribed in the pyramid CABD.
Let us prove its uniqueness. The center of any sphere inscribed in the CABD pyramid is equidistant from its faces, which means it belongs to the bisector planes of the dihedral angles. Therefore, the center of the sphere coincides with point Q. What needed to be proven.
Theorem. In a pyramid in which a circle can be inscribed at the base, the center of which serves as the base of the height of the pyramid, a sphere can be inscribed.
Consequence. You can fit a sphere into any regular pyramid.
Prove that the center of a sphere inscribed in a regular pyramid lies at the height of this pyramid (prove it yourself).
The center of a sphere inscribed in a regular pyramid is the point of intersection of the height of the pyramid with the bisector of the angle formed by the apothem and its projection onto the base.
Task. a, the height is h.
Solve the problem.
Task. 0
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Described sphere.
Definition. The sphere is called circumscribed near a polyhedron if__________________________________________ _______________________________________________________________________________________________________________. The polyhedron is called _______________________________________.
What property does the center of the described sphere have?
Definition. The geometric locus of points in space equidistant from the ends of a certain segment is _________________________________________________________________________________________________________ __________________________________________________________________________________________________________.
Give an example of a polyhedron around which it is impossible to describe a sphere: ________________________ __________________________________________________________________________________________________________ .
Around which pyramid can a sphere be described?
Theorem. ______________________________________________________________________________________________ ______________________________________________________________________________________________________________.
Proof. Consider the triangular pyramid ABCD. Let us construct planes perpendicular to the edges AB, AC and AD, respectively, and passing through their midpoints. Let us denote by O the point of intersection of these planes. Such a point exists, and it is the only one. Let's prove it. Let's take the first two planes. They intersect because they are perpendicular to nonparallel lines. Let us denote the straight line along which the first two planes intersect by l. This straight line perpendicular to plane ABC. A plane perpendicular to AD is not parallel l and does not contain it, since otherwise the line AD is perpendicular l , i.e. lies in the ABC plane. Point O is equidistant from points A and B, A and C, A and D, which means it is equidistant from all vertices of the ABCD pyramid, i.e. a sphere with a center at O of the corresponding radius is a circumscribed sphere for the pyramid.
Let us prove its uniqueness. The center of any sphere passing through the vertices of the pyramid is equidistant from these vertices, which means it belongs to planes that are perpendicular to the edges of the pyramid and pass through the midpoints of these edges. Consequently, the center of such a sphere coincides with point O. The theorem is proven.
What other pyramid can a sphere be described around?
Theorem. ______________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________ _______________________________________________________________________________________________________________.
The center of the sphere circumscribed about the pyramid coincides with the intersection point of a straight line perpendicular to the base of the pyramid passing through the center of the circle circumscribed about the base and a plane perpendicular to any lateral edge drawn through the middle of this edge.
In order to be able to describe a sphere around a polyhedron, it is necessary to ___________________________________ _______________________________________________________________________________________________________________.
In this case, the center of the circumscribed sphere can lie _________________________________________________________________ _______________________________________________________________________________________________________________ and is projected into the center of the circumscribed circle about any face; a perpendicular dropped from the center of a sphere circumscribed about a polyhedron onto an edge of the polyhedron divides this edge in half.
Consequence. ___________________________________________________________________________________________ ___________________________________________________________________________________________________________ .
The center of the sphere described about the regular pyramid lies ________________________________________________ _______________________________________________________________________________________________________________.
Analyze the solution to the problem.
Task. In a regular quadrangular pyramid, the side of the base is equal to a, the height is h. Find the radius of the sphere circumscribed around the pyramid.
Solve the problem.
Task. 0
Preview:
Open lesson on the topic “Inscribed and circumscribed polyhedra”
Lesson topic: A sphere inscribed in a pyramid. A sphere described near a pyramid.
Lesson type: Lesson on introducing new material.
Lesson objectives:
- Developing students' independent work skills.
- Development logical thinking, algorithmic culture, spatial imagination, development of mathematical thinking and intuition, creative abilities at the level necessary for continuing education and for independent activity in the field of mathematics and its applications in future professional activities;
Equipment:
- interactive board
- Presentation “Inscribed and described sphere”
- Conditions of the problems in the drawings on the board.
- Handouts (supporting notes).
- Planimetry. Inscribed and circumscribed circle.
- Stereometry. Inscribed sphere
- Stereometry. Described sphere
Lesson structure:
- Setting lesson goals (2 minutes).
- Preparation for learning new material by repetition (frontal survey) (6 minutes).
- Explanation of new material (15 minutes)
- Understanding the topic when independently compiling notes on the topic “Stereometry. Described area” and application of the topic in solving problems (15 minutes).
- Summing up the lesson by checking knowledge and understanding of the topic studied (frontal survey). Evaluating student responses (5 minutes).
- Setting homework (2 minutes).
- Reserve jobs.
During the classes
1. Setting lesson goals.
- Introduce the concept of a sphere inscribed in a polyhedron; sphere circumscribed about the polyhedron.
- Compare the circumcircle and the circumscribed sphere, the inscribed circle and the inscribed sphere.
- Analyze the conditions for the existence of an inscribed sphere and a circumscribed sphere.
- Develop problem solving skills on the topic.
2. Preparation for learning new material by repetition (frontal survey).
A circle inscribed in a polygon.
- What circle is called inscribed in a polygon?
- What is the name of the polygon in which a circle is inscribed?
- Which point is the center of a circle inscribed in a polygon?
- What property does the center of a circle inscribed in a polygon have?
- Where is the center of a circle inscribed in a polygon?
- Which polygon can be described around a circle, under what conditions?
A circle circumscribed about a polygon.
- What circle is called the circumscribed circle of a polygon?
- What is the name of the polygon around which the circle is circumscribed?
- Which point is the center of the circle circumscribed about the polygon?
- What property does the center of a circle circumscribed about a polygon have?
- Where can the center of a circle circumscribed about a polygon be located?
- Which polygon can be inscribed in a circle and under what conditions?
3. Explanation of new material.
A . By analogy, students formulate new definitions and answer the questions posed.
A sphere inscribed in a polyhedron.
- Formulate the definition of a sphere inscribed in a polyhedron.
- What is the name of a polyhedron into which a sphere can be inscribed?
- What property does the center of a sphere inscribed in a polyhedron have?
- What represents the set of points in space equidistant from the faces of a dihedral angle? (trihedral angle?)
- Which point is the center of a sphere inscribed in a polyhedron?
- In which polyhedron can a sphere be inscribed, under what conditions?
IN . Students prove the theorem.
A sphere can be inscribed into any triangular pyramid.
While working in class, students use supporting notes.
WITH. Students analyze the solution to the problem.
In a regular quadrangular pyramid, the side of the base is equal to a, the height is h. Find the radius of the sphere inscribed in the pyramid.
D. Students solve the problem.
Task. In a regular triangular pyramid, the side of the base is 4, the side faces are inclined to the base at an angle of 60 0 . Find the radius of the sphere inscribed in this pyramid.
4. Understanding the topic when independently compiling notes on “Sphere circumscribed about a polyhedron"and application in problem solving.
A. U Students independently fill out notes on the topic “A sphere described around a polyhedron.” Answer the following questions:
- Formulate the definition of a sphere circumscribed about a polyhedron.
- What is the name of the polyhedron around which a sphere can be described?
- What property does the center of a sphere circumscribed about a polyhedron have?
- What is the set of points in space that are equidistant from two points?
- Which point is the center of the sphere circumscribed about the polyhedron?
- Where can the center of the sphere described around the pyramid be located? (polyhedron?)
- Around which polyhedron can a sphere be described?
IN. Students solve the problem independently.
Task. In a regular triangular pyramid, the side of the base is 3, and the side ribs are inclined to the base at an angle of 60 0 . Find the radius of the sphere circumscribed around the pyramid.
WITH. Checking the compiled outline and analyzing the solution to the problem.
5. Summing up the lesson by checking knowledge and understanding of the topic studied (frontal survey). Evaluating student responses.
A. Students independently summarize the lesson.
IN. Answer additional questions.
- Is it possible to describe a sphere around a quadrangular pyramid, at the base of which lies a rhombus that is not a square?
- Is it possible to describe a sphere around a rectangular parallelepiped? If so, where is its center?
- Where the theory learned in class is applied in real life (architecture, cellular telephony, geostationary satellites, GPS detection system).
6. Setting homework.
A. Make a note on the topic “A sphere described around a prism. A sphere inscribed in a prism." (Look at problems in the textbook: No. 632,637,638)
B. Solve problem No. 640 from the textbook.
S. From the manual of B.G. Ziv “Didactic materials on geometry grade 10” solve problems: Option No. 3 C12 (1), Option No. 4 C12 (1).
D. Additional task: Option No. 5 C12 (1).
7. Reserve tasks.
From the manual of B.G. Ziv “Didactic materials on geometry grade 10” solve problems: Option No. 3 C12 (1), Option No. 4 C12 (1).
Educational and methodological kit
- Geometry, 10-11: Textbook for educational institutions. Basic and profile levels / L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev et al., M.: Education, 2010.
- B.G. Ziv “Didactic materials on geometry grade 10”, M.: Education.
Repetition Circumscribed circle around a polygon What circle is described as circumscribed around a polygon? What is the center of the circle circumscribing the polygon? What property does the center of a circle circumscribed about a polygon have? Where is the center of the circle circumscribed about the polygon? Which polygon can be inscribed in a circle and under what conditions?
Repetition Circle inscribed in a polygon What circle is called inscribed in a polygon? What is the center of a circle inscribed in a polygon? What property does the center of a circle inscribed in a polygon have? Where is the center of a circle inscribed in a polygon? Which polygon can be described around a circle, under what conditions?
A sphere inscribed in a polyhedron Formulate the definition of a sphere inscribed in a polyhedron. What is the name of a polyhedron? What property does the center of an inscribed sphere have? Where are the set of points in space equidistant from the faces of a dihedral angle? (trihedral angle)? In which polyhedron can a sphere be inscribed?
Sphere inscribed in a pyramid
Sphere circumscribed about a polyhedron Formulate the definition of a sphere circumscribed about a polyhedron. What is the name of a polyhedron? What property does the center of the described sphere have? Where are the set of points in space that are equidistant from two points? Where is the center of the sphere described around the pyramid? (polyhedron?) Around which polyhedron can a sphere be described?
Sphere described near a pyramid
Summing up the lesson. Is it possible to describe a sphere around a quadrangular pyramid, at the base of which lies a rhombus that is not a square? Is it possible to describe a sphere around a rectangular parallelepiped? If so, where is its center?
Homework. Make a note on the topic “A sphere described around a prism. A sphere inscribed in a prism." (Look at problems from the textbook: No. 632,637,638) Solve problem No. 640 from the textbook. Solve problems from the manual: Option No. 3 C12 (1), Option No. 4 C12 (1).
The goal of the work is to learn all the theoretical material on the topic “Inscribed and Circumscribed Polyhedra” and learn to apply it in practice.
Polyhedra inscribed in a sphere A convex polyhedron is called inscribed if all its vertices lie on some sphere. This sphere is called described for a given polyhedron. The center of this sphere is a point equidistant from the vertices of the polyhedron. It is the point of intersection of planes, each of which passes through the middle of the edge of the polyhedron perpendicular to it.
Pyramid inscribed in a sphere Theorem: A sphere can be described around a pyramid if and only if a circle can be described around the base of the pyramid.
Formula for finding the radius of a circumscribed sphere Let SABC be a pyramid with equal lateral edges, h is its height, R is the radius of the circumscribed circle around the base. Let's find the radius of the circumscribed sphere. Note the similarity of right triangles SKO 1 and SAO. Then SO 1/SA = KS/SO; R 1 = KS · SA/SO But KS = SA/2. Then R 1 = SA 2/(2 SO); R 1 = (h 2 + R 2)/(2 h); R 1 = b 2/(2 h), where b is the side edge.
Prism inscribed in a sphere Theorem: A sphere can be described around a prism only if the prism is straight and a circle can be described around its base.
A parallelepiped inscribed in a sphere Theorem: A sphere can be described around a parallelepiped if and only if the parallelepiped is rectangular, since in this case it is straight and a circle can be described around its base - a parallelogram (since the base is a rectangle).
A cone and a cylinder inscribed in a sphere Theorem: A sphere can be described around any cone. Theorem: A sphere can be described around any cylinder.
Problem 1 Find the radius of the ball circumscribed by a regular tetrahedron with edge a. about Solution: First, let's construct an image of the center of a circumscribed ball on the image of a regular tetrahedron SABC. Let's draw the apothems SD and AD (SD = AD). In the isosceles triangle ASD, each point of the median DN is equidistant from the ends of the segment AS. Therefore, point O 1 is the intersection of the height SO and the segment DN. Using the formula from R 1 = b 2/(2 h), we obtain: SO 1 = SA 2/(2 SO); SO = SO 1 = a 2/(2 a =a =)=a /4. Answer: SO 1 = a /4.
Problem 2 In a regular quadrangular pyramid, the side of the base is equal to a, and the plane angle at the apex is equal to α. Find the radius of the circumscribed sphere. Solution: Using the formula R 1=b 2/(2 h) to find the radius of the circumscribed ball, we find SC and SO. SC = a/(2 sin(α/2)); SO 2). (a/(2 sin(α/2))2 – (a /2)2 = =). = a 2/(4 sin 2(α/2)) – 2 a 2/4 = = a 2/(4 sin 2(α/2)) (1 – 2 sin 2(α/2)) = = a 2/(4 sin 2(α/2)) · cosα R 1 = a 2/(4 sin 2(α/2)) · 1/(2 a Answer: R 1 = a/(4 sin(α/ 2) /(2 sin(α/2))) = a/(4 sin(α/2)
Polyhedra circumscribed about a sphere A convex polyhedron is called circumscribed if all its faces touch some sphere. This sphere is called inscribed for a given polyhedron. The center of an inscribed sphere is a point equidistant from all faces of the polyhedron.
Position of the center of an inscribed sphere Concept of a bisector plane of a dihedral angle. A bisector plane is a plane that divides a dihedral angle into two equal dihedral angles. Each point of this plane is equidistant from the faces of the dihedral angle. In the general case, the center of a sphere inscribed in a polyhedron is the intersection point of the bisector planes of all dihedral angles of the polyhedron. It always lies inside the polyhedron.
A pyramid circumscribed around a ball A ball is said to be inscribed in an (arbitrary) pyramid if it touches all faces of the pyramid (both lateral and base). Theorem: If the side faces are equally inclined to the base, then a ball can be inscribed in such a pyramid. Since the dihedral angles at the base are equal, their halves are also equal and the bisectors intersect at one point at the height of the pyramid. This point belongs to all bisector planes at the base of the pyramid and is equidistant from all faces of the pyramid - the center of the inscribed ball.
Formula for finding the radius of an inscribed sphere Let SABC be a pyramid with equal lateral edges, h is its height, r is the radius of the inscribed circle. Let's find the radius of the circumscribed sphere. Let SO = h, OH = r, O 1 O = r 1. Then, by the property of the bisector of the internal angle of a triangle, O 1 O/OH = O 1 S/SH; r 1/r = (h – r 1)/ ; r 1 = rh – rr 1; r 1 · (+ r) = rh; r 1 = rh/(+ r). Answer: r 1 = rh/(+ r).
A prism circumscribed around a sphere Theorem: A sphere can be inscribed into a prism if and only if the prism is straight and a circle whose diameter is equal to the height of the prism can be inscribed at the base.
A parallelepiped and a cube described around a sphere Theorem: A sphere can be inscribed into a parallelepiped if and only if the parallelepiped is a straight line and its base is a rhombus, and the height of this rhombus is the diameter of the inscribed sphere, which, in turn, is equal to the height of the parallelepiped. (Of all the parallelograms, only a circle can be inscribed in a rhombus) Theorem: A sphere can always be inscribed in a cube. The center of this sphere is the intersection point of the diagonals of the cube, and the radius is equal to half the length of the edge of the cube.
A cylinder and a cone described around a sphere. Theorem: A sphere can only be inscribed in a cylinder whose height is equal to the diameter of the base. Theorem: A sphere can be inscribed into any cone.
Combinations of figures Inscribed and circumscribed prisms A prism inscribed in a cylinder is a prism in which the planes of the bases are the planes of the bases of the cylinder, and the side edges are the generators of the cylinder. A tangent plane to a cylinder is a plane passing through the generatrix of the cylinder and perpendicular to the plane of the axial section containing this generatrix. A prism described around a cylinder is a prism whose base planes are the planes of the bases of the cylinder, and the side faces touch the cylinder.
Inscribed and circumscribed pyramids A pyramid inscribed in a cone is a pyramid whose base is a polygon inscribed in the circle of the base of the cone, and the apex is the vertex of the cone. The lateral edges of a pyramid inscribed in a cone form the cone. A tangent plane to a cone is a plane passing through the generatrix and perpendicular to the plane of the axial section containing this generatrix. A pyramid circumscribed around a cone is a pyramid whose base is a polygon circumscribed around the base of the cone, and the apex coincides with the apex of the cone. The planes of the side faces of the described pyramid are tangent to the plane of the cone.
Other types of configurations A cylinder is inscribed in a pyramid if the circle of one of its bases touches all the lateral faces of the pyramid, and its other base lies on the base of the pyramid. A cone is inscribed in a prism if its vertex lies on the upper base of the prism, and its base is a circle inscribed in a polygon - the lower base of the prism. A prism is inscribed in a cone if all the vertices of the upper base of the prism lie on the lateral surface of the cone, and the lower base of the prism lies on the base of the cone.
Problem 1 In a regular quadrangular pyramid, the side of the base is equal to a, and the plane angle at the apex is equal to α. Find the radius of the ball inscribed in the pyramid. Solution: Let us express the sides of ∆SOK in terms of a and α. OK = a/2. SK = KC cot(α/2); SK = (a cot(α/2))/2. SO = = (a/2) Using the formula r 1 = rh/(+ r), we find the radius of the inscribed ball: r 1 = OK · SO/(SK + OK); r 1 = (a/2) · (a/2) /((a/2) · ctg(α/2) + (a/2)) = = (a/2) /(ctg(α/2) + 1) = (a/2) Answer: r 1 = (a/2) =
Conclusion The topic “Polyhedra” is studied by students in grades 10 and 11, but in the curriculum there is very little material on the topic “Inscribed and circumscribed polyhedra”, although it arouses great interest among students, since the study of the properties of polyhedra contributes to the development of abstract and logical thinking, which will later be useful to us in study, work, life. While working on this essay, we studied all the theoretical material on the topic “Inscribed and circumscribed polyhedra,” examined possible combinations of figures and learned to apply all the studied material in practice. Problems involving the combination of bodies are the most difficult question in the 11th grade stereometry course. But now we can say with confidence that we will not have problems solving such problems, since in the course of our research work we have established and proved the properties of inscribed and circumscribed polyhedra. Very often, students have difficulty constructing a drawing for a problem on this topic. But, having learned that to solve problems involving the combination of a ball with a polyhedron, the image of the ball is sometimes unnecessary and it is enough to indicate its center and radius, we can be sure that we will not have these difficulties. Thanks to this essay, we were able to understand this difficult but very fascinating topic. We hope that now we will not have any difficulties applying the studied material in practice.
Polyhedra inscribed in a sphere A convex polyhedron is called inscribed if all its vertices lie on some sphere. This sphere is called described for a given polyhedron. The center of this sphere is a point equidistant from the vertices of the polyhedron. It is the point of intersection of planes, each of which passes through the middle of the edge of the polyhedron perpendicular to it.
Formula for finding the radius of a circumscribed sphere Let SABC be a pyramid with equal lateral edges, h is its height, R is the radius of the circle circumscribed around the base. Let's find the radius of the circumscribed sphere. Note the similarity of right triangles SKO1 and SAO. Then SO 1 /SA = KS/SO; R 1 = KS · SA/SO But KS = SA/2. Then R 1 = SA 2 /(2SO); R 1 = (h 2 + R 2)/(2h); R 1 = b 2 /(2h), where b is a side edge.
A parallelepiped inscribed in a sphere Theorem: A sphere can be described around a parallelepiped if and only if the parallelepiped is rectangular, since in this case it is straight and a circle can be described around its base - a parallelogram (since the base is a rectangle) .
Problem 1 Find the radius of a sphere circumscribed about a regular tetrahedron with edge a. Solution: SO 1 = SA 2 /(2SO); SO = = = a SO 1 = a 2 /(2 a) = a /4. Answer: SO 1 = a /4. Let us first construct an image of the center of a circumscribed ball using the image of a regular tetrahedron SABC. Let's draw the apothems SD and AD (SD = AD). In the isosceles triangle ASD, each point of the median DN is equidistant from the ends of the segment AS. Therefore, point O 1 is the intersection of the height SO and the segment DN. Using the formula from R 1 = b 2 /(2h), we get:
Problem 2 Solution: Using the formula R 1 =b 2 /(2h) to find the radius of the circumscribed ball, we find SC and SO. SC = a/(2sin(α /2)); SO 2 = (a/(2sin(α /2)) 2 – (a /2)2 = = a 2 /(4sin 2 (α /2)) – 2a 2 /4 = = a 2 /(4sin 2 ( α /2)) · (1 – 2sin 2 (α /2)) = = a 2 /(4sin 2 (α /2)) · cos α In a regular quadrangular pyramid, the side of the base is equal to a, and the plane angle at the apex is equal to α . Find the radius of the circumscribed ball. R 1 = a 2 /(4sin 2 (α /2)) · 1/(2a/(2sin(α /2))) =a/(4sin(α /2) ·). Answer : R 1 = a/(4sin(α /2) ·).
Polyhedra circumscribed about a sphere A convex polyhedron is called circumscribed if all its faces touch some sphere. This sphere is called inscribed for a given polyhedron. The center of an inscribed sphere is a point equidistant from all faces of the polyhedron.
Position of the center of an inscribed sphere Concept of a bisector plane of a dihedral angle. A bisector plane is a plane that divides a dihedral angle into two equal dihedral angles. Each point of this plane is equidistant from the faces of the dihedral angle. In the general case, the center of a sphere inscribed in a polyhedron is the intersection point of the bisector planes of all dihedral angles of the polyhedron. It always lies inside the polyhedron.
A pyramid circumscribed around a ball A ball is said to be inscribed in an (arbitrary) pyramid if it touches all faces of the pyramid (both lateral and base). Theorem: If the side faces are equally inclined to the base, then a ball can be inscribed in such a pyramid. Since the dihedral angles at the base are equal, their halves are also equal and the bisectors intersect at one point at the height of the pyramid. This point belongs to all bisector planes at the base of the pyramid and is equidistant from all faces of the pyramid - the center of the inscribed ball.
Formula for finding the radius of an inscribed sphere Let SABC be a pyramid with equal lateral edges, h is its height, r is the radius of the inscribed circle. Let's find the radius of the circumscribed sphere. Let SO = h, OH = r, O 1 O = r 1. Then, by the property of the bisector of the internal angle of a triangle, O 1 O/OH = O 1 S/SH; r 1 /r = (h – r 1)/ ; r 1 · = rh – rr 1 ; r 1 · (+ r) = rh; r 1 = rh/(+ r). Answer: r 1 = rh/(+ r).
A parallelepiped and a cube described around a sphere Theorem: A sphere can be inscribed into a parallelepiped if and only if the parallelepiped is straight and its base is a rhombus, and the height of this rhombus is the diameter of the inscribed sphere, which, in turn, is equal to the height of the parallelepiped. (Of all the parallelograms, only a circle can be inscribed in a rhombus) Theorem: A sphere can always be inscribed in a cube. The center of this sphere is the point of intersection of the diagonals of the cube, and the radius is equal to half the length of the edge of the cube.
Combinations of figures Inscribed and circumscribed prisms A prism circumscribed about a cylinder is a prism whose base planes are the planes of the bases of the cylinder, and the side faces touch the cylinder. A prism inscribed in a cylinder is a prism whose base planes are the planes of the bases of the cylinder, and the side edges are the generators of the cylinder. A tangent plane to a cylinder is a plane passing through the generatrix of the cylinder and perpendicular to the plane of the axial section containing this generatrix.
Inscribed and circumscribed pyramids A pyramid inscribed in a cone is a pyramid whose base is a polygon inscribed in the circle of the base of the cone, and the apex is the vertex of the cone. The lateral edges of a pyramid inscribed in a cone form the cone. A pyramid circumscribed around a cone is a pyramid whose base is a polygon circumscribed around the base of the cone, and the apex coincides with the apex of the cone. The planes of the side faces of the described pyramid are tangent to the plane of the cone. A tangent plane to a cone is a plane passing through the generatrix and perpendicular to the plane of the axial section containing this generatrix.
Other types of configurations A cylinder is inscribed in a pyramid if the circle of one of its bases touches all the lateral faces of the pyramid, and its other base lies on the base of the pyramid. A cone is inscribed in a prism if its vertex lies on the upper base of the prism, and its base is a circle inscribed in a polygon - the lower base of the prism. A prism is inscribed in a cone if all the vertices of the upper base of the prism lie on the lateral surface of the cone, and the lower base of the prism lies on the base of the cone.
Problem 1 In a regular quadrangular pyramid, the side of the base is equal to a, and the plane angle at the apex is equal to α. Find the radius of the ball inscribed in the pyramid. Solution: Let's express the sides of SOK in terms of a and α. OK = a/2. SK = KC cot(α /2); SK = (a · ctg(α /2))/2. SO = = (a/2) Using the formula r 1 = rh/(+ r), we find the radius of the inscribed ball: r 1 = OK · SO/(SK + OK); r 1 = (a/2) · (a/2) /((a/2) · ctg(α /2) + (a/2)) = = (a/2) /(ctg(α /2) + 1) = (a/2)= = (a/2) Answer: r 1 = (a/2)
Conclusion The topic “Polyhedra” is studied by students in grades 10 and 11, but the curriculum contains very little material on the topic “Inscribed and circumscribed polyhedra,” although it is of great interest to students, since the study of the properties of polyhedra contributes to the development of abstract and logical thinking, which will later be useful to us in study, work, life. While working on this essay, we studied all the theoretical material on the topic “Inscribed and circumscribed polyhedra,” examined possible combinations of figures and learned to apply all the studied material in practice. Problems involving the combination of bodies are the most difficult question in the 11th grade stereometry course. But now we can say with confidence that we will not have problems solving such problems, since in the course of our research work we have established and proved the properties of inscribed and circumscribed polyhedra. Very often, students have difficulty constructing a drawing for a problem on this topic. But, having learned that to solve problems involving the combination of a ball with a polyhedron, the image of the ball is sometimes unnecessary and it is enough to indicate its center and radius, we can be sure that we will not have these difficulties. Thanks to this essay, we were able to understand this difficult but very fascinating topic. We hope that now we will not have any difficulties in applying the studied material in practice.
Polyhedra inscribed in a sphere A polyhedron is said to be inscribed in a sphere if all its vertices belong to this sphere. The sphere itself is said to be circumscribed about the polyhedron. Theorem. A sphere can be described around a pyramid if and only if a circle can be described around the base of this pyramid.
Polyhedra inscribed in a sphere Theorem. A sphere can be described near a straight prism if and only if a circle can be described near the base of this prism. Its center will be point O, which is the midpoint of the segment connecting the centers of the circles described near the bases of the prism. The radius of the sphere R is calculated by the formula where h is the height of the prism, r is the radius of the circle circumscribed around the base of the prism.
Exercise 3 The base of the pyramid is a regular triangle, the side of which is equal to 3. One of the side edges is equal to 2 and is perpendicular to the plane of the base. Find the radius of the circumscribed sphere. Solution. Let O be the center of the circumscribed sphere, Q the center of the circumscribed circle around the base, E the midpoint of SC. Quadrilateral CEOQ is a rectangle in which CE = 1, CQ = Therefore, R=OC=2. Answer: R = 2.
Exercise 4 The figure shows the pyramid SABC, for which the edge SC is equal to 2 and is perpendicular to the plane of the base ABC, the angle ACB is equal to 90 o, AC = BC = 1. Construct the center of the sphere circumscribed about this pyramid and find its radius. Solution. Through the middle D of edge AB we draw a line parallel to SC. Through the middle E of edge SC we draw a straight line parallel to CD. Their intersection point O will be the desired center of the circumscribed sphere. In the right triangle OCD we have: OD = CD = By the Pythagorean theorem, we find
Exercise 5 Find the radius of a sphere circumscribed about a regular triangular pyramid, the side edges of which are equal to 1, and the plane angles at the apex are equal to 90 degrees. Solution. In the tetrahedron SABC we have: AB = AE = SE = In the right triangle OAE we have: Solving this equation for R, we find
Exercise 4 Find the radius of a sphere circumscribed about a right triangular prism, at the base of which is a right triangle with legs equal to 1, and the height of the prism equal to 2. Answer: Solution. The radius of the sphere is equal to half the diagonal A 1 C of the rectangle ACC 1 A 1. We have: AA 1 = 2, AC = Therefore, R =
Exercise Find the radius of a sphere circumscribed about a regular 6-gonal pyramid, the edges of which are equal to 1, and the side edges are equal to 2. Solution. Triangle SAD is equilateral with side 2. The radius R of the circumscribed sphere is equal to the radius of the circle circumscribed about triangle SAD. Hence,
Exercise Find the radius of the sphere circumscribed about the unit icosahedron. Solution. In rectangle ABCD, AB = CD = 1, BC and AD are the diagonals of regular pentagons with sides 1. Therefore, BC = AD = By the Pythagorean theorem, AC = The required radius is equal to half of this diagonal, i.e.
Exercise Find the radius of a sphere circumscribed about a unit dodecahedron. Solution. ABCDE is a regular pentagon with side In the rectangle ACGF AF = CG = 1, AC and FG are the diagonals of the pentagon ABCDE and, therefore, AC = FG = By the Pythagorean theorem FC = The required radius is equal to half of this diagonal, i.e.
Exercise The figure shows a truncated tetrahedron obtained by cutting off the corners of a regular tetrahedron of triangular pyramids, the faces of which are regular hexagons and triangles. Find the radius of a sphere circumscribed about a truncated tetrahedron whose edges are equal to 1.
Exercise The figure shows a truncated octahedron obtained by cutting off triangular pyramids from the corners of the octahedron, the faces of which are regular hexagons and triangles. Find the radius of the sphere circumscribed about a truncated octahedron whose edges are equal to 1. Exercise The figure shows a truncated icosahedron obtained by cutting off the corners of the icosahedron of pentagonal pyramids, the faces of which are regular hexagons and pentagons. Find the radius of a sphere circumscribed about a truncated icosahedron whose edges are equal to 1.
Exercise The figure shows a truncated dodecahedron obtained by cutting off triangular pyramids from the corners of the dodecahedron, the faces of which are regular decagons and triangles. Find the radius of a sphere circumscribed about a truncated dodecahedron whose edges are equal to 1.
Exercise Find the radius of a sphere circumscribed about a unit cuboctahedron. Solution. Recall that a cuboctahedron is obtained from a cube by cutting off regular triangular pyramids with vertices at the vertices of the cube and side edges equal to half the edge of the cube. If the edge of the octahedron is equal to 1, then the edge of the corresponding cube is equal to The radius of the circumscribed sphere is equal to the distance from the center of the cube to the middle of its edge, i.e. is equal to 1. Answer: R = 1.
Mathematics teacher of secondary school No. 2,
city of Taldykorgan N.Yu.Lozovich
Open lesson on geometry
Lesson topic: “Ball. InscribedAnddescribed polyhedra"
Lesson objectives:
- educational - ensure during the lesson repetition, consolidation and testing of students’ mastery of definitions ball And spheres, and related concepts ( center, radii, diameters,diametrically opposed points, tangent planes And straight); concepts of inscribed and circumscribed polyhedra, knowledge of theorems on the section of a ball by a plane (20.3), on the symmetry of a ball (20.4), on the tangent plane to a ball (20.5), on the intersection of two spheres (20.6), on the construction of the center of a circumscribed (inscribed) sphere a pyramid and the construction of the center of a sphere described around a regular prism;
continue to develop the skills to independently apply the entire body of this knowledge in variable situations based on the model and non-standard ones that require creative activity;
educational - to instill in students responsibility for the results of their studies, perseverance in achieving goals, self-confidence, the desire to achieve great results, a sense of beauty (the beauty of geometric shapes, an elegant, beautiful solution to a problem).
developing - develop in students: the ability for specific and generalized thinking, creative and spatial imagination; associativity (the ability to rely on different connections: by similarity, analogy, contrast, cause-and-effect), the ability to logically and consistently express one’s thoughts, the need for learning and development, to create conditions in the lesson for the manifestation of students’ cognitive activity.
Lesson type
lesson of testing and correction of knowledge and skills.
Teaching methods
Introductory conversation (setting the purpose of the lesson, motivating students' learning activities, creating the necessary emotional and moral atmosphere, instructing students on organizing work in the lesson).
Frontal survey (oral testing of students’ knowledge of basic concepts, theorems, abilities to explain their essence, and to justify their reasoning).
Leveled independent work, based on the principle of a gradual increase in the level of knowledge and skills, i.e. from the reproductive level to the productive and creative level. The essence of the method is the individual independent work of students, constantly controlled and encouraged by the teacher.
Educational visual aids
Stereometric models of geometric bodies, posters, drawings, educational cards for individual independent work.
Update
a) Basic knowledge.
It is necessary to activate the concepts: tangent to a circle, convex polygons inscribed in a circle and circumscribed about a circle, calculation of the radii of inscribed and circumscribed circles for regular polygons from planimetry; from the 10th grade course, the definition of symmetry relative to a plane, the concept of figures that are symmetrical with respect to a point, an axis (straight line), and a plane.
b) Methods of forming motives and arousing interest.
In the introductory conversation, ensure that students are aware of the goal, find out their personal interest in achieving it, reveal the meaning of the goal for the students themselves, emphasize the significance of this topic not only in itself, but also its propaedeutic nature for studying the next topic, saturate the lesson with material of an emotional nature ( beauty of geometric shapes, soap bubbles, Earth and Moon); emphasize the level nature of independent work: on the one hand, this will ensure a high scientific level of the material being studied, and on the other hand, accessibility, the students’ point is that each of them has the right to pedagogical support (“insurance”) for identifying, analyzing real or potential problems of the child, joint design of a possible way out of them; The rating system for assessing knowledge is an additional incentive for the children.
c) Forms of monitoring the progress of work, mutual control. Mutual control (exchange of notebooks) is carried out after students have completed the first part of the 1st (student) level of independent work - students’ written answers to the teacher’s oral questions (mathematical dictation).
After exchanging notebooks, all correct answers are spoken out loud (if possible, visual aids are used: models of stereometric bodies, drawings, posters). Then the guys proceed to the rating assessment of the first part of the independent work: the correct complete answer is scored 1 point, if there are minor comments, then - 0.5 points, otherwise - 0 points. The number of points scored by each student is recorded on the board by the teacher. After which the guys begin to work on individual cards. Those who have completed the tasks of the 1st level and received the go-ahead from the teacher move on to completing the task of the next level. The success of solving the Problem should not be left without attention, encouragement, and praise. At the same time, the teacher carries out correctional work: understanding the student’s strengths and weaknesses, he helps him rely on his own strengths and complements him where the student, no matter how hard he tries, is still objectively unable to cope with something.
When checking operation, the following notation system is used:
The problem is not solved;
The problem is not solved, but there are some reasonable considerations in the work;
Only the answer is given to a problem where one answer is clearly not enough;
± - the problem is solved, but the solution contains minor omissions and inaccuracies;
The problem is completely solved;
+! – the solution to the problem contains unexpected bright ideas.
Great importance is attached to the open record sheet of children’s activities, which is filled out as they complete independent work.
I level
Level II
Level III
IV level
Alipbaeva A
Akhmetkaliev A.
This ensures the indispensable conditions for assessing students' knowledge in the classroom - objectivity, efficiency, goodwill and transparency.
I level
Mathematical dictation.
1) I option. What property do all the vertices of a polyhedron inscribed in a sphere have?
II option. What property does each face of a polyhedron inscribed in a sphere have?
2)I option. If a sphere can be described around some polyhedron, then how can one construct its center?
II option. ABOUT How many parallelepipeds can be used to describe a sphere? Explain your answer.
3) I option. Where lies the center of the sphere described about the correct P- carbon prism?
II option. Where is the center of the sphere described around a regular pyramid?
4)I option. How to construct the center of a sphere inscribed in a regular n-gonal pyramid?
// option. Is it possible to fit a sphere into any regular prism?
Option I
I level
The radius of the ball is 6 cm; a plane is drawn through the end of the radius at an angle of 60° to it. Find the cross-sectional area.
Level II
A regular quadrangular prism is inscribed in a sphere of radius 5 cm. The edge of the base of the prism is 4 cm. Find the height of the prism.
Level III
Calculate the radius of a sphere inscribed in a regular tetrahedron with an edge of 4 cm.
IV level
A ball of radius R is inscribed in a truncated cone. The angle of inclination of the generatrix to the plane of the lower base of the cone is equal to A. Find the radii of the bases and the generatrix of the truncated cone.
Option II
I level
A sphere whose radius is 10 cm is intersected by a plane at a distance of 6 cm from the center. Find the cross-sectional area.
Level II
Find the radius of a sphere circumscribed about a cube with a side of 4 cm.
Level III.
A. Find the radius of the circumscribed sphere.
IV level
A ball of radius R is inscribed in a truncated cone. The angle of inclination of the generatrix to the plane of the lower base of the cone is equal to a. Find the radii of the bases and the generatrix of the truncated cone.
Ш option
I level
A plane perpendicular to it is drawn through the middle of the radius of the ball. How does the area of the great circle relate to the area of the resulting cross-section?
Level II
A regular triangular prism is inscribed in a sphere of radius 4 cm. The edge of the base of the prism is 3 cm. Find the height of the prism.
Level III
In a regular quadrangular pyramid, the side of the base is 4 cm, and the plane angle at the apex is A. Find the radius of the inscribed sphere.
IV level
A regular triangular pyramid with plane corners is inscribed in a ball of radius R A at its top. Find the height of the pyramid.
IV option
I level
Three points are given on the surface of the ball. The straight-line distances between them are 6 cm, 8 cm, 10 cm. The radius of the ball is 11 cm. Find the distance from the center of the ball to the plane passing through these points.
II level
A regular hexagonal prism is inscribed in a sphere of radius 5 cm. The edge of the base of the prism is 3 cm. Find the height of the technique.
Ш level
Find the radius of a sphere circumscribed about a regular n-gonal pyramid if the side of the base is 4 cm and the side edge is inclined to the plane of the base at an angle A.
IV level
A regular triangular pyramid with flat angles a at its vertex is inscribed into a ball of radius R. Find the height of the pyramid.
Lesson summary
The results of independent work are announced and analyzed. Students who need remedial work are invited to remedial classes.
Homework is assigned (with the necessary comments), consisting of mandatory and variable parts.
Mandatory part: paragraphs 187 - 193 - repeat; No. 44,45,39
Variable part No. 35