Geometric figures in space are the object of study of stereometry, the course of which is passed by schoolchildren in high school. This article is devoted to such a perfect polyhedron as a prism. Let us consider in more detail the properties of a prism and give the formulas that serve to describe them quantitatively.
What is a prism?
Everyone imagines what a box or cube looks like. Both figures are prisms. However, the class of prisms is much more diverse. In geometry, this figure is given the following definition: a prism is any polyhedron in space, which is formed by two parallel and identical polygonal sides and several parallelograms. Identical parallel faces of a figure are called its bases (upper and lower). Parallelograms are the side faces of the figure, connecting the sides of the base with each other.
If the base is represented by an n-gon, where n is an integer, then the figure will consist of 2+n faces, 2n vertices and 3n edges. Faces and edges refer toone of two types: either they belong to the lateral surface, or to the bases. As for the vertices, they are all equal and belong to the bases of the prism.
Types of figures of the class under study
Studying the properties of a prism, you should list the possible types of this figure:
- Convex and concave. The difference between them lies in the shape of the polygonal base. If it is concave, then it will also be a three-dimensional figure, and vice versa.
- Straight and oblique. In a straight prism, the side faces are either rectangles or squares. In an inclined figure, the side faces are parallelograms of a general type or rhombuses.
- Wrong and right. In order for the figure to be studied to be correct, it must be straight and have the correct base. An example of the latter are flat figures such as an equilateral triangle or a square.
The name of the prism is formed taking into account the listed classification. For example, the right-angled parallelepiped or cube mentioned above is called a regular quadrangular prism. Regular prisms, due to their high symmetry, are convenient to study. Their properties are expressed in the form of specific mathematical formulas.
Prism area
When considering such a property of a prism as its area, they mean the total area of all its faces. It is easiest to imagine this value if you unfold the figure, that is, expand all the faces into one plane. Below onThe figure shows an example of a sweep of two prisms.
For an arbitrary prism, the formula for the area of its sweep in general form can be written as follows:
S=2So+ bPsr.
Let's explain the notation. The value So is the area of one base, b is the length of the side edge, Psr is the perimeter of the cut, which is perpendicular to the side parallelograms of the figure.
The written formula is often used to determine the areas of inclined prisms. In the case of a regular prism, the expression for S will take on a specific form:
S=n/2a2ctg(pi/n) + nba.
The first term in the expression represents the area of the two bases of a regular prism, the second term is the area of the side rectangles. Here a is the length of the side of a regular n-gon. Note that the length of the side edge b for a regular prism is also its height h, so in the formula b can be replaced by h.
How to calculate the volume of a shape?
Prism is a relatively simple polyhedron with high symmetry. Therefore, to determine its volume, there is a very simple formula. It looks like this:
V=Soh.
Calculating base area and height can be tricky when looking at an oblique irregular shape. Such a problem is solved using sequential geometric analysis involving information about the dihedral angles between the side parallelograms and the base.
If the prism is correct thenthe formula for V becomes quite concrete:
V=n/4a2ctg(pi/n)h.
As you can see, the area S and volume V for a regular prism are uniquely determined if two of its linear parameters are known.
Triangular regular prism
Let's finish the article by considering the properties of a regular triangular prism. It is formed by five faces, three of which are rectangles (squares), and two are equilateral triangles. A prism has six vertices and nine edges. For this prism, the volume and surface area formulas are written below:
S3=√3/2a2+ 3ha
V3=√3/4a2h.
Besides these properties, it is also useful to give a formula for the apothem of the base of the figure, which is the height ha of an equilateral triangle:
ha=√3/2a.
The sides of the prism are identical rectangles. The lengths of their diagonals d are:
d=√(a2+ h2).
Knowledge of the geometric properties of a triangular prism is of not only theoretical but also practical interest. The fact is that this figure, made of optical glass, is used to study the radiation spectrum of bodies.
Passing through a glass prism, light is decomposed into a number of component colors as a result of the dispersion phenomenon, which creates conditions for studying the spectral composition of an electromagnetic flux.
