Studying the properties of polyhedra is an important part of the school course of stereometry. In this article, we will consider the two most famous types of these figures: a pyramid and a prism, and show how to find the volume of a polyhedron, and what formulas should be used.
What is a polyhedron?
Before giving an answer to the question of how to find the volume of a polyhedron, it is necessary to clearly understand what is being discussed. The definition of a polyhedron is quite simple: they put a figure under it in three-dimensional space, which is limited by several polygonal faces. The edges must be flat. Polyhedra in foreign literature are often called polyhedra.
Since the studied class of volumetric figures consists of polygons, it always has vertices and edges that are formed by the intersection of three or two faces, respectively.
In this article, we will study in detail two polyhedra, which are most often found in geometry problems and in the form of which many household objects are made. It will be about prisms and pyramids.
Prism shape and determination of its volume
A prism is understood to mean a polyhedron consisting of two identical and parallel n-gons, the corresponding vertices of which are interconnected. The lateral surface of such a geometric structure is formed by parallelograms.
There are many different types of prisms, for example, straight and inclined, concave and convex, triangular and decagonal. Nevertheless, the answer to the question of how to find the volume of a polyhedron-prism lies in a very specific formula. Here is her:
V = S o * h
Here, the symbol S o reflects the area of the base. Since any prism has two of them, and they are both equal, one can choose any of them to calculate S o . The easiest way is to calculate the area of simple or regular polygons. For example, for a triangle of arbitrary form, it is enough to multiply half the side by the height lowered on it to get its area. If the polygon is correct (sides and angles are equal to each other), then its area S n will be equal to:
S n = n / 4 * ctg (pi / n) * a 2
Where n is the number of angles or sides of the polygon, a is the length of its side.
The height of a prism is the distance between its bases. For a direct or correct prism, it is not difficult to calculate this value, since it is equal to the side edge. If the polyhedron under study is irregular and inclined, then the calculation of the height is complicated. To conduct it, in the general case, it is necessary to know the angular parameters of the figure.
The correct prism is the easiest version of the figure of the class in question in terms of volume calculation. The volume of a regular polyhedron is calculated by the formula:
V = n / 4 * ctg (pi / n) * a 2 * h
Here h can be replaced by the length b of the side rib.
Pyramid figure and calculation of its volume
The pyramid is not only the great construction of the pharaoh Cheops, but also a very specific geometric object. This polyhedron consists of one n-coal base and n triangles. Using triangles, the sides of the base are connected to a single point in space, which is the top of the pyramid.
Like prisms, the class of pyramids includes figures of different types. So, there are oblique and straight pyramids, right and wrong, convex and concave. Nevertheless, all this diversity can be described by a single formula of a general form for their volume.
What is the volume of a polyhedron equal to when it comes to an arbitrary pyramid? The answer to this question will be the following expression:
V = 1/3 * S o * h
This formula is also simple, as for a prism. We see that the volume of the pyramid is three times less than that for the prism, ceteris paribus (the same S o and h).
To calculate the base area S o should adhere to the method described in the previous paragraph. Regarding the height h, we note that in order to calculate it in practice, one often has to resort to the use of trigonometric functions and the Pythagorean theorem.
When considering the correct pyramid, its volume can be calculated by the following formula:
V = n / 12 * ctg (pi / n) * a 2 * h
Hexagonal Prism Challenge
The correct hexagonal prism is specified . The volume of this polyhedron must be calculated if it is known that the length of the side of its hexagon is 6 cm, and the side edge is two times larger.
Since the prism is correct, you can immediately use the formula written above, we have:
V = 6/4 * ctg (pi / 6) * a 2 * 2 * a = 3 * √3 * a 3
When writing this expression, we used the equalities n = 6 and h = 2 * a. Substituting the value a = 6 cm, we get the answer: V = 1122.37 cm 3 .
Quadrangular Pyramid Challenge
Having considered the question of how to find the volume of a polyhedron, we will now solve a somewhat more complicated problem. It is necessary to determine the volume of a regular quadrangular pyramid if the diagonal of its base is 13 cm and the side edge is 20 cm.
The strategy for solving the problem is to calculate the lengths a and h, and then substitute them into the well-known formula. Since the base of the figure is a square, the side a of the diagonal d is defined as follows:
a = d / √2 = 13 / √2 = 9.19 cm
To determine the height, it is necessary to consider a right triangle whose sides are equal to h, d / 2 and b. According to the Pythagorean theorem, we get:
h = √ (b 2 - d 2/4) = √ (20 2 -13 2/4) = 18.91 cm
Now you can use the formula for V:
V = 4/12 * ctg (pi / 4) * a 2 * h = 1/3 * 9.19 2 * 18.91 = 532.35 cm 3
Thus, the volume of the considered pyramid is 532.35 cm 3 .