The sphere is one of the first bodies with high symmetry, the properties of which are studied in the school course of geometry. This article discusses the formula of the sphere, its difference from the ball, and also provides a calculation of the surface area of our planet.
Field: concept in geometry
To better understand the surface formula, which will be given below, you need to get acquainted with the concept of a sphere. In geometry, it is a three-dimensional body that encloses a certain amount of space. The mathematical definition of a sphere is as follows: it is a collection of points that lie at a certain equal distance from one fixed point, called the center. The marked distance is the radius of the sphere, which is denoted by r or R and is measured in meters (kilometers, centimeters and other length units).
The figure below shows the described figure. The lines show the contours of its surface. The black dot is the center of the sphere.
You can get this figure if you take a circle and begin to rotate it around any of the axes passing through the diameter.
Sphere and ball: what is the difference and what are the similarities?
Often, schoolchildren confuse these two figures, which look similar to each other, but have completely different physical properties. A sphere and a ball are primarily distinguished by their mass: a sphere is an infinitely thin layer, while a sphere is a volumetric body of finite density, which is the same at all its points bounded by a spherical surface. That is, the ball has a finite mass and is a very real object. A sphere is an ideal figure without mass, which does not really exist, but it is a successful idealization in geometry when studying its properties.
Examples of real objects, the shape of which almost matches the sphere, are a Christmas toy in the form of a ball for decorating a Christmas tree or a soap bubble.
As for the similarities between the figures in question, we can name the following features:
- both of them have the same symmetry;
- for both, the surface area formula is the same, moreover, they have the same surface area if their radii are equal;
- both figures with equal radii occupy the same volume in space, only the ball fills it completely, and the sphere only limits its surface.
A sphere and a ball of equal radius are shown in the figure below.
Note that a ball, like a sphere, is a body of revolution, so it can be obtained by rotating a circle around the diameter (not a circle!).
Sphere Elements
These are called geometric quantities, the knowledge of which allows us to describe either the entire figure or its individual parts. Its main elements are the following:
- The radius r, which was already mentioned earlier. It is the distance from the center of the figure to the spherical surface. In fact, this is the only value that describes all the properties of the sphere.
- Diameter d, or D. This is a segment whose ends lie on a spherical surface, and the middle passes through the center point of the figure. The diameter of the sphere can be carried out in an infinite number of ways, but all the segments obtained will have the same length, which is equal to twice the radius, that is, D = 2 * R.
- The surface area S is a two-dimensional characteristic, the formula for which will be given below.
- Three-dimensional angles associated with the sphere are measured in steradians. One steradian is an angle whose apex lies in the center of the sphere, and which rests on a part of a spherical surface having an area of R 2 .
Geometric properties of a sphere
From the above description of this figure, you can independently guess about these properties. They are as follows:
- Any line that intersects a sphere and passes through its center is the axis of symmetry of the figure. Rotating a sphere around this axis to any angle takes it into itself.
- The plane that intersects the figure in question through its center divides the sphere into two equal parts, that is, it is a reflection plane.
The surface area of the figure
This value is indicated by the Latin letter S. The formula for calculating the area of a sphere has the following form:
S = 4 * pi * R 2 , where pi ≈ 3.1416.
The formula demonstrates that the area S can be calculated provided that the radius of the figure is known. If its diameter D is known, then the formula of the sphere can be written as follows:
S = pi * D 2 .
The irrational number pi, for which four decimal places are given, can be used in a number of mathematical calculations to the nearest hundredth, that is, 3.14.
It is also interesting to consider how many steradians correspond to the entire surface of the figure in question. Based on the definition of this quantity, we obtain:
Ω = S / R 2 = 4 * pi * R 2 / R 2 = 4 * pi steradian.
To calculate any volume angle, substitute the corresponding value of area S in the expression above.
Surface of planet earth
The sphere formula can be used to determine the surface area of the planet on which we live. Before proceeding with the calculations, a couple of caveats should be made:
- Firstly, the Earth does not have an ideal spherical surface. Its equatorial and polar radii are 6378 km and 6357 km, respectively. The difference between these figures does not exceed 0.3%, so for the calculation you can take the average radius of 6371 km.
- Secondly, the relief of the Earth’s surface is three-dimensional, that is, there are hollows and mountains on it. These characteristic features of the planet lead to an increase in its surface area, however, we will not take them into account in the calculation, since even the largest mountain, Everest, is 0.1% of the Earth's radius (8.848 / 6371).
Using the formula of the sphere, we get:
S = 4 * pi * R 2 = 4 * 3.1416 * 6371 2 ≈ 510.066 million km 2 .
According to official figures, Russia occupies an area of 17.125 million km 2 , which is 3.36% of the planet’s surface. If we consider that only 150.387 million km 2 belong to land, then the area of our country will be 11.4% of the entire territory not covered by water.