Everyone has heard of such voluminous figures as a ball, cylinder, pyramid and prism. The last of them is a perfect polyhedron, whose properties are considered in the high school geometry course. In this article, we will reveal the question of how to find the volume of a direct prism at the base of which lies an arbitrary polygon.
The concept of a direct prism
Before determining the volume of any spatial figure, it is important to clearly understand what to deal with. In our case, we consider a straight, or rectangular prism. It is a figure consisting of two bases and several lateral faces that are perpendicular to these bases. Both bases are represented by the same polygon and are parallel to each other. If the number of faces of the polygon tends to infinity, then such a direct prism passes into the cylinder.
The question of the perpendicularity of the side faces and bases of the figure is of fundamental importance. As will be shown below, the final formula for calculating the volume of a straight or rectangular prism depends precisely on this property. It is called rectangular because its lateral faces are rectangles.
The opposite of a rectangular prism is oblique. As you might guess, the difference between them is only in the angle between the side faces and the bases. In an oblique or inclined prism, this angle is different from 90 o .
The figure above shows two quadrangular prisms, one of which is straight (left), and the second is oblique.
What are rectangular prisms ?
The difference between all direct prisms is the type of polygon that forms the base of the shapes. The polygon with the least number of sides is the triangle. The prism that is built using it will be called triangular. Similarly, a figure with quadrangular bases will be called a quadrangular prism. Arguing in this way, we can say that the n-angular direct prism is formed by two n-gons and n rectangles.
In the previous paragraph, when considering the names of prisms, arbitrary triangles, quadrangles, and n-gons were used. If a regular polygon is considered, then the prism formed by it will also be called regular. For example, for n = 3, the regular polygon is an equilateral triangle, and for n = 4, it is a square.
The figure above shows 6 regular rectangular prisms. Each previous one differs from the next by the presence in the bases of a polygon, which has 1 vertex and 1 side more. The first prism is a regular triangular, and the last is a regular octagonal.
Direct prism volume calculation
Having a clear idea of a rectangular prism, we can proceed to calculate its volume. How to find the volume of a direct prism? Very simple, just determine the area of one of its bases and the height of the figure, and then multiply one value by another. The mathematical formula for the volume is:
V = S o * h
The height of the figure h for a direct prism is nothing but the length of its side edge (the vertical side of the rectangle). This fact greatly simplifies the calculation, because otherwise (when the prism is oblique), to calculate the height h, we would have to build a perpendicular from one base to another and, taking into account the angle of inclination of the prism, calculate the length of this perpendicular.
As for the base area S o , to determine it should use the geometric properties of the corresponding polygon. For example, if it is a parallelogram, then its area is calculated as the product of the base and the height lowered to it. The value of S o is easiest to calculate if the polygon is regular.
Formula for the volume of the correct prism
Suppose that the base of the volumetric figure under consideration is formed by an n-gon. From the general course of geometry it is known that the area of such an n-gon is calculated by the following universal formula:
S n = n / 4 * a 2 * ctg (pi / n)
Where the symbol “a” indicates the length of the side of the n-gon.
The volume of a direct prism, the base of which is a regular polygon with the number of vertices n, is calculated using the expression:
V n = S n * h = n / 4 * ctg (pi / n) * a 2 * h
The cotangent function for each case can be calculated either using the corresponding tables of trigonometric functions or using a calculator.
Further in the article, we will consider several examples of using the above formulas to calculate the volume of a prism.
Task number 1. Volume of a regular heptagonal prism
Let us calculate the volume of the prism rarely used in problems in geometry, the bases of which are regular heptagons. This figure is shown in the figure above. We write for it the corresponding formula:
V 7 = 7/4 * ctg (pi / 7) * a 2 * h
To determine the cotangent of the angle pi / 7, we use a calculator. Then we get:
V 7 = 7/4 * ctg (pi / 7) * a 2 * h = 3.634 * a 2 * h
For example, if a = 10 cm, h = 7 cm, then we get the volume:
V 7 = 3.634 * 10 2 * 7 = 2543.8 cm 3
A container made from this prism is able to accommodate a little more than 2.5 liters of water.
Task number 2. Triangular Prism Volume
Suppose that it is necessary to find the volume of a direct prism with a right-angled triangle at the base. Let the legs of this triangle be equal to a and b, and the height of the figure is h. An example of this figure made of glass is shown in the photo above.
Since the base of the prism is not a regular triangle, it is impossible to use a universal formula. However, you can use the property of a right-angled triangle, for which the area is calculated as half the product of its legs. Given this fact, we obtain the expression:
V = S o * h = 1/2 * a * b * h
If the legs are 5 cm and 9 cm and the height is 8 cm, then the calculated volume will be 180 cm 3 .