The ball is a symmetrical volumetric figure, the properties of which are considered in the school course of geometry in the 8th grade. This article is devoted to the formulas of the radius of the ball, allowing to determine this value, knowing the surface area of the figure or its volume.
What is a ball? Basic formulas
From a mathematical point of view, a ball is a collection of points in space that lie at a certain distance from a fixed point called the center. The surface of this figure is a sphere.
The easiest way to get the ball is as follows: you need to take a circle and rotate it around an axis passing through the diameter. Since the body under consideration is bounded by a spherical surface, the formulas for its area, as well as for the volume that it limits, are also valid for the ball. We write them down:
- Ball surface area: S = 4 * pi * R 2 .
- Ball volume: V = 4/3 * pi * R 3 .
In the above formulas, R is the radius of the ball, pi is the constant called the number Pi, it is approximately 3.1416.
Using these expressions, we can obtain the corresponding formulas for the radius of the ball:
- Through the area S: R = 1 / 2√ (S / pi).
- Through volume V: R = ∛ (3V / 4pi).
Here is another expression that for some reason is not considered in the school course, nevertheless, it allows you to calculate the volume of the ball with an accuracy of 0.03%, without using the number pi. It has the form: V = 67 / 16R 3 . Where: R = ∛ (16V / 67).
Earth Volume Calculations
As you know, our planet is not an ideal ball. Its rotation around its axis over hundreds of millions of years has led to the fact that the Earth is a little "thicker" at equatorial latitudes and a little "thinner" at the poles. The corresponding radii are 6378 and 6357 km. As can be seen from these figures, they differ by a small amount (about 0.3%). Therefore, when considering the Earth in geometry, it is considered an ideal ball with an average radius of 6371 km. We use this value and calculate the volume of our blue planet.
We use the formula of the volume of the ball along the radius. We have: V = 4/3 * 3.1416 * 6371 3 ≈ 1.08 * 10 12 km 3 . Now we calculate V using the formula without pi, we obtain: V = 67/16 * 6371 3 ≈ 1.08 * 10 12 km 3 . That is, we got exactly the same value with an accuracy of 2 decimal places.