Description of harmony algebra. Volume of the ball

The world around us, despite the variety of objects and phenomena occurring with them, is full of harmony due to the clear action of the laws of nature. Behind the seeming freedom with which nature draws outlines and creates forms of things, clear rules and laws hide, involuntarily suggesting the presence in the process of creation of some higher power. On the verge of pragmatic science, which gives a description of the phenomena taking place from the standpoint of mathematical formulas and theosophical worldviews, there is a world that gives us a whole bunch of emotions and impressions from the things that fill it and the events that are happening to them.

A ball as a geometric figure is the most common form in nature for physical bodies. Most of the bodies of the macrocosm and the microworld are in the shape of a ball or they tend to approach it. In fact, the ball is an example of an ideal shape. The generally accepted definition for a ball is considered to be the following: it is a geometric body, a set (set) of all points of space that are from the center at a distance not exceeding a given one. In geometry, this distance is called the radius, and applied to this figure, it is called the radius of the ball. In other words, all points located at a distance from the center not exceeding the length of the radius are enclosed in the volume of the ball.

The ball is still viewed as the result of the rotation of the semicircle around its diameter, which at the same time remains stationary. In addition to these elements and characteristics, like the radius and volume of the ball, the ball's axis (fixed diameter) is added, and its ends are called the poles of the ball. The surface of a sphere is usually called a sphere. If we are dealing with a closed sphere, then it includes this sphere, if it is open, then it excludes it.

Considering the definitions associated with the ball, we must say about the intersecting planes. A secant plane passing through the center of a sphere is usually called a large circle. For other planar sections of the sphere, it is customary to use the term "small circles". When calculating the areas of these sections, the formula πR² is used.

Calculating the volume of the ball, mathematicians have encountered quite fascinating laws and features. It turned out that this value either completely repeats, or is very close in the way of determination to the volume of the pyramid or the cylinder described around the ball. It turns out that the volume of the ball is equal to the volume of the pyramid, if its base has the same area as the surface of the ball, and the height is equal to the radius of the ball. If we consider the cylinder described around the ball, then we can calculate the regularity, according to which the volume of the ball is one and a half times smaller than the volume of this cylinder.

An attractive and original way looks like a way to derive a formula for the volume of a ball using the Cavalieri principle. It consists in finding the volume of any figure by adding the areas obtained by its cross section by an infinite number of parallel planes. For the derivation, we take a half-sphere with radius R and a cylinder having a height R with a base-circle of radius R (the bases of the hemisphere and the cylinder are located in one plane). In this cylinder we inscribe a cone with a vertex in the center of its lower base. Having proved that the volume of the hemisphere and the part of the cylinder that are outside the cone are equal, we easily calculate the volume of the ball. Its formula takes the following form: four third products of a cube of radius by π (V = 4 / 3R ^ 3 × π). This can be easily proved by drawing a common cutting plane through a half-ball and a cylinder. The area of the small circle and the ring, bounded from the outside by the sides of the cylinder and cone, are equal. And, using the Cavalieri principle, it is not difficult to come to the proof of the basic formula by means of which we determine the volume of a sphere.

But not only with the problem of studying natural bodies is the finding of ways to determine their various characteristics and properties. Such a figure of stereometry as a ball is very widely used in the practical activities of a person. The mass of technical devices has, in their designs, details not only of a spherical shape, but also composed of ball elements. It is the copying of ideal natural solutions in the process of human activity that gives the most qualitative results.