Before beginning to study the concept of a sphere, what is the volume of a sphere, to consider the formulas of the calculus of its parameters, it is necessary to recall the concept of a circle studied earlier in the course of geometry. After all, most of the actions in three-dimensional space are analogous or follow from two-dimensional geometry with correction for the appearance of the third coordinate and the third degree.
What is a circle?
The circle is a figure on the Cartesian plane (shown in Figure 1); most often the definition sounds like "the geometric locus of all points on the plane, the distance from which to a given point (center) does not exceed a certain nonnegative number, called the 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 beyond the given radius (do not go beyond the circle shown in Fig. 2).
If the radius is zero, then the circle becomes a point.
Problems with understanding
Students often confuse these concepts. It is easy to remember by drawing an analogy. The hoop that children twist in physical education classes is a circle. Realizing this or remembering that the first letters of both words are "O", the children will understand the difference in a mnemonic way.
Introduction of the concept of "ball"
A ball is a body (Figure 3), bounded by a certain spherical surface. What kind of "spherical surface" becomes clear from its definition: this is the locus of all points on the surface, the distance from which to the given point (center) does not exceed a certain nonnegative number, called the radius. As we see, the concepts of a circle and a spherical surface are similar, only the spaces in which they are located are different. If we represent a sphere in two-dimensional space, we get a circle whose boundary is a circle (the sphere is a spherical surface near the ball). In the figure, we see a spherical surface with radii OA = OB.
The ball is closed and open
The 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 the circle.
- Radius - a segment that connects any point on the boundary of the ball and the point that is the center of the ball.
- Diameter - a segment that connects two points on the border of the ball and passes through its center. Figure 5a clearly demonstrates which segments are the radii of the sphere, and Figure 5b shows the diameters of the sphere (segments passing through point O).
Sections in the sphere (sphere)
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 diameter AB), the other sections by small circles (a circle with a diameter of DC).
The area of these circles is calculated by the following formulas:
Here S is the area designation, R is the radius, and D is the diameter. There is also a constant equal to 3.14. But do not be confused that to calculate the area of a large circle using the radius or diameter of the ball itself (sphere), and to determine the area requires the size of the radius is a small circle.
Such sections, which pass through two points of the same diameter, lying on the boundary of the ball, you can spend countless numbers. As an example, our planet: two points on the North and South Poles, which are the ends of the earth's axis, and in geometric terms - the ends of the diameter, and the meridians that pass through these two points (Figure 7). That is, the number of large circles in the sphere by the number tends to infinity.
Parts of the ball
If you cut off a "piece" from a sphere using a plane (Figure 8), it will be called a spherical or spherical segment. It will have a height - a perpendicular from the center of the cutting plane to a spherical surface O1 K. The point K on the spherical surface into which the altitude comes is called the vertex of the spherical segment. A small circle with radius O1 T (in this case, according to the figure, the plane did not pass through the center of the sphere, but if the cross section passes through the center, the cross section will be large) formed when the ball segment is cut off, will be called the base of our piece of the ball-the spherical segment.
If we connect each point of the base of the spherical segment to the center of the sphere, we get a figure called the "ball sector".
If two planes pass through the sphere, which are parallel to each other, then that part of the sphere that is enclosed between them is called the spherical layer (Fig. 9, where a sphere with two planes is represented and a spherical layer separately).
The surface (the selected part in Figure 9 on the right) of this part of the sphere is called the belt (again, for better understanding, we can draw an analogy with the globe, namely with its climatic zones - arctic, tropical, temperate, etc.), and the circle circles will be the bases the spherical layer. The height of the layer is a part of the diameter drawn perpendicular to the intersecting planes from the centers of the bases. There is also the concept of a spherical sphere. It is formed in the case when the planes that are parallel to each other do not intersect the sphere, but touch it at one point each.
Formulas for calculating the volume of a sphere and its surface area
A ball is formed by rotating around a fixed diameter of a semicircle or circle. To calculate various parameters of this object, not much data is needed.
The volume of the ball, the formula for calculating which is indicated above, is derived by integration. Understand the points.
We consider a circle in a two-dimensional plane, because, as was said above, it is the circle that underlies the construction of the sphere. We use only its fourth part (Figure 10).
We take a circle with unit radius and center at the origin. The equation of such a circle is as follows: X 2 + Y 2 = R 2. We express Y: Y 2 = R2 - X 2.
Be sure to note that the function obtained is nonnegative, continuous and decreasing on the segment X (0; R), since the value of X in the case when we consider a quarter of the circle lies from zero to the value of the radius, that is, to unity.
The next thing we do is rotate our quarter of a circle around the abscissa axis. As a result, we get a half-ball. To determine its volume, we resort to the methods of integration.
Since this is only a hemisphere, we increase the result by a factor of two, from which we obtain that the volume of the ball is equal to:
If it is necessary to calculate the volume of a sphere through its diameter, remember that the radius is half the diameter, and substitute this value in the above formula.
Also, to the formula for the volume of a sphere, you can go 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 this, we also arrive at the above formula for the volume of the sphere. From the same formulas it is possible to express the radius if in the condition of the problem there is a value of volume.