Prism is a polyhedron or polyhedron that is studied in a school course of stereometry. One of the important properties of this polyhedron is its volume. Let us consider in the article how this quantity can be calculated, and also give the formulas for the volume of prisms - regular quadrangular and hexagonal.
Prism in stereometry
By this figure we mean 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 quadrangles or squares. An example of the so-called pentagonal prism is shown below .
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 with the help of parallelograms, we obtain the desired prism.
Every prism consists of faces, vertices and edges. The vertices of the prism, unlike the pyramid, are equal in rights, each of them refers to one of two bases. Facets and ribs are of two types: those that belong to the bases, and those that belong to the lateral sides.
Prisms are of several types (regular, inclined, convex, straight, concave). Let us further consider in the article by what formula the prism volume is calculated taking into account the shape of the figure.
General expression for determining the volume of a prism
Regardless of what type the studied figure belongs to, whether it is straight or inclined, right or wrong, there is a universal expression that allows you to determine its volume. The volume of a spatial figure is the region of space that is enclosed between its faces. The general formula for the prism volume is as follows:
V = S o × h.
Here, S o represents the area of the base. It should be remembered that we are talking about just one basis, and not two. The value of h is the height. By the height of the studied figure, we understand the distance between its identical bases. If this distance coincides with the lengths of the side ribs, then we speak of a direct prism. In a straight shape, all sides are rectangles.
Thus, if the prism is inclined and has an irregular polygon at the base, then the calculation of its volume is complicated. If the figure is straight, then the calculation of the volume is reduced only to determining the area of the base S o .
Determining the volume of the correct figure
A correct prism is any prism that is straight and has a polygonal base with equal sides and angles. For example, such regular polygons are a square and an equilateral triangle. At the same time, the rhombus is not a regular figure, since not all of its angles are equal.
The formula for the correct prism volume uniquely 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 the correct base. Without going into mathematical details, we give a formula for determining the specified area. It is universal in nature for any regular n-gon and has the following form:
S n = n / 4 × ctg (pi / n) × a 2 .
As can be seen from the expression, the area S n is a function of two parameters. The 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 the figure, it is only necessary to multiply the area S n by the height h or by the length of the side rib b (h = b). As a result, we arrive at the following working formula:
V = n / 4 × ctg (pi / n) × a 2 × h.
Note that in order to determine the volume of a prism of arbitrary type, it is necessary to know several quantities (lengths of the sides of the base, height, dihedral angles of the figure), to calculate the value V of the correct prism, we need to know only two linear parameters, for example, a and h.
Volume of a prism quadrangular correct
A quadrangular prism is called a box. If all her faces are equal and are squares, then such a figure will be a cube. Every student knows that the volume of a rectangular box or cube is determined by multiplying its three different sides (length and width). This fact follows from the recorded general expression of volume for the correct figure:
V = n / 4 × ctg (pi / n) × a 2 × h = 4/4 × ctg (pi / 4) × a 2 × h = a 2 × h.
Here the cotangent of 45 ° is 1. Note that the equality of the height h and the length of the base side a automatically leads to the formula for the volume of the cube.
Hexagonal correct prism volume
Now we apply the above theory to determine the volume of a figure with a hexagonal base. To do this, just substitute the value n = 6 in the formula:
V = 6/4 × ctg (pi / 6) × a 2 × h = 3 × √3 / 2 × a 2 × h.
The written expression can be obtained independently without using the universal formula for S n . To do this, 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:
S 3 = √3 / 4 × a 2 .
Multiplying this value by the number of triangles (6) and by height, we get the above formula for the volume.