Stereometry is a section of geometry that studies figures that do not lie in the same plane. One of the objects of study of stereometry are prisms. In the article we will 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 easy. Imagine that there are two identical n-gons, where n is the number of sides or vertices. Let's place them so that they are parallel to each other. After that, the vertices of one polygon should be connected to the corresponding vertices of another. The formed figure will consist of two n-gonal sides, which are called bases, and n quadrangular sides, which in the general case are parallelograms. The set of parallelograms forms the side surface of the figure.
There is one more 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 will get the original polygon. Both polygons and all parallel segments drawn from their vertices form a prism.
The picture above shows a triangular prism. It is called so because its bases are triangles.
Elements that make up the figure
The definition of a prism was given above, from which it is clear that the main elements of a figure are its faces or sides, limiting all the internal points of the prism from the external space. Any face of the figure under consideration belongs to one of two types:
- side;
- grounds.
There are n side pieces, and they are parallelograms or their particular types (rectangles, squares). In general, 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.
Besides the sides, the figure is characterized by its vertices. They are points where three faces touch at the same time. Moreover, two of the three faces always belong to the side surface, and one - to the base. Thus, in a prism there is no specially selected one vertex, as, for example, in a pyramid, all of them are equal. The number of vertices of the figure is 2n (n pieces for eachreason).
Finally, the third important element of a prism is its edges. These are segments of a certain length, which are formed as a result of the intersection of the sides of the figure. Like faces, edges also have two different types:
- or formed only by the sides;
- or appear at the junction of the parallelogram and the side of the n-gonal base.
The number of edges is thus 3n, and 2n of them are of the second type.
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-coal base;
- on side type.
First, let's turn to the second feature and define a straight and oblique prism. If at least one side is a parallelogram of a general type, then the figure is called oblique or oblique. If all parallelograms are rectangles or squares, then the prism will be straight.
The definition of a straight prism can also be given in a slightly different way: a straight figure is a prism whose side edges and faces are perpendicular to its bases. The figure shows two quadrangular figures. The left is straight, the right is oblique.
Now let's move on 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 under consideration contains a polygon with equalsides and angles and is a straight line, then it is called correct. According to this definition, a regular prism at its base can have an equilateral triangle, a square, a regular pentagon, or a hexagon, and so on. The listed correct figures are shown in the figure.
Linear parameters of prisms
The following parameters are used to describe the sizes of the figures under consideration:
- height;
- base sides;
- side rib lengths;
- 3D diagonals;
- diagonal sides and bases.
For regular prisms, all the named 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-gonal regular figure, there are formulas that allow you to determine all the rest by any two linear parameters.
Shape surface
If we refer to the above definition of a prism, then it will not be difficult to understand what the surface of a figure represents. The surface is the area of all the faces. For a straight prism, it is calculated by the formula:
S=2So + Poh
where So is the area of the base, Po is the perimeter of the n-gon at the base, h is the height (distance between the bases).
The volume of the figure
Along with the surface for practice, it is important to know the volume of the prism. It can be determined by the following formula:
V=Soh
Thisthe expression is true for absolutely any kind of prism, including those that are oblique and formed by irregular polygons.
For regular prisms, the volume is a function of the length of the side of the base and the height of the figure. For the corresponding n-gonal prism, the formula for V has a concrete form.
