Many of the bodies that we meet in life or have heard of are spherical in shape, such as a soccer ball, a falling drop of water during rain, or our planet. In this regard, it is relevant to consider how to find the volume of a ball.
Ball shape in geometry
Before answering the question of how to find the volume of a ball, let us consider this body in more detail. Some people confuse it with a sphere. Outwardly, they are really similar, but the ball is an object filled inside, the sphere is only the outer shell of the ball of infinitely small thickness.
From the point of view of geometry, a ball can be represented as a set of points, and those that lie on its surface (they form a sphere) are at the same distance from the center of the figure. This distance is called the radius. In fact, the radius is the only parameter with which you can describe any properties of the ball, such as its surface area or volume.
The figure below is an example of a ball.
If you carefully look at this perfect round object, you can guess how to get it from an ordinary circle. To do this, it is enough to rotate this flat figure around an axis coinciding with its diameter.
One of the famous ancient literary sources, in which the properties of this three-dimensional figure are examined in sufficient detail, is the work of the Greek philosopher Euclid - Elements.
Surface area and volume
Considering the question of how to find the volume of a ball, in addition to this quantity, a formula should be given for its area, since both expressions can be related to each other, as will be shown below.
So, to calculate the volume of the ball, you should apply one of the following two formulas:
- V = 4/3 * pi * R3;
- V = 67/16 * R3.
Here R is the radius of the figure. The first of the above formulas is accurate, however, to take advantage of this, you must use the appropriate number of decimal places for the number pi. The second expression gives a very good result, differing from the first by only 0.03%. For a number of practical tasks, this accuracy is more than enough.
The surface area of ββthe ball is equal to this value for the sphere, that is, it is expressed by the formula S = 4 * pi * R2. If we express the radius from here and then substitute it in the first formula for the volume, then we get: R = β (S / (4 * pi)) => V = S / 3 * β (S / (4 * pi)).
Thus, we examined the issues of how to find the volume of a ball through a radius and through its surface area. These expressions can be successfully applied in practice. Further in the article we give an example of their use.
Task with a drop of rain
Water, when in zero gravity, takes the form of a spherical drop. This is due to the presence of surface tension forces that tend to minimize surface area. The ball, in turn, has its smallest value among all geometric figures with the same mass.
During rain, a falling drop of water is in zero gravity, so its shape is a ball (here we neglect the strength of air resistance). It is necessary to determine the volume, surface area and radius of this drop, if it is known that its mass is 0.05 grams.
The volume is easy to determine, for this it is necessary to divide the known mass by the density of H 2 O (Ο = 1 g / cm 3 ). Then V = 0.05 / 1 = 0.05 cm 3 .
Knowing how to find the volume of the ball, it is necessary to express the radius from the formula and substitute the obtained value, we have: R = β (3 * V / (4 * pi)) = β (3 * 0.05 / (4 * 3.1416)) = 0.2285 cm.
Now we substitute the value of the radius in the expression for the surface area of ββthe figure, we obtain: S = 4 * 3.1416 * 0.22852 = 0.6561 cm 2 .
Thus, knowing how to find the volume of the ball, we got answers to all the questions of the problem: R = 2.285 mm, S = 0.6561 cm 2 and V = 0.05 cm 3 .