A polyhedron whose faces are 4 triangles. Regular polyhedra: elements, symmetry and area

Polyhedra not only occupy a prominent place in geometry, but are also found in Everyday life each person. Not to mention artificially created household items in the form of various polygons, starting with matchbox and ending with architectural elements, crystals in the form of a cube (salt), prism (crystal), pyramid (scheelite), octahedron (diamond), etc. are also found in nature.

The concept of a polyhedron, types of polyhedra in geometry

Geometry as a science contains the section stereometry, which studies the characteristics and properties of volumetric bodies, the sides of which are three-dimensional space formed by limited planes (faces), are called “polyhedra”. There are dozens of types of polyhedra, differing in the number and shape of faces.

Nevertheless, all polyhedra have common properties:

1. All of them have 3 integral components: a face (the surface of a polygon), a vertex (the corners formed at the junction of the faces), an edge (the side of the figure or a segment formed at the junction of two faces).
2. Each edge of a polygon connects two, and only two, faces that are adjacent to each other.
3. Convexity means that the body is completely located on only one side of the plane on which one of the faces lies. The rule applies to all faces of the polyhedron. In stereometry, such geometric figures are called convex polyhedra. The exception is stellated polyhedra, which are derivatives of regular polyhedra. geometric bodies.

Polyhedra can be divided into:

1. Types of convex polyhedra, consisting of the following classes: ordinary or classical (prism, pyramid, parallelepiped), regular (also called Platonic solids), semiregular (another name is Archimedean solids).
2. Non-convex polyhedra (stellate).

Prism and its properties

Stereometry as a branch of geometry studies the properties of three-dimensional figures, types of polyhedra (prism among them). A prism is a geometric body that necessarily has two completely identical faces (they are also called bases) lying in parallel planes, and the nth number of side faces in the form of parallelograms. In turn, the prism also has several varieties, including such types of polyhedra as:

1. A parallelepiped is formed if the base is a parallelogram - a polygon with 2 pairs of equal opposite angles and two pairs of congruent opposite sides.
2. A straight prism has ribs perpendicular to the base.
3. characterized by the presence of indirect angles (other than 90) between the edges and the base.
4. A regular prism is characterized by bases in the form of equal lateral faces.

Basic properties of a prism:

• Congruent bases.
• All edges of the prism are equal and parallel to each other.
• All side faces have the shape of a parallelogram.

Pyramid

A pyramid is a geometric body that consists of one base and the nth number of triangular faces connecting at one point - the apex. It should be noted that if the side faces of the pyramid are necessarily represented by triangles, then at the base there can be a triangular polygon, a quadrangle, a pentagon, and so on ad infinitum. In this case, the name of the pyramid will correspond to the polygon at the base. For example, if at the base of a pyramid there is a triangle - this is a quadrilateral, etc.

Pyramids are cone-shaped polyhedra. The types of polyhedra in this group, in addition to those listed above, also include the following representatives:

1. A regular pyramid has a regular polygon at its base, and its height is projected at the center of a circle inscribed in the base or circumscribed around it.
2. A rectangular pyramid is formed when one of the side edges intersects the base at a right angle. In this case, this edge can also be called the height of the pyramid.

Properties of the pyramid:

• If all the lateral edges of the pyramid are congruent ( same height), then they all intersect with the base at the same angle, and around the base you can draw a circle with the center coinciding with the projection of the top of the pyramid.
• If a regular polygon lies at the base of the pyramid, then all the side edges are congruent, and the faces are isosceles triangles.

Regular polyhedron: types and properties of polyhedra

In stereometry special place occupy geometric bodies with absolutely equal faces, at the vertices of which the same number of edges are connected. These bodies are called Platonic solids, or regular polyhedra. There are only five types of polyhedra with these properties:

1. Tetrahedron.
2. Hexahedron.
3. Octahedron.
4. Dodecahedron.
5. Icosahedron.

Regular polyhedra owe their name to the ancient Greek philosopher Plato, who described these geometric bodies in his works and associated them with the natural elements: earth, water, fire, air. The fifth figure was awarded similarity to the structure of the Universe. In his opinion, the atoms of natural elements are shaped like regular polyhedra. Thanks to their most fascinating property - symmetry, these geometric bodies represented big interest not only for ancient mathematicians and philosophers, but also for architects, artists and sculptors of all times. The presence of only 5 types of polyhedra with absolute symmetry was considered a fundamental find, they were even associated with the divine principle.

Hexahedron and its properties

In the form of a hexagon, Plato's successors assumed a similarity with the structure of the atoms of the earth. Of course, at present this hypothesis has been completely refuted, which, however, does not prevent the figures from attracting minds in modern times famous figures its aesthetics.

In geometry, a hexahedron, also known as a cube, is considered a special case of a parallelepiped, which, in turn, is a type of prism. Accordingly, the properties of the cube are related to the only difference that all the faces and corners of the cube are equal to each other. The following properties follow from this:

1. All edges of the cube are congruent and lie in parallel planes with respect to each other.
2. All faces are congruent squares (there are 6 of them in the cube), any of which can be taken as the base.
3. All interhedral angles are equal to 90.
4. Each vertex has an equal number of edges, namely 3.
5. The cube has 9 which all intersect at the point of intersection of the diagonals of the hexahedron, called the center of symmetry.

Tetrahedron

A tetrahedron is a tetrahedron with equal faces in the shape of triangles, each of the vertices of which is the connecting point of three faces.

Properties of a regular tetrahedron:

1. All faces of a tetrahedron - this means that all faces of a tetrahedron are congruent.
2. Since the base is represented by a regular geometric figure, that is, it has equal sides, then the faces of the tetrahedron converge at the same angle, that is, all angles are equal.
3. The sum of the plane angles at each vertex is 180, since all angles are equal, then any angle of a regular tetrahedron is 60.
4. Each vertex is projected to the point of intersection of the heights of the opposite (orthocenter) face.

Octahedron and its properties

When describing the types of regular polyhedra, one cannot fail to note such an object as the octahedron, which can be visually represented as two quadrangular regular pyramids glued together at the bases.

Properties of the octahedron:

1. The very name of a geometric body suggests the number of its faces. An octahedron consists of 8 congruent equilateral triangles, at each of the vertices of which an equal number of faces converge, namely 4.
2. Since all the faces of the octahedron are equal, its interface angles are also equal, each of which is equal to 60, and the sum of the plane angles of any of the vertices is thus 240.

Dodecahedron

If we imagine that all the faces of a geometric body are a regular pentagon, then we get a dodecahedron - a figure of 12 polygons.

Properties of the dodecahedron:

1. Three faces intersect at each vertex.
2. All faces are equal and have same length ribs, as well as an equal area.
3. The dodecahedron has 15 axes and planes of symmetry, and any of them passes through the vertex of the face and the middle of the edge opposite to it.

Icosahedron

No less interesting than the dodecahedron, the icosahedron figure is a three-dimensional geometric body with 20 equal faces. Among the properties of the regular 20-hedron, the following can be noted:

1. All faces of the icosahedron are isosceles triangles.
2. Five faces meet at each vertex of a polyhedron, and the sum adjacent corners tops is 300.
3. The icosahedron, like the dodecahedron, has 15 axes and planes of symmetry passing through the midpoints of opposite faces.

Semiregular polygons

In addition to the Platonic solids, the group of convex polyhedra also includes the Archimedean solids, which are truncated regular polyhedra. The types of polyhedra in this group have the following properties:

1. Geometric bodies have pairwise equal faces of several types, for example, a truncated tetrahedron has, like a regular tetrahedron, 8 faces, but in the case of an Archimedean body, 4 faces will be triangular in shape and 4 will be hexagonal.
2. All angles of one vertex are congruent.

Star polyhedra

Representatives of non-volumetric types of geometric bodies are stellate polyhedra, the faces of which intersect with each other. They can be formed by the fusion of two regular three-dimensional bodies or as a result of the extension of their faces.

Thus, such stellated polyhedra are known as: stellated forms of octahedron, dodecahedron, icosahedron, cuboctahedron, icosidodecahedron.

Lesson 7 on the topic: “Polyhedra. Vertices, edges, faces of a polyhedron"

Purpose of the lesson: introduce students to one of the types of polyhedra - the cube; by measuring and observing, find as many properties of the cube as possible.

Lesson type: learning new material

Methods:

By sources of knowledge: verbal, visual;

According to the degree of teacher-student interaction: heuristic conversation;

Regarding didactic tasks: preparation for perception;

Regarding the nature of cognitive activity:reproductive, partially search.

Equipment: Textbook:Mathematics: Visual geometry. 5-6 grades I.F. Sharygin, multimedia projector, computer.

Learning outcomes:

Personal: capacity for emotional perception mathematical objects, the ability to express one’s thoughts clearly and accurately.

Metasubject: ability to understand and use visual aids.

Subject: learn to draw scans and make shapes using them.

Equipment: textbook “Visual Geometry. 5th - 6th grade" S. Sharygin, interactive board, scissors.

UUD:

educational: analysis and classification of objects

regulatory: goal setting; identifying and realizing what is already known and what needs to be learned

communicative: educational cooperation with the teacher and peers.

During the classes

Organizing time.

Updating and recording basic knowledge.

On the table are polyhedra, which the students became acquainted with in primary school. What shapes can you name? Which figures are there the most?

It is difficult to find a person who is not familiar with the cube. After all, cubes are a favorite game for kids. It seems that we know everything about the cube. But is it?

The cube is a representative of a large family of polyhedra. You have already met some - this is a pyramid, cuboid. Meeting others awaits you ahead.

Despite all their differences, polyhedra have a number of common properties.

The surface of each of them consists of flat polygons, which are calledpolyhedron faces . Two adjacent flat polygons have a common side -polyhedron edge . The ends of the ribs arepeaks polyhedron.

In the last lesson you were interested in the types of polyhedra and here are 5 representatives of regular polygons.

Tetrahedron octahedron icosahedron hexahedron dodecahedron

Generalization and systematization of knowledge

Look at the image of the cube in the figure, draw it in your notebook and write the names of the main elements of the cube. Remember and use these terms in the future.

A cube is a regular polyhedron whose faces are squares and three edges and three faces meet at each vertex. It has: 6 faces, 8 vertices and 12 edges.

Working with models.

Working with sweeps.

№ 2 (Mathematics: Visual geometry. Grades 5-6 I.F. Sharygin) On a piece of paper, draw the development of a cube. Cut it out and roll it into a cube, glue it together.

The cut out figure is calledcube scan . Think about why it is named that way.

№ 3 (Mathematics: Visual geometry. Grades 5-6 I.F. Sharygin) Try to assemble a cube from the proposed developments and transfer them to your notebook.

№ 5 (Mathematics: Visual geometry. Grades 5-6 I.F. Sharygin) The development of a cube is given. Which of the cubes in Figure 30, a-c can be glued together from it? Choose a cube and justify your choice.

№ 12 (Mathematics: Visual geometry. Grades 5-6 I.F. Sharygin) There is a strip of paper measuring 1*7. How to make a single cube out of it?

№ 15 (Mathematics: Visual geometry. Grades 5-6 I.F. Sharygin) A spider and a fly sit at opposite vertices of the cube. What is the shortest way for a spider to crawl to a fly? Explain your answer

Reflection on educational activities.

today I found out...

it was interesting…

it was difficult…

I completed tasks...

I purchased...

I learned…

I managed …

I was able...

I will try…

I was surprised...

gave me a lesson for life...

Homework. Make a cube model from cardboard.

Subject."Polyhedron. Elements of a polyhedron - faces, vertices, edges."

Goals. Create conditions for expansion theoretical knowledge about spatial figures: introduce the concepts of “polyhedron”, “faces”, “vertex”, “edge”; ensure the development in schoolchildren of the ability to highlight the main thing in cognitive object; promote development spatial imagination students.

Educational materials. Textbook “Mathematics. 4th grade" (author V.N. Rudnitskaya, T.V. Yudacheva); computer; projector; presentation "Polygons"; printed forms “Coordinate angle”, “Polygons”, “Problem”; models of polyhedra, development of polyhedra; mirrors; scissors.

DURING THE CLASSES

Before the start of the lesson, children are divided into three groups according to their level of knowledge - high, average, low.

I. Organizational moment

Teacher. My dear restless people, once again I invite you to fascinating world mathematics. And I am sure that in this lesson you will learn new things, consolidate what you have learned and be able to apply the acquired knowledge in practice.

Today I would like to begin our lesson with the words of the English philosopher Roger Bacon about mathematics: “He who does not know mathematics cannot study other sciences and cannot understand the world.” I think that in the lesson we will certainly find confirmation of the words of this philosopher.

II. Repetition of covered material. Constructing polygons by coordinates

U. In mathematics lessons in grades 1, 2, 3, we studied various flat geometric figures and also learned how to build them. I suggest you build in coordinate angle flat figures according to these coordinates.

The task is completed on printed forms.

Group 1

Construct a figure if the coordinates are known A (0; 2), IN (2; 5), WITH(9; 2). What kind of figure did you get?

Group 2

Construct a rectangle if the points A(3; 2) and IN(6; 5) are its opposite vertices. Give the coordinates of the opposite vertices. What is another name for this figure?

Group 3

Construct a figure if the coordinates of its vertices are known A (2; 3), IN (2; 6), WITH (5; 8), D (8; 6), K (8; 3), M(5; 1). What kind of figure did you get?

– What can you call all these figures?

Children. These are polygons.

Slide 1

U. We know that all polygons have vertices and sides. Name and show them.

One person from the group completes the task at the board.

III. Getting to know new material

U. Today I will introduce you to volumetric geometric shapes, which are called polygons. Their models are presented on your tables.

Students have three-dimensional figures on their tables: cubes, parallelepipeds, pyramids, prisms.

– Sit back, look carefully, listen carefully and remember.

Introduction to the concepts of “polyhedron”, “face”, “vertex”, “edge”

– If you take 4 triangles, you can create three-dimensional figure – pyramid. From squares you can get another figure - a cube, from rectangles - a parallelepiped. You have another figure on your table - a prism, which is made up of rectangles and triangles. All these figures are called polyhedra .

Each of the polygons (in in this case triangles) are called edge polyhedron. And the sides of polygons are called ribs polyhedron. And, of course, the vertices of the polygon will be peaks polyhedron. This is what a drawing of a polyhedron looks like on a sheet of paper.

Slide 2

– It seems that the figure is made of glass. What do you think is shown by the dotted line in the drawing?

D. Invisible ribs.

Children work according to a drawing at the board.

U. So what is it?

D. Polyhedron.

U. Name and show the faces of the polyhedron, its edges and vertices.

Children point with a pointer and list.

– If you cut the pyramid from the top to the base along the edges, you will get something like this.
And now, my dear fidgets, find a form on the table with a picture of a polygon, carefully read the instructions:

1. Carefully examine the drawing of the polygon.
2. Find the desired development of the polygon (models on the board).
3. Assemble the polygon model.
4. Indicate the number of vertices __, faces __, edges __ of the polygon.
5. Name each vertex __, edge __, face __ of the polygon.

Group 1

Group 2

Group 3

– The board shows developments of polyhedra. Try to find the development of your figure from the drawing and assemble a polyhedron. Work together and I think you will succeed.

Checking the completion of the task (slides 3, 4, 5).

peaks – 8; ribs – 12; faces – 6; vertices – M, B, C, A, X, K, O, T; ribs – MB, MA, MT, TX, TO, XK, XA, KO, KC, CB, AC, BO; faces – MBOT, MBCA, KCBO, TXKO, ACKX, MAXT.

peaks – 8; ribs – 12; faces – 6; vertices – M, B, C, A, X, K, O, T; ribs – MB, MA, MT, TX, TO, XK, XA, KO, KC, CB, AC, BO; faces – MBOT, MBCA, KCBO, TXKO, ACKX, MAXT.

peaks – 12; ribs – 18; faces – 8; vertices – Y, B, A, X, N, M, P, E, D, F, L, C; ribs – YB, YX, BA, XA, XN, NM, AM, ME, EP, NP, ED, PF, DF, FL, LC, CD, LY, CB; faces – BAMEDC, YXNPFL, YBAX, XAMN, NMEP, EDFP, DFLC, CLYB.

IV. Generalization and systematization of knowledge

U. Tell me, are there objects in the world around us that have the shape of polyhedra?

The children's answers are listened to. An impromptu “walk” is held around the school yard. Children “examine” models of the school building and utility rooms, which look like polyhedra.

– Complete the task:

The Wolf and the Hare glued together a house from colored paper. How many faces of each color were needed? What polygon shape does the edge of each color have?

Slide 6

V. Consolidation of previously learned

U. Guys, imagine yourself as architects, designers or builders and try to solve problems.

Group 1 assignment

Find the area that the new school building will occupy if its length is 74 m and its width is 13 m. ( Answer: 962 sq. m.)

Group 2 assignment

The area of ​​the playground in the courtyard of our school is 1080 square meters. m. This is for 1320 sq. m less than the area of ​​the hockey rink. Calculate the area of ​​the hockey rink. ( Answer: 2400 sq. m)

Group 3 assignment

An area of ​​2,500 square meters has been allocated for the construction of a new building for our school. m. It is known that the building will be 13 m wide and 74 m long. What area of ​​the site will be left for flower beds and paths after the building is built? ( Answer: 1) 962 sq. m; 2) 1538 sq. m)

Children check solutions to problems and explain how they solved them.

VI. Lesson summary

U. It turns out that Roger Bacon was right when he said: “He who does not know mathematics cannot learn other sciences and cannot understand the world.”

The teacher evaluates the work of the groups.

1. In Figure 1, indicate convex and non-convex polyhedra.

Answer: Convex - b), d); non-convex - a), c), d).

2. Give an example of a non-convex polyhedron in which all faces are convex polygons.

Answer: Figure 1, a).

3. Is it true that the union of convex polyhedra is convex polyhedron?

Answer: No.

4. Can the number of vertices of a polyhedron be equal to the number of its faces?

Answer: Yes, near the tetrahedron.

5. Establish a connection between the number of flat angles P of a polyhedron and the number of its edges P.

Answer: P = 2P.

6. The only faces of a convex polyhedron are triangles. How many vertices B and faces D does it have if it has: a) 12 edges; b) 15 ribs? Give examples of such polyhedra.

7. Three edges emerge from each vertex of a convex polyhedron. How many vertices B and faces D does it have if it has: a) 12 edges; b) 15 ribs? Draw these polyhedra.

Answer: a) B = 8, D = 6, cube; b) B = 10, G = 7, pentagonal prism.

8. Four edges converge at each vertex of a convex polyhedron. How many vertices B and faces D does it have if the number of edges is 12? Draw these polyhedra.

9. Prove that any convex polyhedron has a triangular face or three edges meet at some vertex.

10. Think about where the convexity of the polyhedron was used in the reasoning showing the validity of Euler’s relation.

11. What is the value of B - P + G for the polyhedron shown in Figure 6?

Regular polyhedra

A convex polyhedron is called regular if its faces are equal regular polygons, and all polyhedral angles are equal.

Let us consider possible regular polyhedra and, first of all, those whose faces are regular triangles. The simplest such regular polyhedron is a triangular pyramid, the faces of which are regular triangles (Fig. 7). Three faces meet at each of its vertices. Having only four faces, this polyhedron is also called a regular tetrahedron, or simply tetrahedron, which is translated from Greek language means tetrahedron.

A polyhedron whose faces are regular triangles and four faces meet at each vertex is shown in Figure 8. Its surface consists of eight regular triangles, which is why it is called an octahedron.

A polyhedron, at each vertex of which five regular triangles meet, is shown in Figure 9. Its surface consists of twenty regular triangles, which is why it is called an icosahedron.

Note that since more than five regular triangles cannot converge at the vertices of a convex polyhedron, there are no other regular polyhedra whose faces are regular triangles.

Similarly, since only three squares can converge at the vertices of a convex polyhedron, then, except for the cube (Fig. 10), there are no other regular polyhedra whose faces are squares. A cube has six faces and is therefore also called a hexahedron.

A polyhedron whose faces are regular pentagons and three faces meet at each vertex is shown in Figure 11. Its surface consists of twelve regular pentagons, which is why it is called a dodecahedron.

Let us consider the concept of a regular polyhedron from the point of view of the topology of science, which studies the properties of figures that do not depend on various deformations without discontinuities. From this point of view, for example, all triangles are equivalent, since one triangle can always be obtained from any other by appropriate compression or expansion of the sides. In general, all polygons with the same number of sides are equivalent for the same reason.

How to define the concept of a topologically regular polyhedron in such a situation? In other words, which properties in the definition of a regular polyhedron are topologically stable and should be retained, and which properties are not topologically stable and should be discarded.

In the definition of a regular polyhedron, the number of sides and the number of faces are topologically stable, i.e. not changing under continuous deformations. The regularity of polygons is not a topologically stable property. Thus, we come to the following definition.

A convex polyhedron is called topologically regular if its faces are polygons with the same number of sides and converges at each vertex same number faces.

Two polyhedra are said to be topologically equivalent if one can be obtained from the other by continuous deformation.

For example, everyone triangular pyramids are topologically regular polyhedra, equivalent to each other. All parallelepipeds are also equivalent topologically regular polyhedra. For example, quadrangular pyramids are not topologically regular polyhedra.

Let us clarify the question of how many topologically regular polyhedra that are not equivalent to each other exist.

As we know, there are five regular polyhedra: tetrahedron, cube, octahedron, icosahedron and dodecahedron. It would seem that there should be much more topologically regular polyhedra. However, it turns out that there are no other topologically regular polytopes that are not equivalent to the already known regular ones.

To prove this, we will use Euler's theorem. Let a topologically regular polyhedron be given, the faces of which are n-gons, and m edges converge at each vertex. It is clear that n and m are greater than or equal to three. Let us denote, as before, B the number of vertices, P the number of edges, and G the number of faces of this polyhedron. Then

nГ = 2P; Г = ; mB = 2P; B = .

According to Euler's theorem, B - P + G = 2 and, therefore,

Where does P = .

From the resulting equality, in particular, it follows that the inequality 2n + 2m - nm > 0 must hold, which is equivalent to the inequality (n - 2)(m - 2)< 4.

Let's find all possible values ​​of n and m that satisfy the found inequality and fill in the following table

	tetrahedron	H=6, P=12, D=8	H=12, P=30, D=20 icosahedron
	H=8, P=12, D=4	Does not exist	Does not exist
	H=20, P=30, D=12 dodecahedron	Does not exist	Does not exist

For example, the values ​​n = 3, m = 3 satisfy the inequality (n - 2)(m - 2)< 4. Вычисляя значения Р, В и Г по приведенным выше формулам, получим Р = 6, В = 4, Г = 4.

The values ​​n = 4, m = 4 do not satisfy the inequality (n - 2)(m - 2)< 4 и, следовательно, соответствующего многогранника не существует.

Check the other cases yourself.

From this table it follows that the only possible topologically regular polyhedra are the regular polyhedra listed above and the polyhedra equivalent to them.

Definition. A polyhedron is called regular if: 1) it is convex; 2) all its faces are regular polygons equal to each other; 3) the same number of edges converge at each of its vertices; 4) all its dihedrals are equal.

An example of a regular polyhedron is a cube: it is a convex polyhedron, all its faces are equal squares, three edges meet at each vertex, and all dihedral angles of the cube are right. A regular tetrahedron is also a regular polyhedron.

The question arises: how many are there? various types regular polyhedra?

Five types of regular polyhedra:

Consider an arbitrary regular polyhedron M , which has B vertices, P edges and G faces. According to Euler’s theorem, the following equality holds for this polyhedron:

B - P + G = 2. (1)

Let each face of a given polyhedron contain m edges (sides), and converge at each vertex n ribs Obviously,

Since the polyhedron B has vertices, and each of them has n edges, we get n edges. But any edge connects two vertices of a polyhedron, so each edge will appear in the product n twice. This means that the polyhedron has various ribs Then

From (1), (3), (4) we obtain - P + = 2, whence

+ = + > . (5)

Thus we have

From inequalities 3 and 3 it follows that the faces of a regular polyhedron can be either regular triangles, or regular quadrilaterals, or regular pentagons. Moreover, in cases m = n = 4; m = 4, n = 5; m = 5, n = 4; m = n = 5 we come to a contradiction with the condition. Therefore, five cases remain possible: 1) m = n = 3; 2) m = 4, n = 3; 3) m = 3, n = 4; 4) m = 5, n = 3; 5) m = 3, n = 5. Let us consider each of these cases using relations (5), (4) and (3).

1) m = n = 3(each face of the polyhedron is a regular triangle. This is the one we know regular tetrahedron (« tetrahedron"means tetrahedron).

2) m = 4, n = 3(each face is a square, and three edges meet at each vertex). We have

P = 12; B = 8; G = 6.

We get a regular hexagon, each face of which is a square. This polyhedron is called regular hexahedron and is a cube (" hexahedron"- hexagon), any parallelepiped is a hexahedron.

3) m = 3, n = 4(each face is a regular triangle, four edges meet at each vertex). We have

P = 12; B = =6; G = =8.

We get a regular octahedron, each face of which is a regular triangle. This polyhedron is called regular octahedron ("octahedron" -- octahedron).

4) m = 5, n = 3(each face is a regular pentagon, three edges meet at each vertex). We have:

P = 30; B = = 20; G = = 12.

We get a regular dodecahedron, each face of which is a regular pentagon. This polyhedron is called regular dodecahedron (« dodecahedron"- dodecahedron).

5) m = 3,n = 5(each face is a regular triangle, five edges meet at each vertex). We have

P = 30; B = =12; G = = 20.

We get the correct twenty-sided figure. This polyhedron is called regular icosahedron (« icosahedron" - twenty-sided).

Thus, we have obtained the following theorem.

Theorem. There are five different (up to similarity) types of regular polyhedra: regular tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron and regular icosahedron.

This conclusion can be reached somewhat differently.

Indeed, if the face of a regular polyhedron is a regular triangle, and they converge at one vertex k ribs, i.e. all flat angles convex k-faceted angles are equal, then. Hence, natural number k can take values: 3;4;5. in this case Г = , Р = . Based on Euler's theorem we have:

B+-= 2 or B (6 - k) = 12.

Then at k= 3 we get: B = 4, G = 4, P = 6 (regular tetrahedron);

at k = 4 we get: B = 6, G = 8, P = 12 (regular octahedron);

at k = 5 we get: B = 12, G = 20, P = 30 (regular icosahedron).

If the face of a regular polyhedron is a regular quadrilateral, then. This condition corresponds to a single natural number k= 3. Then: Г = , Р= ; B + - = 2 or. This means B = 8, G = 6, P = 12 - we get a cube (regular hexahedron).

If the face of a regular polyhedron is a regular pentagon, then. This condition is also met only by k= 3 and Г = ; P = . Likewise previous calculations we get: and B = 20, G = 12, P = 30 (regular dodecahedron).

Starting with regular hexagons, presumably the faces of a regular polyhedron, plane angles become no smaller and narrower k= 3 their sum becomes no less, which is impossible. Consequently, there are only five types of regular polyhedra.

The figures show the layout of each of the five regular polyhedra.

Regular tetrahedron

Regular octahedron

Regular hexahedron

Regular icosahedron

Regular dodecahedron

Some properties of regular polyhedra are given in the following table.

Face type	Flat apex angle	View of a polyhedral vertex angle	Sum of plane angles at a vertex				Polyhedron name
Correct triangle	3-sided				Regular tetrahedron
Correct triangle	4-sided				Regular octahedron
Correct triangle	5-sided				Regular icosahedron
	3-sided				Correct hexahedron (cube)
Correct pentagon	3-sided					Correct dodecahedron

For each of the regular polyhedra, in addition to those already indicated, we will most often be interested in:

• 1. Its size dihedral angle with an edge (with an edge length a).
• 2. Its area full surface(with rib length a).
• 3. Its volume (with rib length a).
• 4. The radius of the sphere described around it (with an edge length a).
• 5. The radius of the sphere inscribed in it (with the length of the edge a).
• 6. The radius of a sphere touching all its edges (with an edge length a).

The simplest solution is to calculate the area of ​​the total surface of a regular polyhedron; it is equal to G, where G is the number of faces of a regular polyhedron, and is the area of ​​one face.

Let us recall that sin =, which gives us the opportunity to write in radicals: ctg =. Taking this into account, we create tables:

a) for the area of ​​a face of a regular polyhedron

b) for the total surface area of ​​a regular polyhedron

Now let's move on to calculating the dihedral angle of a regular polyhedron at its edge. For a regular tetrahedron and cube, you can easily find the value of this angle.

In a regular dodecahedron, all plane angles of its faces are equal, therefore, applying the cosine theorem for trihedral angles to any trihedral angle of a given dodecahedron at its vertex, we obtain: cos, from which

On the regular octahedron ABCDMF shown, you can verify that the dihedral angle at the edge of the octahedron is equal to 2arctg.

To find the value of the dihedral angle at the edge of a regular icosahedron, we can consider the trihedral angle ABCD at vertex A: its plane angles BAC and CAD are equal, and the third plane angle BAD, opposite which the dihedral angle B(AC)D = lies, is equal to (BCDMF - regular pentagon ). By the cosine theorem for trihedral angle ABCD we have: . Considering that, we get where from. Thus, the dihedral angle at the edge of the icosahedron is equal.

So, we obtain the following table of values ​​of dihedral angles at the edges of regular polyhedra.

Before finding the volume of a particular regular polyhedron, we first discuss how to find the volume of regular polyhedra in general form.

Try first to prove that if the center of each face of any regular polyhedron is a straight line, perpendicular to the plane this face, then all the drawn lines intersect at a certain point ABOUT, removed from all faces of a given polyhedron at the same distance, which we denote by r. Dot ABOUT will be the center of a sphere inscribed in a given polyhedron, and r- its radius. By connecting the resulting point ABOUT with all the vertices of a given polyhedron, we will divide it into Γ equal pyramids (Γ is the number of faces of a regular polyhedron): the bases of the formed pyramids are equal r. Then the volume of this polyhedron equal to the sum volumes of all these pyramids. Since the polyhedron is regular, its volume is V can be found using the formula:

It remains to find the length of the radius r.

To do this, connect the dot ABOUT with the middle TO edges of the polyhedron, try to make sure that the slant KO to the face of a polyhedron containing an edge, makes an angle with the plane of this face equal to half the dihedral angle at this edge of the polyhedron; oblique projection KO on the plane of this face belongs to its apothem and is equal to the radius of the circle inscribed in it. Then

where p is the semi-perimeter of the face. Then from (1) and (2) we obtain a general formula for calculating their volumes for all regular polyhedra:

This formula is completely unnecessary for finding the volumes of a cube, regular tetrahedron and octahedron, but it makes it quite easy to find the volumes of a regular icosahedron and dodecahedron.