Geometric volumetric figures are solids that occupy a nonzero volume in Euclidean (three-dimensional) space. These figures are studied by a branch of mathematics called spatial geometry. Knowledge of the properties of three-dimensional figures is used in engineering and in the sciences of nature. Consider the question in the article, geometric volumetric figures and their names.
Geometric volumetric bodies
Since these bodies have finite dimension in three spatial directions, a system of three coordinate axes is used to describe them in geometry. These axes have the following properties:
- They are orthogonal to each other, that is, perpendicular.
- These axes are normalized, that is, the basis vectors of each axis have the same length.
- Any of the coordinate axes is the result of the vector product of the other two.
Speaking of geometric volumetric figures and their names, it should be noted that they all belong to one of 2 large classes:
- Polyhedron class. These figures, based on the name of the class, have straight edges and flat faces. A face is a plane that borders a shape. The junction of two faces is called an edge, and the junction of three faces is a vertex. Polyhedra include the geometric shape of a cube, tetrahedra, prisms, and pyramids. For these figures, the Euler theorem holds, which establishes a relationship between the number of sides (C), edges (P) and vertices (B) for each polyhedron. Mathematically, this theorem is written as follows: C + B = P + 2.
- A class of round bodies or bodies of revolution. These figures have at least one surface that forms them, curved. For example, ball, cone, cylinder, torus.
As for the properties of three-dimensional figures, two most important of them should be distinguished:
- The presence of a certain volume, which the figure occupies in space.
- The presence of each volumetric figure surface area.
Both properties for each figure are described by specific mathematical formulas.
Let us consider below the simplest geometric volumetric figures and their names: cube, pyramid, prism, tetrahedron and ball.
Cube Shape: Description
By a geometric figure, a cube is understood to mean a volumetric body, which is formed by 6 square planes or surfaces. This figure is also called the regular hexahedron, because it has 6 sides, or a rectangular parallelepiped, since it consists of 3 pairs of parallel sides that are mutually perpendicular to each other. A cube is also called a rectangular prism, in which the base is a square, and the height is equal to the side of the base.
Since the cube is a polyhedron or polyhedron, we can apply Euler's theorem to it to determine the number of its edges. Knowing that the number of sides is 6, and the vertices of the cube are 8, the number of edges is equal to: P = C + B - 2 = 6 + 8 - 2 = 12.
If we denote by the letter “a” the length of the side of the cube, then the formulas for its volume and surface area will have the form: V = a 3 and S = 6 * a 2 , respectively.
Pyramid figure
A pyramid is a polyhedron that consists of a simple polyhedron (the base of the pyramid) and triangles that connect to the base and have one common vertex (the top of the pyramid). Triangles are called the side faces of the pyramid.
The geometric characteristics of a pyramid depend on which polygon lies at its base, and also on whether the pyramid is straight or oblique. By a straight pyramid is meant a pyramid for which a straight line perpendicular to the base, drawn through the top of the pyramid, intersects the base at its geometric center.
One of the simple pyramids is a quadrangular straight pyramid, at the base of which lies a square with side "a", the height of this pyramid is "h". For this pyramid figure, the volume and surface area will be equal: V = a 2 * h / 3 and S = 2 * a * √ (h 2 + a 2/4) + a 2 , respectively. Applying Euler's theorem for it, taking into account the fact that the number of faces is 5 and the number of vertices is 5, we get the number of edges: P = 5 + 5 - 2 = 8.
Tetrahedron Figure: Description
By a geometric figure, a tetrahedron is understood as a volumetric body formed by 4 faces. Based on the properties of space, only triangles can represent such faces. Thus, the tetrahedron is a special case of a pyramid with a triangle at its base.
If all 4 triangles forming the faces of the tetrahedron are equilateral and equal to each other, then such a tetrahedron is called regular. This tetrahedron has 4 faces and 4 vertices, the number of edges is 4 + 4 - 2 = 6. Using standard formulas from flat geometry for the figure in question, we get: V = a 3 * √2 / 12 and S = √3 * a 2 , where a is the length of the side of an equilateral triangle.
It is interesting to note that in nature, some molecules have the shape of a regular tetrahedron. For example, a methane molecule CH 4 , in which hydrogen atoms are located at the vertices of the tetrahedron, and are connected to the carbon atom by covalent chemical bonds. The carbon atom is located in the geometric center of the tetrahedron.
The tetrahedron shape that is easy to manufacture is also used in engineering. For example, the tetrahedral shape is used in the manufacture of anchors for ships. Note that the NASA space probe, Mars Pathfinder, which landed on the surface of Mars on July 4, 1997, also had the shape of a tetrahedron.
Prism figure
This geometric figure can be obtained by taking two polyhedra, arranging them parallel to each other in different planes of space, and connecting their vertices in an appropriate manner to each other. The result is a prism, two polyhedra are called its bases, and the surfaces connecting these polyhedra will have the form of parallelograms. A prism is called a straight line if its sides (parallelograms) are rectangles.
Prism is a polyhedron, therefore Euler's theorem is true for it . For example, if there is a hexagon at the base of the prism, then the number of sides of the prism is 8, and the number of vertices is 12. The number of edges will be: P = 8 + 12 - 2 = 18. For a straight prism of height h, at the base of which lies the correct hexagon with side a, volume is: V = a 2 * h * √3 / 4, surface area is: S = 3 * a * (a * √3 + 2 * h).
Balloon figure
Speaking of simple geometric volumetric figures and their names, we should mention the ball. A volumetric body called a ball means a body that is limited by a sphere. In turn, a sphere is a set of points in space equidistant from one point, which is called the center of the sphere.
Since the ball belongs to the class of round bodies, for it there is no concept of sides, edges and vertices. The surface area of the sphere bounding the ball is found by the formula: S = 4 * pi * r 2 , and the volume of the ball can be calculated by the formula: V = 4 * pi * r 3/3, where pi is the number pi (3,14), r is the radius of the sphere (ball).