Prism is a polyhedron or polyhedron, which is studied in the school course of solid geometry. One of the important properties of this polyhedron is its volume. Let's consider in the article how this value can be calculated, and also give the formulas for the volume of prisms - regular quadrangular and hexagonal.
Prism in stereometry
This figure is understood as a polyhedron, which consists of two identical polygons located in parallel planes, and of several parallelograms. For certain types of prisms, parallelograms can represent rectangular quadrilaterals or squares. Below is an example of a so-called pentagonal prism.
To build a figure as in the figure above, you need to take a pentagon and carry out its parallel transfer to a certain distance in space. Connecting the sides of two pentagons using parallelograms, we get the desired prism.
Every prism consists of faces, vertices and edges. The vertices of the prismunlike the pyramid, are equal, each of them refers to one of the two bases. Faces and edges are of two types: those that belong to the bases and those that belong to the sides.
Prisms are of several types (correct, oblique, convex, straight, concave). Let us consider later in the article by what formula the volume of a prism is calculated, taking into account the shape of the figure.
General expression for determining the volume of a prism
Regardless of what type the figure under study belongs to, whether it is straight or oblique, regular or irregular, there is a universal expression that allows you to determine its volume. The volume of a spatial figure is the area of space that is enclosed between its faces. The general formula for the volume of a prism is:
V=So × h.
Here So represents the area of the base. It should be remembered that we are talking about one basis, and not about two. The h value is the height. The height of the figure under study is understood as the distance between its identical bases. If this distance coincides with the lengths of the side ribs, then one speaks of a straight prism. In a straight figure, all sides are rectangles.
Thus, if a prism is oblique and has an irregular base polygon, then calculating its volume becomes more complicated. If the figure is straight, then the calculation of the volume is reduced only to determining the area of \u200b\u200bthe base So.
Determining the volume of a regular figure
Regular is any prism that is straight and has a polygonal base with sides and angles equal to each other. For example, such regular polygons are a square and an equilateral triangle. At the same time, a rhombus is not a regular figure, since not all of its angles are equal.
The formula for the volume of a regular prism unambiguously follows from the general expression for V, which was written in the previous paragraph of the article. Before proceeding to write the corresponding formula, it is necessary to determine the area of \u200b\u200bthe correct base. Without going into mathematical details, we present the formula for determining the indicated area. It is universal for any regular n-gon and has the following form:
S=n / 4 × ctg (pi / n) × a2.
As you can see from the expression, the area Sn is a function of two parameters. An integer n can take values from 3 to infinity. The value a is the length of the side of the n-gon.
To calculate the volume of a figure, it is only necessary to multiply the area S by the height h or by the length of the side edge b (h=b). As a result, we arrive at the following working formula:
V=n / 4 × ctg (pi / n) × a2 × h.
Note that to determine the volume of a prism of an arbitrary type, you need to know several quantities (lengths of the sides of the base, height, dihedral angles of the figure), but to calculate the value V of a regular prism, we need to know only two linear parameters, for example, a and h.
The volume of a quadrangular regular prism
A quadrangular prism is called a parallelepiped. If all its faces are equal and are squares, then such a figure will be a cube. Every student knows that the volume of a rectangular parallelepiped or cube is determined by multiplying its three different sides (length, height and width). This fact follows from the written general volume expression for a regular figure:
V=n/4 × ctg (pi / n) × a2 × h=4/4 × ctg (pi / 4) × a2× h=a2 × h.
Here the cotangent of 45° is equal to 1. Note that the equality of the height h and the length of the side of the base a automatically leads to the formula for the volume of a cube.
Volume of hexagonal regular prism
Now apply the above theory to determine the volume of a figure with a hexagonal base. To do this, you just need to substitute the value n=6 in the formula:
V=6/4 × ctg (pi / 6) × a2 × h=3 × √3/2 × a2 × h.
The written expression can be obtained independently without using the universal formula for S. To do this, you need to divide the regular hexagon into six equilateral triangles. The side of each of them will be equal to a. The area of one triangle corresponds to:
S3=√3/4 × a2.
By multiplying this value by the number of triangles (6) and by the height, we get the above formula for the volume.
