Prism is a fairly simple geometric volumetric figure. Nevertheless, some schoolchildren encounter problems in determining its basic properties, the cause of which, as a rule, is associated with incorrectly used terminology. In this article, we consider what prisms are, what they are called, and also describe in detail the correct quadrangular prism.
Prism in geometry
The study of volumetric figures is the task of stereometry - an important part of spatial geometry. In stereometry, a prism is understood to mean such a figure, which is formed by the parallel transfer of an arbitrary flat polygon to a certain distance in space. Parallel transfer involves such a movement in which rotation around an axis perpendicular to the plane of the polygon is completely excluded.
As a result of the described method for producing a prism, a figure is formed, bounded by two polygons having the same dimensions, lying in parallel planes, and a certain number of parallelograms. Their number coincides with the number of sides (vertices) of the polygon. Identical polygons are called prism bases, and their surface area is the area of the bases. Parallelograms connecting the two bases form a side surface.
Prism Elements and Euler's Theorem
Since the considered three-dimensional figure is a polyhedron, that is, it is formed by a set of intersecting planes, it is characterized by a certain number of vertices, edges, and faces. All of them are elements of a prism.
In the mid-18th century, Swiss mathematician Leonhard Euler established a connection between the number of basic elements of a polyhedron. This relationship is written in the following simple formula:
Number of edges = number of vertices + number of faces - 2
For any prism, this equality is true. We give an example of its use. Suppose there is a regular quadrangular prism. She is depicted in the figure below.
It can be seen that the number of vertices for it is 8 (4 for each quadrangular base). The number of sides or faces is 6 (2 bases and 4 side rectangles). Then the number of edges for it will be equal to:
Number of ribs = 8 + 6 - 2 = 12
All of them can be counted if we turn to the same figure. Eight ribs lie in the bases, and four ribs are perpendicular to these bases.
Complete prism classification
It is important to understand this classification so as not to get confused in terminology and use the correct formulas to calculate, for example, the surface area or the volume of figures.
For any prism of arbitrary shape, 4 features can be distinguished that will characterize it. We list them:
- By the number of corners of the polygon at the base: triangular, pentagonal, octagonal and so on.
- By type of polygon. It may be right or wrong. For example, a right triangle is wrong, and an equilateral triangle is right.
- By the type of convexity of the polygon. It can be concave or convex. Convex prisms are most commonly found.
- At the corners between the bases and lateral parallelograms. If all these angles are equal to 90 o , then they speak of a direct prism, if not all of them are straight, then such a figure is called oblique.
Of all these points, I would like to dwell on the latter in more detail. A direct prism is also called a rectangular prism. This is due to the fact that for it parallelograms are rectangles in the general case (in some cases they can be squares).
For example, the figure above shows a pentagonal concave rectangular, or straight figure.
Regular quadrangular prism
The base of this prism is a regular quadrangle, i.e. a square. The figure above has already shown what this prism looks like. In addition to the two squares that border it above and below, it also includes 4 rectangles.
We denote the side of the base of the regular quadrangular prism by the letter a, the length of its lateral edge by the letter c. This length is also the height of the figure. Then the surface area of this prism is expressed by the formula:
S = 2 * a 2 + 4 * a * c = 2 * a * (a + 2 * c)
Here, the first term reflects the contribution of the bases to the total area, the second term is the area of the lateral surface.
Given the introduced notation for the lengths of the sides, we write the formula for the volume of the figure in question:
V = a 2 * c
That is, the volume is calculated as the product of the area of the square base and the length of the side rib.
Cube figure
Everyone knows this ideal volumetric figure, but few people thought that it is a regular quadrangular prism, the side of which is equal to the length of the side of the square base, that is, c = a.
For a cube, the formulas for the total surface area and volume will take the form:
S = 6 * a 2
V = a 3
Since the cube is a prism consisting of 6 identical squares, any parallel pair of them can be considered the base.
A cube is a highly symmetrical figure, which in nature is realized in the form of crystal lattices of many metallic materials and ionic crystals. For example, the lattices of gold, silver, copper and sodium chloride are cubic.