Polyhedra. Types of polyhedra and their properties

Polyhedra not only occupy a prominent place in geometry, but also occur in the everyday life of each person. Not to mention the artificially created objects of everyday use in the form of various polygons, starting with the matchbox and ending with architectural elements, in nature there are also crystals in the form of a cube (salt), prisms (crystal), pyramids (scheelite), octahedra (diamond) and t E.

The concept of a polyhedron, the types of polyhedra in geometry

Geometry as a science contains a section of stereometry that studies the characteristics and properties of three-dimensional figures. Geometric bodies, whose sides in the three-dimensional space are formed by bounded planes (faces), are called "polyhedra." Types of polyhedra number more than one dozen representatives, differing in the number and shape of faces.

Nevertheless, all polyhedra have common properties:

1. All of them have 3 integral components: the face (polygon surface), the vertex (the corners formed at the junction of the faces), the edge (the side of the figure or the segment formed at the junction of the two faces).
2. Each edge of a polygon joins two, and only two faces that are adjacent to each other.
3. Convexity means that the body is completely located only on one side of the plane on which one of the faces lies. The rule is applicable to all faces of the polyhedron. Such geometric figures in stereometry are called convex polyhedra. The exception is star polyhedra, which are derivatives of regular polyhedral 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 bodies), semi-regular (second name - Archimedean bodies).
2. Nonconvex polyhedra (stellate).

Prism and its properties

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

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

The basic properties of the prism:

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

Pyramid

A pyramid is a geometric body that consists of one base and the n-th number of triangular faces joining at one point - the vertex. It should be noted that if the side faces of the pyramid are represented by triangles, then in the base there can be both a triangular polygon, a quadrilateral, and a pentagon, and so on ad infinitum. The name of the pyramid will correspond to the polygon at the bottom. For example, if there is a triangle at the bottom of the pyramid, it is a triangular pyramid, a quadrilateral is quadrangular, and so on.

Pyramids are cone-like polyhedra. Types of polyhedra of this group, besides the above, also include the following representatives:

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

Properties of the pyramid:

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

Correct polyhedron: types and properties of polyhedra

In stereometry, a special place is occupied by geometric bodies with absolutely equal sides, at the vertices of which the same number of edges are connected. These bodies are called Platonic bodies, or regular polyhedra. Types of polyhedra with these properties have only five figures:

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

By their name, the correct polyhedra are due to the Greek philosopher Plato, who described these geometric bodies in his works and connected them with the natural elements: earth, water, fire, air. The fifth figure was awarded a similarity to the structure of the universe. In his view, the atoms of natural elements resemble in shape the kinds of regular polyhedra. Due to its most fascinating property - symmetry, these geometric bodies were of great interest not only for ancient mathematicians and philosophers, but also for architects, artists and sculptors of all time. The presence of only 5 types of polyhedra with absolute symmetry was considered a fundamental find, they were even awarded a connection with the divine beginning.

Hexahedron and its properties

In the shape of a hexagon, Plato's successors assumed a similarity to the structure of the atoms of the earth. Of course, at the present time this hypothesis is completely refuted, which, however, does not prevent the figures from attracting the minds of well-known figures in their aesthetics.

In geometry, a hexahedron, also a cube, is considered a particular case of a parallelepiped, which in turn is a kind of prism. Accordingly, the properties of the cube are related to the properties of the prism with the only difference that all the faces and angles of the cube are equal to each other. This implies the following properties:

1. All the 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 in the cube), any of which can be taken as the base.
3. All the interfacial angles are equal to 90.
4. From each vertex comes an equal number of edges, namely 3.
5. The cube has 9 axes of symmetry, which all intersect at the point of intersection of the diagonals of the hexahedron, called the center of symmetry.

Tetrahedron

The tetrahedron is a tetrahedron with equal faces in the form of triangles, each of whose vertices is a point of connection of three faces.

Properties of a regular tetrahedron:

1. All faces of a tetrahedron are equilateral triangles, from which it follows 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 the angles are equal.
3. The sum of the planar angles at each of the vertices is 180, since all the angles are equal, then any angle of the regular tetrahedron is 60.
4. Each of the vertices is projected to the intersection point of the heights of the opposite (orthocenter) face.

Octahedron and its properties

Describing the types of regular polyhedra, one can not fail to note an object such as an octahedron that can be visually represented in the form of two quadrilateral regular pyramids stuck together by bases.

Properties of an octahedron:

1. The very name of the geometric body tells us the number of its faces. The octagon consists of 8 congruent equilateral triangles, in each vertex of which there is an equal number of faces, namely 4.
2. Since all the faces of an octahedron are equal, its inter-angle angles, each equal to 60, are equal, and the sum of the planar angles of any of the vertices is, therefore, 240.

Dodecahedron

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

Properties of the dodecahedron:

1. Each vertex intersects three faces.
2. All faces are equal and have the same length of edges, 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 opposite edge.

Icosahedron

No less interesting than a dodecahedron, the icosahedron is a voluminous geometric body with 20 equal faces. Among the properties of a regular dentagon, the following can be noted:

1. All faces of an icosahedron are isosceles triangles.
2. Each vertex of the polyhedron converges to five faces, and the sum of the adjacent vertex angles is 300.
3. The icosahedron has, like the dodecahedron, 15 axes and planes of symmetry passing through the centers of opposite faces.

Semi-regular polygons

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

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

Star polyhedra

Representatives of non-existent types of geometric bodies are star polyhedra whose faces intersect each other. They can be formed by the fusion of two regular three-dimensional bodies or as a result of the continuation of their faces.

Thus, such star polyhedra as are known: star-shaped octahedron, dodecahedron, icosahedron, cuboctahedral, icosododecahedron.