Before starting to study the concept of a ball, what is the volume of a ball, consider the formulas for calculating its parameters, we need to recall the concept of a circle studied earlier in the geometry course. After all, most of the actions in three-dimensional space are similar or stem from two-dimensional geometry, corrected for the appearance of the third coordinate and the third degree.
What is a circle?
A circle is a figure on a Cartesian plane (shown in Figure 1); most often, the definition sounds like “the geometric locus of all points on a plane, the distance from which to a given point (center) does not exceed a certain non-negative number, called a radius”.
As we see from the figure, the point O is the center of the figure, and the set of absolutely all points that fill the circle, for example, A, B, C, K, E, are not further than a specified radius (do not go beyond the circumference shown in Figure 2).
If the radius is zero, then the circle turns into a point.
Pupils often confuse these concepts. Easy to remember by drawing an analogy. The hoop that children twist in physical education classes is a circle. Understanding this or remembering that the first letters of both words are “O”, the children will mnemonically understand the difference.
Introduction of the concept of "ball"
A ball is a body (Fig. 3), bounded by a certain spherical surface. What a “spherical surface” will become clear from its definition: this is the geometric location of all points on the surface, the distance from which to a given point (center) does not exceed a certain non-negative number, called the radius. As you can see, the concepts of a circle and a spherical surface are similar, only the spaces in which they are located differ. If we draw a ball in two-dimensional space, we get a circle, the boundary of which is a circle (at the ball the boundary is a spherical surface). In the figure we see a spherical surface with radii OA = OB.
Ball closed and open
Concepts you need to know for the following calculations
- Radius and diameter.
- The radius of the ball and its diameter are determined in the same way as a circle.
- Radius - a segment connecting any point on the boundary of the ball and a point that is the center of the ball.
- Diameter - a segment connecting two points on the border of the ball and passing through its center. Figure 5a clearly demonstrates which segments are the radii of the ball, and Figure 5b shows the diameters of the sphere (the segments passing through the point O).
Sections in sphere (ball)
Any section of a sphere is a circle. If it passes through the center of the ball, it is called a large circle (a circle with a diameter AB), the remaining sections are small circles (a circle with a diameter DC).
The area of these circles is calculated by the following formulas:
Here S is the area designation, R is the radius, D is the diameter. Also present is a constant of 3.14. But do not confuse that for calculating the area of a large circle using the radius or diameter of the ball (sphere), and to determine the area, the dimensions of the radius are required just for the small circle.
There are countless such sections that pass through two points of the same diameter that lie on the boundary of the ball. As an example, our planet: two points at the North and South Poles, which are the ends of the earth's axis, and in a geometrical sense - the ends of the diameter, and the meridians that pass through these two points (Figure 7). That is, the number of large circles in a sphere tends to infinity in number.
If you cut off a sphere from a sphere with the help of a certain “slice” plane (Figure 8), then it will be called a spherical or spherical segment. It will have a height - perpendicular from the center of the cutting plane to the spherical surface O1 K. The point K on the spherical surface to which the height comes is called the vertex of the spherical segment. A small circle with a radius of O1 T (in this case, according to the figure, the plane did not pass through the center of the sphere, but if the section passes through the center, then the section circle will be large) formed when the ball segment is cut off, will be called the base of our piece of a sphere - a spherical segment.
If we connect each point of the base of the spherical segment with the center of the sphere, we will get a shape called the “spherical sector”.
If two planes that are parallel to each other pass through the sphere, then that part of the sphere that is enclosed between them is called a spherical layer (Figure 9, which shows a sphere with two planes and a spherical layer separately).
The surface (the highlighted part in Figure 9 on the right) of this part of the sphere is called the belt (again, for a better understanding, you can draw an analogy with the globe, namely with its climatic zones - arctic, tropical, moderate, etc.), and the section circles will be the bases ball layer. The height of the layer is the part of the diameter that is perpendicular to the cutting planes from the centers of the bases. There is also the concept of a sphere. It is formed in the case when the planes, which are parallel to each other, do not intersect the sphere, but touch it at one point each.
Formula for calculating the volume of the ball and its surface area
A ball is formed by rotating around a fixed diameter of a semicircle or circle. To calculate the various parameters of this object will not need too much data.
The volume of the ball, the formula for calculating which is indicated above, is derived by means of integration. We will understand the points.
We consider a circle in a two-dimensional plane, because, as mentioned above, it is the circle that underlies the construction of the ball. We use only its fourth part (Figure 10).
Take a circle with a unit radius and center at the origin. The equation of such a circle is as follows: X 2 + Y 2 = R 2. From here we express Y: Y 2 = R 2 - X 2.
Be sure to note that the resulting function is non-negative, continuous and decreasing on the segment X (0; R), because the value of X in the case when we consider a quarter of a circle, lies from zero to the value of the radius, that is, to one.
The next thing we do is rotate our quarter of a circle around the x-axis. As a result, we get a hemisphere. To determine its volume, we resort to the methods of integration.
Since this volume is only a hemisphere, we double the result, whence we get that the volume of the ball is equal to:
If it is necessary to calculate the volume of the ball through its diameter, remember that the radius is half the diameter, and substitute this value in the above formula.
Also, the formula for the volume of the ball can be reached through the area of its bordering surface - the sphere. Recall that the area of the sphere is calculated by the formula S = 4πr 2. Integrating that, we also arrive at the above formula for the volume of the ball. From these same formulas, one can express the radius if the condition of the problem contains a value of volume.