Sphere and ball properties. Ball as a geometric figure

To get a competent answer to the question in the title, the reader of the article will need to properly strain his abilities for abstract thinking and how to delve deeply into certain sections of mathematics that he has studied in school. And to stimulate the imagination, it will be useful to recall that “Education is what remains after everything that we have been taught is forgotten” (the authorship of the phrase is attributed to A. Einstein).

A short dive into one of the sections of mathematics

To begin with, you need to remember the existence of the science of geometry (in a somewhat loose translation from Greek this word means "surveying") - a separate branch of mathematics specializing in the study of spatial structures, their relationships with each other and various generalizations arising from this. It is important that despite such a "mundane" origin of the name, this science operates purely abstract concepts, which in the world we are accustomed to do not exist in direct physical embodiment.

One of these basic concepts- it geometric point... Stretch your imagination: unlike "a point with a pencil", "a point from a pin" and so on, this point is a completely abstract object in an imaginary space without any measurable characteristics such as "thickness", "color" and so on (mathematics they like to say the phrase "zero-dimensional object"). Basically, everything else in geometry will be further defined in terms of this abstraction.

The next concept is needed for further reasoning - this is the "ritual" mathematical phrase "geometric place of points" (GMT). With its help, a certain set (set) of points is described that fall under a certain relation (property) - thus, it is set “ geometric figure". Example: a sphere (from the ancient Greek σφαῖρα, originally denoting a ball / ball) is a geometric place of such points in space, which can be described as equidistant (located at exactly the same distance) from some set point commonly referred to as the "center of the sphere".

The distance from the center of the sphere to this GMT is usually called the "radius of the sphere". During all these manipulations, it is important to continue to remember that the sphere is a more ephemeral concept than even everyone is familiar and familiar. soap bubble: any soap bubble still has a quite perceptible wall made of a water-soap film of microscopic thickness, which can be physically measured (and even pierced), but a sphere cannot!

Now let's turn to the definition of a ball: a ball is understood as a set of all such points in space that are located from a certain point (the center of the ball) at a distance not greater than a given (radius of the ball). In other words, a ball is a "geometric body" - something that, according to the primary definition of Euclid, "has length, width and depth" (in modern textbooks this definition is less clear: "a part of space, limited by its formed form").

In passing, we note that the methods of defining a sphere and a ball through the center and radius used here are not the only ones: for example, defining a sphere / ball in space can be performed by rotating a circle, circle, etc. (Those deeply interested in this issue are strongly advised to familiarize themselves with a separate section of geometry called "Shapes and bodies of revolution", since this is a frequently used way of defining a wide variety of geometric shapes and bodies in space).

Thus, both in the case of a sphere and in the case of a ball, one has to deal with a certain way of a given geometrical place of points (that is, a geometric figure), but only in the case of a ball can one speak of a geometric body. It is curious to note that, strictly speaking, the sphere can be "subtracted" from the ball: in this case, mathematicians speak of an "open ball". However, "by default" there is a "closed ball", where the sphere is its natural boundary and its part.

Summary

Both the ball and the sphere are abstract geometric objects (geometric shapes) defined through some geometric place of points in space - for example, using the concept of the center of the ball / sphere and the radius of the ball / sphere. However, only the ball is a full-fledged geometric body, since it includes not only the description of the surface bounding it, but also the entire part of the space that this surface contains. From this point of view, the sphere is only the outer abstract boundary (surface) of the sphere specified in space.

A ball is a body consisting of all points in space that are at a distance not more than a given one from a given point. This point is called the center of the ball, and this distance is called the radius of the ball. The boundary of a ball is called a spherical surface or sphere. The points of the sphere are all points of the sphere that are removed from the center by a distance equal to the radius. Any segment that connects the center of the ball with a point on the ball surface is also called the radius. The segment passing through the center of the ball that connects two points of the ball surface is called the diameter. The ends of any diameter are called diametrically opposite points of the ball.

A ball is a body of revolution, just like a cone and a cylinder. The ball is obtained by rotating a semicircle around its diameter as an axis.

The surface area of a ball can be found by the formulas:

where r is the radius of the ball, d is the diameter of the ball.

The volume of the ball is found by the formula:

V = 4/3 πr 3,

where r is the radius of the ball.

Theorem. Any section of a sphere by a plane is a circle. The center of this circle is the base of the perpendicular dropped from the center of the ball onto the cutting plane.

Based on this theorem, if a ball with center O and radius R is intersected by the plane α, then a circle of radius r with center K is obtained in the section. The radius of the ball's section by the plane can be found by the formula

It can be seen from the formula that the planes equidistant from the center intersect the ball in equal circles. The radius of the section is the greater, the closer the cutting plane is to the center of the ball, that is, the smaller the distance OK. The largest radius has a section by a plane passing through the center of the ball. The radius of this circle is equal to the radius of the ball.

The plane passing through the center of the ball is called the diametrical plane. The section of a ball by the diametrical plane is called a large circle, and the section of a sphere is called a large circle, and the section of a sphere is called a large circle.

Theorem. Any diametrical plane of the ball is its plane of symmetry. The center of the ball is its center of symmetry.

The plane that passes through point A of the spherical surface and is perpendicular to the radius drawn to point A is called the tangent plane. Point A is called the tangency point.

Theorem. The tangent plane has only one common point with the ball - the tangency point.

A straight line that passes through point A of the spherical surface perpendicular to the radius drawn to this point is called a tangent line.

Theorem. An infinitely many tangents pass through any point of the spherical surface, and all of them lie in the tangent plane of the ball.

A spherical segment is the part of a ball that is cut off from it by a plane. Circle ABC is the base of the ball segment. The segment MN of the perpendicular drawn from the center N of the circle ABC to the intersection with the spherical surface is the height of the spherical segment. Point M is the apex of the spherical segment.

The surface area of the spherical segment can be calculated using the formula:

The volume of the spherical segment can be found by the formula:

V = πh 2 (R - 1 / 3h),

where R is the radius of the large circle, h is the height of the spherical segment.

The spherical sector is obtained from a spherical segment and a cone, as follows. If a spherical segment is less than a hemisphere, then the spherical segment is complemented by a cone, whose apex is in the center of the ball, and the base is the base of the segment. If the segment is larger than a hemisphere, then the specified cone is removed from it.

A spherical sector is a part of a ball bounded by a curved surface of a spherical segment (in our figure it is AMCB) and a conical surface (in the figure it is OABC), the base of which is the base of the segment (ABC), and the top is the center of the ball O.

The volume of the spherical sector is found by the formula:

V = 2/3 πR 2 H.

A spherical layer is a part of a sphere enclosed between two parallel planes (in the figure, planes ABC and DEF), intersecting a spherical surface. The curved surface of a spherical layer is called a spherical belt (zone). Circles ABC and DEF are the bases of the ball belt. The distance NK between the bases of the ball belt is its height.

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- (Greek sphaira ball). one) solid, in which all points of the surface are equally distant from the inner point, called the center of the ball; the image of the earth in the form of a globe. 2) part of the space in which the planet makes its way. 3) in the figurative ... Dictionary foreign words Russian language

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sphere- s, w. sphère f. gr. sphaira. 1. geom. A closed surface, all points of which are equally distant from one point (center /. BAS 1. | transfer. Ten spheres flying air, I saw a drinking house in the distance. I. Naumov Yason. // Iroi comic poem 560. 2. ... ... Historical Dictionary gallicisms of the Russian language

Spheres, women [greek. sphaira ball]. 1. The same as the ball (mat.). 2. transfer. The area, place, limits, in which it exists, acts, develops, applies what n. (book). “Depending on the nature of the poetic talent and the degree of its development, the sphere ... Ushakov's Explanatory Dictionary

SPHERE, s, wives. 1. The area, the limits of the distribution of which n. C. activities. C. influence. 2. Wednesday, social environment. In my field. Higher spheres (about the ruling, aristocratic circles). 3. Closed surface, all points to the swarm are equally distant ... ... Ozhegov's Explanatory Dictionary

See area ... Synonym dictionary

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Sphere component compound words, meaning: 1) one of the shells of planets and stars: asthenosphere atmosphere barisphere biosphere geosphere heterosphere hydrosphere homosphere ionosphere lithosphere magnetosphere mesosphere stratosphere substratosphere ... ... Wikipedia

- (from the Greek sphaira ball), 1) the area of action, the limits of the spread of something (for example, the sphere of influence). 2) Public environment, environment, setting ... Modern encyclopedia

- (from the Greek sphaira ball) 1) scope, limits of distribution of something (for example, sphere of influence). 2) Public environment, environment, setting ...

A closed surface, all points of which are equally distant from one point (the center of the sphere). The segment connecting the center of the sphere with any of its points (as well as its length) is called the radius of the sphere. The surface area of the sphere is S = 4? R2, where R is the radius of the sphere ... Big Encyclopedic Dictionary

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The sphere is one of the first bodies with high symmetry, the properties of which are studied in school course geometry. This article discusses the formula of a sphere, its difference from a sphere, and also provides a calculation of the surface area of our planet.

Sphere: concept in geometry

To better understand the surface formula, which will be given below, it is necessary to get acquainted with the concept of a sphere. In geometry, it is a three-dimensional body that contains a certain volume 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 units of length).

The figure below shows the described figure. The lines show the contours of its surface. The black point is the center of the sphere.

You can get this shape if you take a circle and start rotating it around any of the axes passing through the diameter.

Sphere and ball: what is the difference and what is the similarity?

Often schoolchildren confuse these two figures, which are outwardly similar to each other, but have completely different physical properties... The sphere and the ball are primarily distinguished by their mass: the sphere is infinite 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 that has no mass, which in reality does not exist, but it is a successful idealization in geometry when studying its properties.

Examples of real objects, the shape of which almost corresponds to a sphere, are a Christmas ball-shaped toy for decorating a Christmas tree or a soap bubble.

As for the similarities between the figures under consideration, the following features can be called:

they both 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 its diameter (not a circle!).

Sphere elements

This is the name of the geometric quantities, the knowledge of which makes it possible to describe either the whole figure or its individual parts. Its main elements are as follows:

The radius r, which was already mentioned earlier. It is the distance from the center of the shape to the spherical surface. In fact, this is the only quantity that describes all the properties of the sphere.
Diameter d, or D. This is a line segment, the ends of which lie on a spherical surface, and the middle passes through the center point of the figure. The diameter of the sphere can be drawn 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.
Surface area S is a two-dimensional characteristic, the formula for which will be given below.
The 3D angles associated with a sphere are measured in steradians. One steradian is the angle, the vertex of which lies in the center of the sphere, and which rests on the part of the spherical surface having the area R 2.

Geometric properties of the sphere

From the above description of this figure, you can independently guess about these properties. They are as follows:

Any straight line that crosses the sphere and passes through its center is the axis of symmetry of the figure. Rotating the sphere around this axis at any angle translates it into itself.
The plane that intersects the figure under consideration through its center divides the sphere into two equal parts, that is, it is the plane of reflection.

Surface area of a figure

This value is denoted by the Latin letter S. The formula for calculating the area of a sphere is as follows:

S = 4 * pi * R 2, where pi ≈ 3.1416.

The formula demonstrates that the area S can be calculated if the radius of the figure is known. If its diameter D is known, then the formula for the sphere can be written as follows:

Irrational number pi, for which four decimal places are given, in a number of mathematical calculations can be used with an accuracy of hundredths, that is, 3.14.

It is also interesting to consider the question of how many steradians the entire surface of the figure under consideration corresponds to. Based on the definition of this quantity, we get:

Ω = S / R 2 = 4 * pi * R 2 / R 2 = 4 * pi steradian.

To calculate any volumetric angle, substitute the corresponding value of the area S in the expression above.

Surface of planet earth

The sphere formula can be applied to determine which one we live on. Before starting the calculations, a couple of caveats should be made:

First, the Earth does not have a perfect 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 the average radius of 6371 km can be taken for the calculation.
Secondly, the relief is three-dimensional, that is, there are depressions and mountains on it. These characteristics planets lead to an increase in its surface area, however, we will not take them into account in the calculation, since even the most big mountain, Everest, is 0.1% of the earth's radius (8.848 / 6371).

Using the sphere formula, we get:

S = 4 * pi * R 2 = 4 * 3.1416 * 6371 2 ≈ 510.066 million km 2.

Russia, according to official data, covers an area of 17.125 million km 2, which is 3.36% of the planet's surface. If we take into account 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 with water.