In geometry, when studying a particular class of figures, a number of formulas are given that allow you to quantitatively characterize the object in question from different angles. In this article, the prism figure will be considered, and the surface area and volume formulas for it will be given.
What is understood by a prism in stereometry?
When talking about a prism, we mean a three-dimensional polyhedron, which is formed by two identical flat n-gons and n parallelograms. Both n-gons must be in parallel planes, they are called the sides of the base of the figure. As for parallelograms, they form a lateral surface and can be rhombuses, rectangles and squares.
Like any polyhedron, a prism consists of vertices, edges and faces. We have already mentioned the facets. The vertices of the prism, unlike the vertices of the pyramid, are equal in rights and belong to the bases. The edges of the prism are of two types: belonging to the bases and related to the side faces.
When forming the name of a particular prism, four types of classification are included in it. These types are easier to understand if you give a couple of examples.
The figure above shows the hexagonal convex regular straight shape. It is called hexagonal and convex because its base is a convex hexagon. The correct and straight figure is because the hexagon itself has equal sides and angles, and the side surface is represented by the same rectangles.
This figure depicts a pentagonal convex irregular oblique prism. We invite the reader to independently understand why she was given such a name.
Prism linear and angular parameters
For an arbitrary prism, an unambiguous description of its characteristics requires knowledge of linear parameters and dihedral angles between its faces. The main linear parameters are the lengths of the sides a i of the base, the length of the side rib b and the height h. Note that the sides of the base can have different lengths, while the lateral ribs are equal to each other even in an inclined figure. For a straight and regular prism, the length of the side edge b coincides with the height h (b = h). To calculate the height of the inclined prism, it is necessary to know the values of the angles of dihedral between the base planes and side parallelograms.
Knowing the listed linear parameters allows you to calculate the surface area of the figure and its volume.
Shape surface
One of the important formulas of a prism is the expression that allows you to calculate its area. To determine the area of all faces of a figure, it is convenient to expand it on a plane. An example of one of the scans is given below.
For an arbitrary prism, the surface area formula is defined as follows:
S = 2 * S o + S b = 2 * S o + b * P sr
Where S o - the area of one base, P sr - the perimeter perpendicular to all side faces of the slice.
In the case of a correct prism, the formula for S takes the form:
S = n / 2 * ctg (pi / n) * a 2 + n * a * h
Here n is the number of vertices (sides) of the polygon at the base. The first term here reflects two base areas, the second term is the sum of the areas of all the lateral rectangles of the prism.
For example, the formula of a triangular correct prism will take the form:
S 3 = √3 / 2 * a 2 + 3 * a * h
Note that to determine the area of any regular figure, it is enough to know any two of its parameters. In this case, these parameters are the side of the base a and the height h.
Volume Formulas
By the volume of a polyhedron we understand the part of the space that is enclosed between its faces. To calculate the volume of an arbitrary type of prism, it is necessary to use the same formula as for the volume of the cylinder. It has the following form:
V = S o * h
Despite the simplicity of this expression, the calculation can be complicated by the fact that first you need to calculate the height and area of the base. For an inclined prism with an irregular convex or concave base, this problem is not trivial and does not have a general solution. In this case, you should use the general approach: knowing the dihedral angle at the base and one of the diagonals of the base or side edge, you can calculate the height of the figure; the area of the polygonal base consists of the areas of elementary figures, the formulas for which are known.
If the prism under study is correct, then a specific expression can be written for it. The formula for the correct prism volume takes the form:
V = n / 4 * ctg (pi / n) * a 2 * h
For example, for a triangular figure we get:
V 3 = √3 / 4 * a 2 * h
As for area S, to determine the volume V of the correct figure, you need to know the height and side of the base.