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Text content of presentation slides: Polyhedra and bodies of revolution Evgenia Valentinovna Ponarina MBOU Secondary School No. 432016 Voronezh Polyhedra A body that is limited by flat polygons is called a polyhedron. The polygons that form the surface of a polyhedron are called faces. The sides of these polygons are the edges of the polyhedra. The vertices of polygons are the vertices of polyhedra. Polyhedra Polyhedra PrismParallelepipedPyramid Elements of polyhedra Faces: ABCD, AA1B1B, AA1D1D, CC1B1B, CC1D1D, A1B1C1D1 Edges: AB, BC, CD, DA, AA1, BB1, CC1, DD1, A1B1, B1C1, C1D1, D1A1 Ver tires:A, B, C, D, A1, B1, C1, D1 PrismOp: A prism is a polyhedron consisting of two equal polygons located at parallel planes and n parallelograms. Polygons - bases of a prism Parallelograms - faces of a prism Parallel segments connecting the vertices of polygons - lateral edges of a prism Prism Straight prism Oblique prism Regular prism Def: A prism is called straight if its lateral edges are perpendicular to the bases Def: A prism is called oblique if its lateral edges are not perpendicular to the bases and inclined to them at a certain angle. Def: A prism is called regular if it is straight and has a regular polygon at its base. a right parallelepiped with a rectangle at its base. Definition: A cube is a rectangular parallelepiped, all of whose edges are equal. Pyramid Def: an n-gonal pyramid is a polyhedron, one face of which is an arbitrary n-gon, and the remaining faces are triangles that have a common vertex. The polygon A1A2...An is called the base. Point S is the vertex of the pyramid. Segments SA1, SA2 ... SAn are the side edges pyramids.ΔA1SA2 ... ΔAn-1SAn – lateral faces of the pyramid. Regular pyramid Def: A pyramid is called regular if its base is a regular polygon, and the segment connecting the vertex to the center of the base is its height. (SO - height) Def: The height of a pyramid is a perpendicular segment drawn from the top of the pyramid to the plane of the base, as well as the length of this segment. Def: The center of a regular polygon is the center of a circle inscribed in it or circumscribed about it. Def: The height of the side face of a regular polygon of a pyramid drawn from its top is called the apothem of this pyramid.h - apothem Task Some of the figures in the picture are polyhedra, and some are not. What numbers are the polyhedra shown under? Assignment: Some of the polyhedra in the picture are pyramids, and some are not. What numbers are the pyramids shown under? Bodies of revolutionA body of revolution is a figure obtained by rotating a flat polygon around an axis. Bodies of rotationCylinderConeBall, sphere CylinderDef: A right circular cylinder is a figure formed by two equal circles, the planes of which are perpendicular to the line passing through their centers, as well as all segments parallel to this line, with ends on the circumferences of these circles. Elements of a cylinder: The two circles that form the cylinder are called the bases. Def: The radius of the base of a cylinder is called the radius of this cylinder. Def: The straight line passing through the centers of the bases of the cylinder is called its axis. Def: The segment connecting the centers of the bases, as well as the length of this segment, is called the height of the cylinder. Def: The segment parallel to the axis of the cylinder, with ends on the circles of its bases is called the generator of the given cylinder. Sections of a cylinder ConeOp: Consider a circle L with center O and a segment OP perpendicular to the plane of this circle. We connect each point of the circle with a segment to a point P. The surface formed by these segments is called a conical surface, and the segments themselves are the generators of this surface. A body bounded by a conical surface and a circle with boundary L is called a cone. A cone is obtained by rotation right triangle ABC around leg AB ConeOp: The conical surface is called the lateral surface, and the circle is the base of the cone. The segment OP is called the height, the straight line OP is the axis of the cone. Point P is called the vertex of the cone. The generators of a conical surface are also called generators of the cone, the radius of the circle R is called the radius of the cone. Sections of a coneSection of a cone by plane α perpendicular to its axis Axial section of a cone is an isosceles triangle Sphere Definition: A sphere is a set of points in space equidistant from given point. This point is called the center of the sphere. Def: The segment connecting any point of the sphere and its center, as well as the length of this segment, is called the radius of the sphere. A ball is a figure consisting of a sphere and the set of all its internal points. The sphere is called the boundary or surface of the ball, and the center of the sphere is the center of the ball. Sphere Points whose distance to the center of the sphere is less than its radius are called internal points of the sphere. Points whose distance to the center of the sphere is greater than its radius are called external points of the sphere. SphereA segment connecting two points on a sphere is called a chord of a sphere (sphere). Any chord passing through the center of a sphere is called the diameter of a sphere (sphere).
Lesson Plan
Topic: “Polyhedra, figures of revolution, their surface areas and volumes”
Lesson type – combined lesson.
Target: to form in students an understanding of polyhedra, figures of rotation, and also teach them to find their surface areas and volumes.
Tasks:
Define the concepts of polyhedron, figure of rotation;
Introduce students to basic polyhedra and figures of rotation;
To develop students’ skills in calculating the surface areas of polyhedra and figures of revolution;
develop students’ thinking when performing exercises;
Formation of interest and positive motivation of students to study geometry;
Preservation, consolidation and development of students’ spatial concepts.
Lesson structure :
Organizational moment(1-2 minutes)
Examination homework ( 10-15 minutes)
Reporting the topic of the lesson, updating (1-2 minutes)
Learning new material ( 17-20 minutes)
Consolidating new material(45-55 minutes)
Lesson summary, reflection (3-4 minutes)
Homework (1 minute)
Progress of the lesson
1. Organizational moment
Before the start of the lesson, the teacher checks the readiness of the classroom for the lesson.
Greeting students, identifying absentees, filling out a group log.
2. Checking homework:
Finds out whether there were any difficulties with completing homework. Answers student questions as needed. Asks some students to turn in their notebooks to check their homework.
3. Reporting the topic of the lesson, updating
The topic and purpose of the lesson are communicated. He says thatThe topic “Polyhedra and bodies of rotation” is important as it is related to a number of subjects school curriculum: fine arts, drawing, labor training, computer science.
4. Learning new material:
Polyhedron , more precisely three-dimensional polyhedron - a collection of a finite number of flat polygons in three-dimensional space such that:
each side of any of the polygons is simultaneously a side of another (but only one), calledadjacent with the first (on this side);
connectivity : from any of the polygons that make up the polyhedron, you can reach any of them by going to the one adjacent to it, and from this, in turn, to the one adjacent to it, etc.
These polygons are callededges, their sides - ribs, and their vertices are peaks polyhedron.
Types of polyhedra:
Pyramid is a polyhedron, one face of which is a polygon, and the remaining faces are triangles with a common vertex. A pyramid is called regular if its base is a regular polygon and the height of the pyramid passes through the center of the polygon. A pyramid is called truncated if its vertex is cut off by a plane.
Prism - a polyhedron, two faces of which (the bases of the prism) are equal polygons with mutually parallel sides, and all other faces are parallelograms. A prism is called straight if its edges are perpendicular to the plane of the base. If the base of the prism is a rectangle, the prism is called a parallelepiped.
Parallelepiped - a prism whose base is a rectangle.
Cube - a parallelepiped, all dimensions of which are equal to each other.
Bodies of revolution - volumetric bodies that arise during the rotation of a flat geometric figure, bounded by a curve, around an axis lying in the same plane.
Examples of bodies of revolution:
Ball - formed by a semicircle rotating around the diameter of the cut
Cylinder - formed by a rectangle rotating around one of its sides
Cone - formed by a right triangle rotating around one of the legs
Formulas for finding the surface areas of polyhedra and bodies of rotation, as well as their volumes.
Figure
S basic
S side
S full
Parallelepiped:
rectangular
cube
arbitrary
S basic = ab
S basic = a 2
S basic = ab* sinα
l- side . edge
S side =2(a+b)H
S side = 4a 2
S side =P slaughter l
S full = S side +2S basic
V=abc
V=a 3
V=S basic H
Prism
S side =P slaughter l
S full = S side +2S basic
V= Ql (Q-perpendicular cross-sectional area)
Pyramid
S side =P basic l, l-apothem
S full = S side +S basic
V= 1/3* S basic H
Truncated pyramid
S side =( P 1 + P 2) l, l-apothem
S full = S side +S 1 + S 2
V=1/3* H(S 1 + +S 2
Cylinder
S basic = πR 2
S side = 2 πRH
S full = 2 πR(H+ R)
V=πR 2 H
Cone
S basic = πR 2
S side = πRl, l- generatrix
S full = πR(l+ R)
V=1/3*πR 2 H
truncated cone
S basic = πR 2
S side = π (R+ r) l, l-forming
S full = π (R 2 + r 2 )+ R+ r) l
V=1/3*πH(R 2 +Rr+r 2 )
Ball
S full =4πR 2
V=4/3*πR 3
5. Consolidating new material:
1. The generatrix of a straight cone is 4 cm and inclined to the plane of the base at an angle of 30 0 . Find the volume of the cone.
2. Base rectangular parallelepiped- square. Find the volume of a parallelepiped if its height is 4 cm and the diagonal of the parallelepiped forms an angle of 45 with the plane of the base
7. The base of the pyramid is a square. The side of the base is 20 dm, and its height is 21 dm. Find the volume of the pyramid.
8. The diagonal of the axial section of the cylinder is 13 cm, the height is 5 cm. Find the volume of the cylinder.
9. Measurements of a rectangular parallelepiped 15 m, 50 m, 36 m. Determine the edge of a cube equal in size to the rectangular parallelepiped.
10. Find the volume of a rectangular parallelepiped if its length is 6 cm, width is 7 cm, and diagonal is 11 cm.
11. The height of the cylinder is 6 dm, the radius of the base is 5 dm. Find the lateral surface and volume of the cylinder.
6. Summing up the lesson, reflection
Announces the outcome of the lesson and names the grades.
As a reflectionStudents are encouraged to complete the sentences and express their opinions.
This activity is for me...
I felt that...
In the future I...
Today work was for me...
I would like to change...
For the next lesson I would like...
7. Homework assignment
1) The diagonal of the cube is 15 cm. Find the volume of the cube.
2) The diagonal of the side face of a regular triangular prism forms an angle of 30 with the base 0 . Find the volume of the prism if the area of the lateral surface of the prism is equal to cm 2.
A polyhedron is a body bounded on all sides by planes.Elements of a polyhedron: faces, edges, vertices. The set of all the edges of a polyhedron is called its mesh. A polyhedron is called convex if all of it lies on one side of the plane of any of its faces; Moreover, its faces are convex polygons. For convex polyhedra, Leonhard Euler proposed a formula:
Г+В-Р=2, where Г is the number of faces; B – number of vertices; P – number of ribs.
Among the many convex polyhedra, the most interesting are regular polyhedra (Platonic solids), pyramids and prisms. A polyhedron is called regular if all its faces are equal regular polygons. These include (Fig. 26): a - tetrahedron; b - hexahedron (cube); c - octahedron; g - dodecahedron; d - icosahedron.
a) b) c) d) e)
Rice. 26
Parameters of regular polyhedra (Fig. 26)
| Correct polyhedron (Plato's body) | Number | Angle between adjacent ribs, deg. | |||||||||||||
| faces | peaks | ribs | sides each face | Number of edges at each vertex | |||||||||||
| Tetrahedron | 4 | 4 | 6 | 3 | 60 | 3 | |||||||||
| Hexahedron (cube) | 6 | 8 | 12 | 4 | 90 | 3 | |||||||||
| Octahedron | 8 | 6 | 12 | 3 | 60 | 4 | |||||||||
| Dodecahedron | 12 | 20 | 30 | 5 | 72 | 3 | |||||||||
| Icosahedron | 20 | 12 | 30 | 3 | 60 | 5 | |||||||||
The table shows that the number of faces and vertices of the cube and octahedron, respectively, is 6.8 and 8.6. This allows them to be inscribed (described) into each other ad infinitum (Fig. 27).
Large group make up the so-called semi-regular polyhedra (Archimedes solids). These are convex polyhedra whose faces are regular polygons different types. Archimedes' solids are truncated Platonic solids. Appearance some of them are shown in Fig. 28, and below their parameters are in the table.
a) b) c) d)
Rice. 27 Fig. 28
Parameters of semiregular polyhedra (Fig. 28)
A polyhedron may occupy a general position in space, or its elements may be parallel and/or perpendicular to the projection planes. The initial data for constructing a polyhedron in the first case are the coordinates of the vertices, in the second - its dimensions. Constructing projections of a polyhedron comes down to constructing projections of its mesh. The outer outline of the projection of the polyhedron is called the contour of the body.
Prism
─ a convex polyhedron whose lateral edges are parallel to each other. The lower and upper faces ─ equal polygons that determine the number of side edges are called the bases of the prism. A prism is called regular if its base is a regular polygon, and right if its side edges are perpendicular to the base. Otherwise the prism is inclined. The lateral faces of a straight prism are rectangles, and the inclined ones are parallelograms. Lateral surface the direct prism refers to the projecting objects and degenerates into a polygon onto the projection plane perpendicular to the side edges. The projections of points and lines located on the lateral surface of the prism coincide with its degenerate projection.
Typical task 3 (Fig. 29) : Construct a complex drawing of a straight prism with dimensions: l - side of the base (length of the prism); b- height of the isosceles triangle of the base (width of the prism); h is the height of the prism. Determine the position of edges and faces relative to the projection planes. On the faces ABB’A’ and ACC’A’, set the frontal projections of point M and straight line n, respectively, and construct their missing projections.
1. Mentally position the polyhedron in the system of projection planes so that its base is D ABC║P 1; and its edge is AC║P 3 (Fig. 29, a).
2. Mentally introduce the base planes: S║P 1 and coinciding with the base (D ABC); D║P 2 and coinciding with the rear edge ACC’A’. We build the base lines S 2, S 3, D 1, D 3 (Fig. 29, b).
3. We build horizontal, then frontal and, finally, profile projections of the prism, using the base lines D 1, D 3 (Fig. 29, c).
Ribs: AB, BC ─ horizontal; AC ─ profile-projecting; AS, SC, SB ─ horizontally projecting. Edges: ABC A"B'C' ─ horizontal levels; ABB'A', BCC'B' ─ horizontally projecting; ACC"A' ─frontal level..
5. The construction of horizontal projections of points lying on the lateral faces of the prism is carried out using the collective property of the projecting object: all projections of points and lines located on the lateral surface of the prism coincide with its degenerate (horizontal) projection. We build profile projections of points (for example M) by plotting along the horizontal lines of connection of their depth (Y M) from D 3, which are measured on the horizontal projection from D 1 (see also pp. 8, 17). On straight line n we set points 1, 2 and construct these points on the surface of the prism, similarly to point M. We determine visibility using the method of competing points. To complete the task “Prism with a cutout,” see.
a) b) c)
Rice. 29
Pyramid
a polyhedron, one of whose faces is a polygon (the base of the pyramid), which determines the number of lateral faces, and the remaining faces (sides) are triangles with a common vertex, called the vertex of the pyramid. The segments connecting the top of the pyramid with the vertices of the base are called lateral edges. The perpendicular dropped from the top of the pyramid to the plane of its base is called the height of the pyramid. A pyramid is regular if the base is a regular polygon and straight if the vertex is projected into the center of the base. The lateral edges of a regular pyramid are equal, and the lateral faces are isosceles triangles. The height of the side face of a regular pyramid is called apothem. If the top of the pyramid is projected outside its base, then the pyramid is inclined.
Typical problem 4(Fig. 30-32) : Construct a complex drawing of a straight regular pyramid with dimensions: l - side of the base (length); b- height of the base triangle (width); h is the height of the pyramid. Determine the position of edges and faces relative to the projection planes. Set the frontal and horizontal projections of points M and N belonging to the faces ASB and ASC, respectively, and construct their missing projections.
1. Mentally position the polyhedron in the system of projection planes so that its base is D ABC║P 1; and its edge is AC║P 3 (Fig. 31).
2. Mentally introduce the base planes: S║P 1 and coinciding with the base (D ABC);
D║P 2 and coinciding with the edge AC. We build the base lines S 2, S 3, D 1, D 3 (Fig. 32).
3. We build horizontal, then frontal and, finally,
profile projection of the pyramid (see Fig. 32).
4. We analyze the position of the edges and faces in the complex drawing of the pyramid, taking into account the initial data and classifiers of the position of straight lines and planes (p. 11,14).
Ribs: AB, BC ─ horizontal; AC ─ profile-projecting; AS, SC ─ general position; SB ─ profile level. Faces: ASB, BSC ─ general position; ABC ─horizontal level; ASC ─ profile-projecting.
5. We construct the missing projections of points lying on the faces of the pyramid using the “belonging of points to a plane” attribute. We use horizontal lines or arbitrary lines as auxiliary lines. We construct profile projections of points by plotting along horizontal connection lines the depths of points (in the direction of the Y axis), which are measured on the horizontal projection (see pp. 8, 17).
Rice. 30 Fig. 31 Fig. 32
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Polyhedra and bodies of revolution
Within the framework of the USP “First steps into space”
Team "Fur Seals", Novokuznetsk
"Navy Seals"?
Fur seals are not only cute, but also very smart. They are easy to train. Cats have a great built-in navigation system. Despite the fact that they are school animals, fur seals go hunting alone and generally display individualism. We called ourselves these animals because we want to be like them in many ways, to be brave and smart, because these animals are often underestimated.
Team motto:
We are Navy SEALs Active and smart Our motto is just three words, Smiling is cool!
Poems about geometric shapes
There is a pyramid in the world -
Amazing object
It was built in Egypt
But here's a secret for everyone.
So I walk around the apartment and look around me, And bodies of rotation surround me everywhere. There is a toy in the shape of a cone on the window. But the tea can took the shape of a cylinder.
There is a refrigerator in the kitchen It is shaped like a parallelepiped. Like his square Six facets on the face However there are differences
The cube has equal sides
And he has the opposite.
I confess to you prism, Well, very capricious. I'll say it without deception But so multifaceted (author Natalya U.)
And the best figure is a cube!
I'll put my tooth on the line
And all the edges and edges in it,
Straight at right angle
Polyhedra and bodies of revolution in objects of the surrounding world
Hypothesis: In many objects of the surrounding world, you can see polyhedra and bodies of revolution
Polyhedron -
A geometric body whose surface consists of a finite number of planar polygons.
Prism -
A polyhedron, two of whose faces are n-gons and the remaining faces are parallelograms.
Parallelepiped -
A prism whose bases are parallelograms.
Cube -
Rectangular parallelepiped with equal dimensions. All faces of a cube are equal squares.
Pyramid -
A polyhedron whose base is a polygon and whose remaining faces are triangles that have a common vertex.
Truncated pyramid -
A polyhedron whose vertices are the vertices of the base and the vertices of its section by a plane parallel to the base.
Bodies of revolution -
Volumetric bodies that arise when a flat geometric figure bounded by a curve rotates around an axis lying in the same plane.
Cylinder -
A figure obtained by rotating a rectangle around an axis containing its side.
Cone -
A figure obtained by rotating a right triangle around an axis.
Conclusion
During the study, we confirmed our hypothesis and made sure that many objects in the world around us have the shape of bodies of rotation and polyhedra.
Hypothesis:
THERE IS NO BOARD BETWEEN THE WORLD OF ART
AND THE WORLD OF GEOMETRY.
The famous artist, who was fond of geometry, Albrecht Durer (1471-1528), in a famous engraving "Melancholy"
in the foreground
depicted a stone polyhedron .
Dutch artist Moritz Cornilis Escher (1898-1972) has created unique and captivating works that use or display a wide range of mathematical ideas.
Regular geometric bodies - polyhedra - had a special charm for Escher. In many of his works, polyhedra are the main figure and in even more works they appear as auxiliary elements.
"Four Bodies" Escher depicted the intersection of the main regular polyhedra located on the same axis of symmetry; in addition, the polyhedra look translucent, and through any of them you can see the rest.
An elegant example of a star dodecahedron can be found in his work "Order and chaos." IN in this case a star-shaped polyhedron is placed inside a glass sphere. The ascetic beauty of this design contrasts with the trash randomly scattered on the table.
Most interesting work Escher - engraving "Stars" on which you can see the bodies obtained by combining tetrahedrons, cubes and octahedra.
If Escher had depicted in this work only various options polyhedra, we would never have known about it. But for some reason he placed chameleons inside the central figure to make it difficult for us to perceive the entire figure.
In the picture "Gravity" depicted dodecahedron , formed by twelve flat five-pointed stars. On each of the sites lives a long-necked, four-legged, tailless fantastic animal; its body is in a pyramid, into the holes of which it sticks out its limbs; the top of the pyramid is one of the walls of the dwelling of the neighboring monster .
In the artist's painting Salvador Dali "The Last Supper" Christ and his disciples are depicted against the background of a huge transparent dodecahedron.
According to the ancients, the UNIVERSE had the shape of a dodecahedron, i.e. they believed that we live inside a vault shaped like the surface of a regular dodecahedron.
Conclusion:
THE HYPOTHESIS HAS BEEN PROVEN, GEOMETRIC FIGURES, POLYHEDES ARE AN ESSENTIAL PART OF GEOMETRY. USING THE WORKS OF GREAT ARTISTS, WE HAVE PROVED THAT THERE IS NO DIMENSION BETWEEN ART AND GEOMETRY.
What contribution does geometry make to the development of human culture?
Art is special way knowledge and reflection of reality. Art develops a person’s spiritual culture. Having analyzed the works of great artists, we can say without a doubt that there is no boundary between the world of art and the world of geometry. This means that geometry also develops intellectual, creativity human, figurative and spatial thinking, therefore this science is an integral part of human culture.
Mind map “Polyhedra and bodies of rotation in the products of enterprises in my city”
Where does geometry live in your city?
Geometry lives everywhere in our city!!! No matter what architectural structure you look at, it always contains polyhedra and bodies of rotation. Collected together in one building they create unique, inimitable, ingenious buildings!!!
Literature used:
- http://www.uzluga.ru/potrb/Polyhedron+–+is+a+body whose surface+consists+of+a+finite+number+of+flat+polygonsb/part-5.html
- http://kamensky.perm.ru/proj/mng/01.htm
- http://www.liveinternet.ru/tags/%FD%F8%E5%F0/page3.html
- http://www.distedu.ru/mirror/_math/www.tmn.fio.ru/works/26x/304/d9_3.htm
- https://ru.wikipedia.org/wiki/Escher,_Maurits_Cornelis
- http://www.propro.ru/graphbook/graphbook/book/001/027.htm
- http://math4school.ru/mnogogranniki.html
