What is the surface area of sphere ?
Geometry is a branch of mathematics that deals with exploring everything related to geometry figures, including the surface area of sphere. In order to know how to deal with calculating the surface area of sphere, it is important to have some theoretical knowledge in regard to this aspect and a bit of practice, of course. By a sphere, one should mean a geometrical figure the main characteristic of which is that is it completely and perfectly round. It is, on the one hand, very similar to a circle, which from the point of view of geometry is a figure with two dimensions. On the other hand, it is defined differently in terms of mathematics, as a set of points, all of which are located at the equal distance one from another. While outside mathematics we can often hear that the two terms such as «ball» and «sphere» have the same meaning and can change each other, in mathematics, particularly in spherical geometry, they have different meanings. In this article, you will find the explanation of the term «sphere», get historical background concerning this issue and learn how to calculate the surface area of sphere.
In terms of mathematics and geometry particularly, when talking of the surface area of sphere, one should mean a measure of the overall area of the geometrical figure. Apart from the mathematical definition, there are also others, which you don’t have to mix if you are going to deal with the geometry problems. The definition of the surface area of sphere that is known and commonly accepted today was used for the first time in the beginning of the twentieth century. The definition of the surface area of sphere was a part of geometric measure theory that has developed a lot since that time.
The term surface area of sphere is also used in other sciences such as chemistry and biology. In the first case, this term is important in the aspect of chemical kinetics. As for the biology, the surface area of sphere is important for a number of considerations. However, in biology, the surface area is measured in a different way, not like it is done from the point of view of geometry.
The formula by means of which we can calculate the surface area of sphere was discovered long time ago, more than two thousand years ago. At that time, there was a rise of several sciences in Greece, including geometry. The formula was discovered by an outstanding philosopher, Archemedes, whose achievements had a significant impact on all the other discoveries that were made after that. Apart from the above mentioned formula, the ancient philosopher also concluded that the surface area of sphere is absolutely the same that the area of the curved wall of the smallest cylinder that a particular sphere can contain. This kind of cylinder is called the circumscribed cylinder.
The history of spherical geometry
Spherical geometry is the aspect of classical geometry that deals with a surface with two dimensions of a sphere. This aspect of geometry has two different practical sides, each of which serves as an application of the principles to such areas as astronomy and navigation.
Spherical geometry has come through a number of periods indicating different stages of its development. Since the times when people concluded that the Earth was not flat but spherical, there has began a significant increase in the development of spherical geometry. Even long before America was discovered, scientist in ancient Greece and Rome had numerous ideas concerning the concept of spheres, which they applied with the purpose of exploration the world and surroundings. Despite the commonly accepted belief that the man who discovered the Earth was spherical not flat was Columbus, the truth is that it was not him. In fact, there were Phoenician who manifested the fact of spherical surface of the Earth. This happened more than two thousand of years ago, which means that the history of spherical geometry has also more than two thousand. Since that time, the overall understanding of all issues concerning the Earth, the organization of the world and so on has changed.
Taking into consideration how long the history of spherical geometry exists, it is logical to admit that all the maps that have been created since the beginning of this story were created exclusively due to the understanding of a sphere and its properties.
Spherical geometry as a science about spheres
The key element and object that the spherical geometry is dedicated to is the sphere, which is a surface that has three dimensions. The dimensions of a sphere are made of the set of points that are located at certain distance one from another and from a constant point that is considered to be the center of the figure. Apart from it, there is specific property characterizing the sphere, which is called «the great circles». They are the figures of intersection that are characterized by equal radiuses and have the same centers. These great circles are meant to divide the sphere into two parts of the same size and with the same properties. As a matter of fact, the great circle is considered to be the largest circle. It is located in the very center of the sphere and looks like the equator of the planet Earth.
Spherical geometry as a part of classical geometry that is dedicated to the studies of geometry figures and their properties, is sometimes viewed as a specific kind of planar geometry. These two aspects of classical geometry are very close to each other, even though they are absolutely different. The majority of students who study geometry at high schools or colleges usually face the essentials of planar geometry. The common facts that they deal with are some basic properties of geometry figures: that two parallel lines in a space can never intersect; if to put all the angles of a triangle together we will get the number that is equal to one hundred and eighty degrees; if to draw the shortest line from one point in a space to another we will get a straight line. Having said that, it is important to remember that all these concepts are not always considered when it comes to spherical geometry.
Basic properties that you should know about spherical geometry
- In contrast to what we have said above concerning the concepts of planar geometry, in spherical geometry, parallel lines do not actually exist. As a matter of fact, you will not even face straight lines in this aspect of geometry. For this reason, in spherical geometry the role of a line plays the great circle.
- Since there are no straight lines that can be drawn within a sphere, we can draw a line that would be the shortest distance from one point on the sphere to another that are all located on the surface of the great circle.
- On the surface of the sphere, you can measure the angle that is located between two arcs by means of forming an angle from the lines intersection that lie tangent to the given arcs.
- In case if the three arcs of the sphere intersect each other, you will not get two triangles that would be equal. However, these spherical triangles will be congruent, if only they have one angle and share it together.
- In spherical geometry, “antipodal points” are any two lines that are drawn in such a way that they intersect each other in an opposite way.
- There are three natural units in spherical geometry, each of which is based on particular conditions. The first one is the natural unit of the measurement of the angle, which is based on the revolution. The second one is the natural unit of the length, which is grounded on the great circle’s circumference. Finally, it is the natural unit of the area, which is grounded on the spherical area of the figure.
- Every line that can be drawn on the sphere need to be associated with two antipodal points. These specific points serve as the poles of the drawn line and they are the lines that are intersected by a set of other lines that have to be perpendicular to the drawn line.
- Every point that can be drawn on the sphere should be associated with the unique line. This specific line is usually called the polar line of the given point.
These are only some of the properties that spherical geometry has, but they are the most essential to be aware of in order to deal with the spherical geometry problems. Apart from them, it is important to mention the properties of the Euclidean line. The name of the line has originated from the line of an outstanding Greek mathematician, who is known as Euclid. The properties of the Euclidean line are characterized by the ways in which a line can interact with the sphere. These ways are the following:
- The line doesn’t intersect or meet the sphere under any circumstances.
- The line can meet the sphere in only only one point that is drawn on the sphere.
- There are two points in which the line can meet the sphere.
Euclidean has payed a significant role in the development of spherical geometry. He discovered and defended a great number of theories, algorithms, domains, divisions and so on. Apart from it, there are aspects of spherical geometry that are called after Euclidean, such as the Euclidean space, distance, ball, etc.
Why studying of a sphere is important
If people did not understand the concept of the sphere, they would not understand the fact that the Earth is spherical, because they wouldn’t have any proofs to conclude this fact. However, more than two thousand years ago there was a breakthrough in the area of mathematical sciences in ancient Greece which led to the achievements that played a considerable role in the life and history of humanity.
As it is widely known, the concept of the spherical Earth was not the first concept concerning people’s beliefs about the form of the planet. Earlier, people tended to think that the Earth was flat like a disk and that it was surrounded by the space called sky and by water. The conclusion about the real form of the planet was not accepted instantly, there had a long time to pass before people were ready to accept the information, which is now is obvious to everybody.
The concept of the sphere was the object of exploration and research not only for the mathematicians, but also for ancient philosophers, such as Plato, Aristotle, Pythagoras and many others. As for Pythagoras, his concept of the form of the planet suggested an approach that was easy to deal with from a mathematical point of view. As for Aristotle, he played a special role in this area for a number of reasons. First of all, he concluded that there are stars that can be seen in only one part of the planet, while from the other part you will see other stars. This conclusion led him to further observations, which served to support the concept of the spherical form of the Earth.
Apart from the above-provided concepts explaining the form and shape of the planet, there are also such as the ellipsoid concept, which started from the ellipsoid revolution, the geoid concept and other more complicated approaches to describing the shape of the planet Earth.
All the information that is provided in the article is meant to help you deal with the issues of geometry related to the spheres and their properties, as well as cope with geometry problems concerning surface area of sphere.
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