Geometric figures in space are an object of study of stereometry, the course of which students in high school take. This article is devoted to such a perfect polyhedron as a prism. Let us consider in more detail the properties of the prism and give formulas that serve for their quantitative description.
What is a prism?
Each represents what a box or cube looks like. Both figures are prisms. However, the prism class is much more diverse. The following definition is given in the geometry of this figure : a prism is any polyhedron in space that is formed by two parallel and identical polygonal sides and several parallelograms. The identical parallel faces of a figure are called its bases (upper and lower). Parallelograms, on the other hand, are the lateral faces of a 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, 2 * n vertices and 3 * n edges. The faces and edges are one of two types: either they belong to the side surface, or the bases. As for the peaks, they are all equal and relate to the bases of the prism.
Types of figures of the studied class
Studying the properties of a prism, you should list the possible types of this figure:
- Convex and concave. The difference between them is in the form of a polygonal base. If it is concave, then a volumetric figure will also be such, and vice versa.
- Direct and inclined. At a direct prism, the side faces are represented by either rectangles or squares. In an inclined figure, the side faces are parallelograms of a general type or rhombs.
- Wrong and right. For the figure to be studied to be correct, it must be straight and have the right foundation. An example of the latter is flat figures such as an equilateral triangle or square.
The name of the prism is formed taking into account the above classification. For example, the above parallelepiped with right angles or a cube are 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. This value is easiest to imagine if you sweep the figure, that is, decompose all the faces on one plane. The figure below shows an example of a scan of two prisms.
For an arbitrary prism, the formula for its sweep area in general form can be written as follows:
S = 2 * S o + b * P sr .
Let us explain the notation. The value of S o is the area of one base, b is the length of the side rib, P sr is the perimeter of the slice, which is perpendicular to the side parallelograms of the figure.
The recorded formula is often used to determine the areas of inclined prisms. In the case of a correct prism, the expression for S will take on a specific form:
S = n / 2 * a 2 * ctg (pi / n) + n * b * a.
The first term in the expression represents the area of the two bases of the regular prism, the second term is the area of the side rectangles. Here a is the side length of a regular n-gon. Note that the length of the side edge b for a regular prism is also its height h, therefore, in the formula b can be replaced by h.
How to calculate the volume of a figure?
Prism is a comparative simple polyhedron with high symmetry. Therefore, to determine its volume, there is a very simple formula. It has the following form:
V = S o * h.
Calculating the base area and height can be difficult if an oblique irregular figure is considered. This problem is solved using sequential geometric analysis using information on dihedral angles between lateral parallelograms and the base.
If the prism is correct, then the formula for V takes on a very specific form:
V = n / 4 * a 2 * ctg (pi / n) * h.
As can be seen, the area S and volume V for the correct prism are uniquely determined if its two linear parameters are known.
The prism is triangular correct
We conclude the article by considering the properties of the triangular prism correct. It is formed by five faces, three of which are rectangles (squares), and two - equilateral triangles. Prism has six vertices and nine ribs. For this prism, the formulas for volume and surface area are written below:
S 3 = √3 / 2 * a 2 + 3 * h * a
V 3 = √3 / 4 * a 2 * h.
In addition to these properties, it is also useful to give a formula for the apothem of the base of the figure, which represents the height h a of an equilateral triangle:
h a = √3 / 2 * a.
The sides of the prism are the same rectangles. The lengths of their diagonals d are equal to:
d = √ (a 2 + h 2 ).
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, the light decomposes into a number of constituent colors as a result of the dispersion phenomenon, which creates the conditions for studying the spectral composition of the electromagnetic flux.