Stereometry is a section of geometry that studies shapes that do not lie on the same plane. One of the objects of studying stereometry is prism. In the article, we give a definition of a prism from a geometric point of view, and also briefly list the properties that are characteristic of it.
Geometric figure
The definition of a prism in geometry is as follows: it is a spatial figure consisting of two identical n-gons located in parallel planes connected to each other by their vertices.
Getting a prism is not difficult. Imagine that there are two identical n-gons, where n is the number of sides or vertices. We place them so that they are parallel to each other. After this, the vertices of one polygon should be connected with the corresponding vertices of another. The formed figure will consist of two n-angular sides, which are called bases, and n quadrangular sides, which are generally parallelograms. The set of parallelograms forms the side surface of the figure.
There is another way to geometrically obtain the figure in question. So, if we take an n-gon and transfer it to another plane using parallel segments of equal length, then in the new plane we get the original polygon. Both polygons and all parallel segments drawn from their vertices form a prism.
The figure above shows a triangular prism. So it is called because its bases are triangles.
Elements of which the figure consists
The prism definition was given above, from which it is clear that the main elements of the figure are its faces or sides, which limit all the internal points of the prism from external space. Any face of the figure in question belongs to one of two types:
Lateral n pieces, and they are parallelograms or their private types (rectangles, squares). In the general case, the side faces differ from each other. There are only two faces of the base; they are n-gons and are equal to each other. Thus, every prism has n + 2 sides.
In addition to the sides, the figure is characterized by its peaks. They represent the points where three faces are in contact at the same time. Moreover, two of the three faces always belong to the side surface, and one to the base. Thus, in the prism there is no specially selected single vertex, as, for example, in the pyramid, they are all equal. The number of vertices of the figure is 2 * n (n pieces for each base).
Finally, the third important element of the prism is its edges. These are segments of a certain length that are formed as a result of the intersection of the sides of the figure. Like faces, edges also have two different types:
- either formed only by the sides;
- or arise at the junction of the parallelogram and the side of the n-carbon base.
The number of edges, therefore, is 3 * n, and 2 * n of them belong to the second of these types.
Prism Types
There are several ways to classify prisms. However, they are all based on two features of the figure:
- on the type of n-carbon base;
- on the side type.
To begin with, we turn to the second feature and give a definition of a prism inclined and straight. If at least one side is a parallelogram of a general type, then the figure is called oblique, or oblique. If all the parallelograms are rectangles or squares, then the prism will be straight.
The prism of a straight line can also be defined in a slightly different way: a straight figure is that prism in which the side edges and faces are perpendicular to its bases. The figure shows two quadrangular figures. The left one is straight, the right one is sloping.
Now we pass to the classification according to the type of n-gon lying in the bases. It can have the same sides and angles or different. In the first case, the polygon is called regular. If the figure in question contains a polygon with equal sides and angles at the base and is a straight line, then it is called regular. According to this definition, a regular prism at the base can have an equilateral triangle, square, regular pentagon or hexagon, and so on. The listed correct figures are shown in the figure.
Prism linear parameters
The following parameters are used to describe the sizes of the figures in question:
- height;
- sides of the foundation;
- lengths of side ribs;
- volume diagonals;
- diagonals of the sides and bases.
For the correct prisms, all of these quantities are related to each other. For example, the lengths of the side ribs are the same and equal to the height. For a specific n-coal correct figure, there are formulas that allow one to determine all the others by any two linear parameters.
Shape surface
If we turn to the definition of prism given above, then it will be easy to understand what the surface of the figure represents. A surface is the area of all faces. For a direct prism, it is calculated by the formula:
S = 2 * S o + P o * h
where S o is the area of the base, P o is the perimeter of the n-gon in the base, h is the height (distance between the bases).
Figure volume
Along with the surface, it is important for the practitioner to know the volume of the prism. You can determine it by the following formula:
V = S o * h
This expression is valid for absolutely any kind of prism, including those that are inclined and formed by irregular polygons.
For correct prisms, volume is a function of the length of the side of the base and the height of the figure. For the corresponding n-angle prism, the formula for V has a specific form.