The sphere is one of the first bodies with high symmetry whose properties are studied in the school course of geometry. This article discusses the formula of the sphere, its difference from the ball, and also provides the calculation of the surface area of \u200b\u200bour planet.

Sphere: concept in geometry

To better understand the formula of the surface, which will be given below, it is necessary to get acquainted with the concept of sphere. In geometry, it is a three-dimensional body that contains some amount of space. The mathematical definition of the sphere is as follows: this is a set of points that lie at a certain distance from one fixed point called the center. The noted distance is the sphere radius, which is denoted by R or R and is measured in meters (kilometers, centimeters and other length units).

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 figure if you take a circle and start rotating 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 that are outwardly similar to each other, but possess completely different physical properties. The sphere and the ball first of all differ in their mass: the sphere is an infinitely thin layer, the ball is the volumetric body of the final density, which is the same in all its points bounded by a spherical surface. That is, the ball has the ultimate mass and is quite a real object. The sphere is an ideal figure that does not have a mass, which does not actually exist, but it is a successful idealization in geometry in the study of its properties.

Examples of real objects, the form of which practically corresponds to the sphere, are a Christmas toy in the form of a ball for decorating a Christmas tree or a soap bubble.

With regard to the similarity between the figures under consideration, the following signs can be called:

• both of them possess the same symmetry;
• for both, the surface area formula is the same, moreover, they have an equal surface area if their radii is equal;
• both figures with equal radii occupy the same amount in space, only the ball fills it completely, and the sphere only limits its surface.

The sphere and the ball of equal radius are shown in the figure below.

Note that the ball, as well as the sphere, is the body of rotation, so it can be obtained if you rotate around the diameter of the circle (not a circle!).

Elements of the sphere

So the geometric values \u200b\u200bare called, the knowledge of which allows you to describe either the entire figure or its individual parts. The main elements are the following:

• Radius R, which has already been mentioned earlier. It is the distance from the center of the figure to the spherical surface. In essence, this is the only value that describes all the properties of the sphere.
• Diameter D, or D. This is a segment, the ends of which lie on the spherical surface, and the middle passes through the central point of the figure. The diameter of the sphere can be carried out by an infinite number of methods, but all obtained segments will have the same length, which is equal to a double radius, that is, D \u003d 2 * R.
• The surface area S is a two-dimensional characteristic, the formula for which will be shown below.
• Spring-related three-dimensional angles are measured in steradians. One Steradian is an angle, the top of which lies in the center of the sphere, and which relies onto a portion of a spherical surface having an area R 2.

Geometric properties of the sphere

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

• Any direct, which crosses the sphere and passes through its center, is the axis of symmetry of the figure. Rotate the sphere around this axis to any angle translates it in itself.
• The plane that crosses the figure under consideration through its center divides the sphere into two equal parts, that is, it is a reflection plane.

Surface Surface Figure

This value is indicated by the Latin letter S. The formula for calculating the area of \u200b\u200bthe sphere has the following form:

S \u003d 4 * Pi * R 2, where pi ≈ 3,1416.

The formula demonstrates that the square S can be calculated subject to the knowledge of the radius of the figure. If its diameter D is known, then the formula of the sphere can be written as:

The irrational number PI, for which four signs after a comma are given, in a number of mathematical calculations can be used with accuracy to hundredths, that is, 3.14.

It is curious to also consider how many steradians correspond to the entire surface of the figure under consideration. Based on the definition of this value, we get:

Ω \u003d S / R 2 \u003d 4 * PI * R 2 / R 2 \u003d 4 * PI Steradian.

To calculate any surround angle, it is necessary to substitute the corresponding value of S.

The surface of the planet Earth

The formula of the sphere can be applied to determine which we live. Before proceeding with the calculations, you should make a couple of reservations:

• First, the Earth does not have an ideal spherical surface. Its equatorial and polar radii is equal to 6378 km and 6357 km, respectively. The difference between these numbers does not exceed 0.3%, so it is possible to take the average radius of 6371 km to calculate.
• Secondly, the relief is three-dimensional, that is, there are depressions and mountains. These characteristic features of the planet lead to an increase in its surface area, nevertheless, we will not consider it, 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 \u003d 4 * PI * R 2 \u003d 4 * 3,1416 * 6371 2 ≈ 510.066 million km 2.

Russia, according to official data, covers an area of \u200b\u200b17.125 million km 2, which is 3.36% of the surface of the planet. If you consider that only 150.387 million km 2 include the land of our country, then the area of \u200b\u200bour country will be 11.4% of the entire territory not covered with water.

The ball is a body consisting of all the points of space that are at a distance, not more than this point. This point is called the ball center, and this distance is a radius of the ball. The border of the ball is called a ball surface or sphere. Spheres points are all points of the ball, which are removed from the center to the distance equal to the radius. Any segment that connects the center of the ball with a ball surface point is also called a radius. Passing through the center of the ball of a segment, which connects two points of the ball surface, is called a diameter. The ends of any diameter are called diametrically opposite points of the ball.

The ball is the body of rotation, as well as a cone and cylinder. The ball is obtained when the semicircular rotates around its diameter as an axis.

The surface area of \u200b\u200bthe ball can be found by formulas:

where R is the radius of the ball, D - the diameter of the ball.

The bulk of the ball is by the formula:

V \u003d 4/3 πr 3,

where R is a ball radius.

Theorem. Any section of the ball with a plane is a circle. The center of this circle is the base of the perpendicular, lowered from the ball center to the securing plane.

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

From the formula it is clear that the planes that are equidistant from the center, crosses the ball along the equal circles. The radius of the section is the greater, the closer the sequential plane to the ball center, that is, the smaller the distance is OK. The largest radius has a cross section to the plane passing through the center of the ball. The radius of this circle is equal to the ball radius.

The plane passing through the center of the ball is called a diametral plane. The cross section of a bowl with a diametral plane is called a large circle, and the cross section of the sphere is a large circle, and the cross section of the sphere is a large circle.

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

The plane that passes through the point A of the ball surface and is perpendicular to the radius spent in point A, is called a tangent plane. Point A is called a touch point.

Theorem. The tangent plane has only one common point with a ball - a touch point.

Direct, which passes through the point and the ball surface perpendicular to the radius spent at this point is called tangent.

Theorem. Through any point of the ball surface, there is an infinitely much tangent, and all of them lie in the tangent plane of the ball.

The ball segment is called a part of the ball, which cut from it with a plane. The ABC circle is the base of the ball segment. Cut Mn perpendicular, conducted from the center N circle of ABC before intersection with a spherical surface, is the height of the ball segment. Point M is the top of the ball segment.

The surface area of \u200b\u200bthe ball segment can be calculated by the formula:

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

V \u003d πh 2 (R - 1 / 3H),

where R is the radius of a large circle, H is the height of the ball segment.

The ball sector is obtained from a ball segment and cone as follows. If the ball segment is smaller than the semisphere, then the ball segment is complemented by a cone, which has a vertex in the center of the ball, and the base is the base of the segment. If the segment is greater than the semitter, the specified cone from it is removed.

The ball sector is part of the ball, bounded by the surface of the spherical segment (on our figure it is an AMCB) and a conical surface (in the figure it is OABC), the base of which serves as the base of the segment (ABC), and the top of the ball of the bowl O.

The volume of the ball sector is in the formula:

V \u003d 2/3 πR 2 H.

The ball layer is part of the ball, concluded between two parallel planes (in the figure of ABC and DEF planes) crossing the spherical surface. The curve surface of the ball layer is called a ball belt (zone). Circles ABC and DEF - the base of the ball belt. The distance NK between the bases of the ball belt is its height.

the site, with full or partial copying of the material reference to the original source is required.

In Chapter 2, we will continue the "construction geometry" and tell about the structure and properties of the most important spatial figures - ball and spheres, cylinders and cones, prisms and pyramids. Most of the items created by the hands of a person - buildings, cars, furniture, dishes, etc. ., etc., consists of parts having the shape of these figures.

§ 4. Sphere and ball

After the straight and planes, the sphere and the ball are the simplest, but very important and rich in various properties of spatial figures. On the geometric properties of the ball and its surface - sphere - whole books are written. Some of these properties were also known to the ancient Greek geometers, and some found quite recently, in last years. These properties (together with the laws of natural science) explain why, for example, the shape of the ball has heavenly bodies and fish eggs, why the ball in the shape of a ball make batisysphs and soccer balls, why are so common in the technique of ball bearings, etc. We can prove only the simplest properties of the ball. Proof of others, albeit very important properties, often require the use of at all elementary methods, although the formulation of such properties can be very simple: for example, among all bodies that have the same surface area, the greatest volume of the ball.

4.1. Definitions of sphere and ball.

The sphere and ball in space are defined in the same way as a circle and a circle on the plane. The sphere is called a figure consisting of all points of space remote from this

point per and the same (positive) distance.

This point is called the center of the sphere, and the distance is its radius (Fig. 4.1).

So, the sphere with the center of O and Radius R is a figure formed by all points X of space for which

The ball is called a figure formed by all the points of space located at a distance of no longer this (positive) distance from this point. This point is called the ball center, and this distance is its radius.

So, the ball with the center of O and Radius R is a figure formed by all points X of space for which

The points x of the ball with the center of O and Radius R for which the sphere is formed. It is said that this sphere limits this ball or that it is its surface.

‌‌‌V district scientific and practical conference research, design and creative student students "First steps in science"

Research on this topic:

"Sphere and ball - ordinary geometric bodies."

Performed: 9th grade MBOU student

"Kochetovskaya Secondary School" of the Romanov Dima.

Leader: Teacher of Mathematics and Physics Tremaskina V.S.

Introduction _______________________________________________________3

1. History of study of geometric bodies: ball, sphere _______________________ 3

2. Sphere and ball.

2.1. The concept of sphere and ball ___________________________________________ 3-4

2.2. Equation of the sphere ________________________________________________ 4

2.3. Mutual arrangement spheres and planes _________________________ 4-6

2.4. Tangential plane to the sphere ____________________________________ 6-7

2.5. Area of \u200b\u200bsphere and ball volume ________________________________ 7

2.6. Receipt of the sphere ___________________________________________ 7-8

2.7. Finding the sphere and ball in nature ______________________________ 9-13

2.8.Sfer and ball in everyday life_________________________________14-15

2.9. Replacing the sphere and ball in architecture ____________________________ 16-22

2.10. Application of the sphere and ball in geodesy ______________________________ 23

2.11 Provision of the sphere and ball in astronomy and geography _________________ 24

2.12. Sphere and ball in art _________________________________________ 25

Conclusion _______________________________________________________ 25.

Literature _______________________________________________________ 26.

The urgency of the chosen topic.

Over the centuries, mankind has not ceased to replenish its scientific knowledge in a particular field of sciences. Many geometry scientists, and ordinary peopleInterested in such a figure as a ball and his "shell", which is called the sphere. Many real objects in physics, astronomy, biology and other natural sciences have a ball shape. Therefore, the issues of studying the properties of the ball was assigned to various historical epochs and was given a significant role in our time.

Purpose of the study:examine geometric bodies ball and sphere, consider their use in different fields of science, in everyday life, in nature, create a presentation "Sphere and a ball - ordinary geometric bodies."

Tasks:

1. Collect material about the ball and sphere using various sources of information, including Internet resources.

2. Systematize the material about the ball and sphere.

4. Create a presentation " Sphere and ball - ordinary geometric bodies».

5. Submit work in the geometry lesson when studying the topic "Sphere and Ball".

Object of study : sphere and ball

Subject of study : Elements and properties of the sphere and ball

Hypothesis: We need balls in order to make our world more diverse and volume.

Methods: partial search, research, comparative analysis, synthesis, practical.

Result Research: the knowledge gained is needed not only to astronomers, navigations of sea ships, aircraft, spacecraftwhich stars determine their coordinates, but also builders of mines, metro, tunnels, architects, as well as during geodetic shooting of large areas of the Earth's surface, when it becomes necessary to take into account its shag-likeness, in everyday life.

Scientific novelty: Theoretical material is presented in the form of high school students accessible to understand.

Practical significance:this material can be used as a basis for electively course In the classes of the physico-mathematical profile, in the lessons when studying the "sphere and ball".

Introduction

For many centuries, humanity has not ceased to replenish its scientific knowledge in a particular field of science. Stereometry, as a science of figures in space, is inherently connected with many of scientific disciplines. Such disciplines include: mathematics, physics, computer science and programming, as well as chemistry and biology. In the latter there is a problem of studying a microworld, which is a complex combination of various particles in space relative to each other. The architecture constantly uses theorems and effects from stereometry.

A lot of scientists geometras, and ordinary people, were interested in such a figure as a ball and its "shell", which is the name of the sphere. Surprisingly, the ball is the only body with a larger surface area with an amount equal to the volume of other compared bodies, such as a cube, prism, or other all sorts of polyhedra. With balls, we are dealing daily. For example, almost every person uses a ball with a handle to the end of the rod of which a metal ball is mounted, rotating under the action of friction forces between it and paper and in the process of rotation on its surface, the ball "takes out" the next portion of ink. In the automotive industry, ball supports are made, which are a very important detail in the car and providing the right turn of the wheels and the stability of the machine on the road. Elements of machines, aircraft, rockets, motorcycles, shells, swimming trips that are subjected to constant water or air influences, mainly have any spherical surfaces called fairies.

History of study of geometric bodies: ball, sphere

The ball is made to call the body limited to the sphere, i.e. Ball and sphere are different geometric bodies. However, both words "ball" and "sphere" originate from the same Greek word "sfyra" - the ball. At the same time, the word "ball" was formed from the transition of consonant SF in sh.

In the XI book "Beginning" Euclidean determines the ball as a figure described by a rotating near the fixed diameter by a semicircle. In antiquity, the sphere was in great honor. Astronomical observations over the celestial arch invariably caused the image of the sphere.

The sphere has always been widely used in various fields of science and technology.

2.1. The concept of sphere and ball

The sphere is called a surface consisting of all points of space located at a given distance from this point.

The body limited to the sphere is called a ball.

This point is called the center of the sphere, and this distance is the radius of the sphere.

Cut connecting two points of the sphere and passing

through its center, is called the diameter of the sphere.

The center, the radius, the diameter of the sphere is also called the center, the radius and the diameter of the ball.

2.2. Equation sphere

Set up rectangular system Coordinates ABOUTxyz.

We construct the sphere c center at point C (x 0; y 0; z 0)

and radius R.

Ms \u003d (x - x 0) 2 + (y - y 0) 2 + (z - z 0) 2

MS \u003d R, or MS2 \u003d R2

consequently equation

the spheres looks like:

(X - X 0 ) 2 + (Y - Y 0 ) 2 + (z - z 0 ) 2 \u003d R. 2

2.3. Mutual location of the sphere and plane

Given:

The sphere of radius R with the center C (x 0; y 0; z 0), point M (x; y; z) lies in the sphere.

What is the distance of the MS?

T. K. MS \u003d R, T.

M.

R.

from

FROM FROMSS

Danar: Plane α, sphere (C; R),

d - distance from the center C to the plane α.

We introduce the coordinate system where the point C (x 0; y 0; z 0). Make the equation of the sphere and plane α.

z.

P
ust dot C lies on the z axis. Then its coordinates (0; 0; D).

Equation of the sphere:

Plane equation α: z. = 0

We explore the system of equations:

z \u003d 0.

Then

Depending on the ratio D and R, 3 cases are possible ...

1
) D.< R .

Then

circle equation (O; R)

Section of the Plane - Circle

2
) D \u003d R.

Then

IN eroNo

x \u003d 0 and y \u003d 0

Sphere and plane have one common point.

3
) D\u003e R.

Then

no solutions.

Sphere and plane do not have common points.

2.4. Tangential

The plane having only one common point with a sphere is called a tangent plane to the sphere, and their common point is called a point of touching the plane and sphere.

Theorem. The radius of the sphere spent on the point of touching the sphere and the plane is perpendicular to the tangent plane.

Dano: sphere with the centerABOUT and radiusR. , α - tangent to the sphere at the pointBUT plane.

Prove Oa. but .

Proof: Let Oa. Not perpendicular to the plane but , then Oa. is inclined to plane, it means the distance from the center to the plane d. < R. . Those. The sphere should intersect with a plane around the circumference, but this does not satisfy the theorem condition. It means Oa. but .

We prove the reverse theorem.

If the radius of the sphere is perpendicular to the plane passing through its end lying on the sphere, then this plane is tangent to the sphere.

Dano: sphere with the centerABOUT and radius Oa. , but, Oa. but .

Provebut - tangent plane.

Proof: Because Oa. but , The distance from the center of the sphere to the plane is equal to the radius. So, the sphere and the plane have one common point. By definition, the plane is tangent to the sphere.

2.5. Sphere area and ball

and Bowl of radius defined by formulas:

Evidence

Take a quarter of the radius circle R with the center at the point. Equation of the circumference of this circle:From!.

The function is continuous, increasing, non-negative. When a quarter of a circle is rotated around the OX axis, a half-hour is formed, therefore:

Where did it come from?

Evidence

Ch. t. D.

Part of the ball, [ ] Cowned by some plane from him, called ball or spherical segment. The base of the ball segment is called a circle Abcd. . The height of the ball segment is called a segment Nm. . The length of the perpendicular restored from the center N. The bases to the intersection with the surface of the ball. Point M. called the top of the ball segment.

The volume of the ball segment it is expressed by the formula:

V. = π h. 2 ( R. − 1/3 h)

Ball layer - This is part of the ball [ ], concluded between two secant parallel planes. Ball belt or Ball zone - This is the curve surface of the ball layer. Circles ABC and DEF. this is the base of the ball belt. Distance between baseson - This is the height of the ball layer.

Volume of ball layer it is expressed by the formula:

V. = 1/6 π h. 3 + 1/2 π( r. 1 2 + r. 2 2 ) h.

Ball sector- This is part of the ball [ ], limited by the curve with the surface of the ball segment and the conical surface of which serves as the base of the segment, and the vertex is the center of the ball.

Ball sector volume raven , the basis of which has the same area as part of the ball surface cut by the sector, and the height is equal to the radius

V. = 1/3 R S. = 2/3 π R. 2 h.

2.6. Obtaining a sphere

The sphere can be obtained by the rotation of the semicircle of the CCD around the diameter of the AV

2.7. Finding the sphere and ball in nature

Z. magages of nature - balls-messages.These mysterious stone formations of perfect round form were discovered in the late 1940s in the jungle of the Central American Republic of Costa Rica. Balls have dimensions from 10 cm to 3-4 meters in diameter. At the air survey, it turned out that they are scattered over the surface of the Earth, it is not by chance that they make geometric shapes. It is even possible that the balls are not scattered, but are decomposed in the form of a huge star card; Each ball is a star with the appropriate description.

Among the hypotheses of the origin of the balls there are only exotic versions: from the aliens to the sculptors of Atlantis. There is a version that the balls cut out (based on future dividends from tourism) who bored Nazi migrants flooded Latin America After the collapse of the Third Reich. Natural reasons explain the abundance of balls and strange drawings on them failed. In Kazakhstan, when developing a sandy career at a sufficiently large depth, several large copies of such boulders were also discovered ... This Nakhodka reported the Phenomenon Commission; Alas, photos of findings are not survived.

Crystal ball. Macro. On a branch of some tree lies a ball of glass, it reflects the surrounding nature. Very cute yellow flowers and green juicy grass.

FROM severed balls

in the photo in places of power - the result of the decay of uranium or plasmoid form of life?

Temple of the Holy Sepulcher and other places of Israel

AND
relevant natural phenomenon thousands of regular ice balls were formed on the shore of Michigan Lake

Sea algae in the form of unusual balls

Strange balls appeared on the coast of Hampton, which is on the east coast of the United States, in June 2002. The tidal wave began to endure the inconspicuous number of such greenish balls - soft, remotely resembling a sponge and size with a ball for tennis or golf. At a distance of about 300 meters or more all The sandy beach was literally littered with such balls. Immediately the disputes began - what is it and where? Biologists were also involved in the debates, and resting on the beach, and random passers-by. Before, no one seen nothing of the kind here.

Nature is afraid of symmetry, nature does not know the ideal geometric shapes. But the person can make nature acquire these alien shapes. Visual example this is the work of the Korean artist Lee Jae-Hyo, which creates fromtree trunks perfect spheres

T.

oschi small purple balls strangely found themselves in the center of the desert in Arizona, USA. Residents of the city of Tucson Geralddin Vargas and her husband discovered an inexplicable accumulation of incomprehensible balls a couple of weeks ago while walking around the surrounding area. "We photographed the nature of the desert when they came across this strange place ... I do not understand how we didn't immediately notice him immediately?" Said Geraldin to journalists. - It just sparkled in the sun. " Photographers sent photos from strange objects With his familiar zoologist, but she could not say what it was, she didn't even have any assumptions on this.

Balls from minerals.

Amethyst.Brasilia.

Mountain crystal. Yule. Schelob.Ordaz.

Amazonit. Skolsky P-Ozrodan.

2.8 sphere and ball in everyday life

N.
and the geometric ball is similar globe, soccer ball, New Year's toys.

Ball from foam with her own hands

Zorbing (Zorbing) - This is one of the most fashionable extreme entertainment today. Zorbing will allow you to experience new, unusually bright and powerful sensations and shake on the everyday life of everyday life.

What is a bowl zorb

Z. orb (Zorb) It is a transparent sphere (ball) with a diameter of 3.2 meters inside which the sphere with a diameter of 1.8 meters is located in which zorbonavt (passenger Zorba). The space between these spheres is filled with air, the pressure of which spheres are opened between themselves, and with pins, on the contrary, are held. Such a system is very well absorbs, smoothes the unevenness of the track and makes riding safe.

2.9.Application of the sphere and ball in architecture

Such a house is called Wigwam. Such houses are building Indians.

Stainless steel balls and hemispheres

Fountain "Rotatingball "In St.

Petersburg -

Modern houses

What ifhouse not just on the tree, but also in the shape of a ball.

This village is from the most realround houses .

FROM
round round houses

Montreal biosphere - US Exhibition Pavilion at Expo-67 in Canada,

created by architect Richard Fuller.

Hotel in the form of transparent balls

IN
about the French city of Rubare (Roubaix) in one of the parks opened portable hotel rooms Hotel Bolha. Made it specifically for people who even in the center of the city jungle wish to be closer to nature.Bubble concept came up with designer Pierre Stefan Duma. Such an advanced design was created to temporarily access the guests to the unknown. After all, not many can afford to sleep under a round ceiling.

Dress from the balls.

Country office Soon spring (and there and summer) and many will begin to ride the cottage to rest.
But sometimes you need to work at the cottage (so that you!). No place to leave?
You can here in such a small spherical structure "Archipod":

Energy Efficiency B.architecture . Smart home - molecule.

In the park of science and technology, La Vilette, built on the place of slaughterhouse on the eastern outskirts of Paris, rushes a giant ball, in the mirror surface of which the Paris sky and the surrounding landscape are reflected. To date, this building is considered the most perfect construction of spherical form in the world. Parisians call him "Aepex" (GEODE). This is panoramic

cinema with the largest in Europe screen. House-bowl Mirror

Such balls from the threads can simply hang to the tree branches, if your holiday passes in nature, or to the ceiling. As well as you can make a banquet table, adding the composition with candles and flowers.

2.10. The use of the sphere and ball in geodesy.

Cartographic projections

displays the entire surface of the earth ellipsoid (see ) or any part of it on the plane, obtained mainly to build a map.

Scale.K.p. are built on a certain scale. Reducing mentally earthly ellipsoid inM.once, for example, 10,000,000 times, it receives its geometric model - , the image of which is already in a natural value on the plane gives the map of the surface of this ellipsoid. Value 1:M.(In Example 1: 10 000 000), defines the main, or general, the scale of the card. T. K. The surface of the ellipsoid and ball can not be deployed on the plane without breaks and folds (they do not belong to the class of deploying surfaces (see )), any K. p. Inherent in distortion of lengths of lines, angles, etc., peculiar to any map. The main characteristic of K. p. At any point is a private scale μ. This is the value, the inverse attitude of the infinitely small segmentds.on the earth's ellipsoid to its imagedσ.on surface: μ MIN. ≤ μ ≤ μ Max, and equality here is possible only at regular points or along some lines on the map. T. about., The main scale of the card characterizes it only in general terms, in some annealed form. Attitude μ / m are called a relative scale, or an increase in length, difference M \u003d 1.

1. Networks of spherical coordinate lines.

2.11. The use of sphere and ball in astronomy and geography.

FROM fera and ball, as well as circumference and circle, considered in deep antiquity. The discovery of the softenness of the Earth, the emergence of ideas about the heavenly sphere gave impetus to the development of special science - spherika studying the figures located on the sphere.

By traveling around the world, the navigaters noticed that when returning at the same place, there was a loss or winning of the whole day, which would be absolutely impossible if the Earth had a disk form.

So, the evidence of the shag-formation of the Earth is currently served:

Always a circular figure of the horizon in the ocean and in open lowlands or plateales;

Around the world travel.

Gradual approximation or removal of objects;

AND
slaughter of various geographic Mapswe found that there are in geography geographical namesassociated with a ball. For example, there is a strait between the Northern and Southern Isles of the New Earth, which connects the Barents and Kara Sea, which is called Musian Ball, or the shed between the shores of Waygach Island and the Mainland Eurasia - Ugra Ball. We think that these straits are called balls due to the fact that their dimensions, the bottom form resemble a ball surface.

2.12. Sphere and ball in art

Mathematics Escher

In addition, the "game" with the logic of space are the paintings of the Escher, which depict various "impossible figures"; Escher depicted them both separately and in plot lithographs and engravings

Three spheres. 1946.

Hand with reflective sphere. 1935.

Conclusion

I think that I assembled material and knowledge, obtained during the work done can be used in the lessons of geometry, labor, in everyday life, as the basis for the elective course in the classes of the physico-mathematical profile, as well as on extracurricular activities to expand the horizons of students.

Literature

Adamar J. Elementary geometry. Part 2. M. Stockedgiz, 1958. Andreev

Atanasyan L.S. Geometry. Part 2. - M: Education, 1987. - 352c.

Basilev V.T. Geometry. M: Education, 1975.

Basilev V.T. Collection of tasks in geometry. M: Education, 1980. -240c.

Egorov I.P. Geometry. - M: Education, 1979. - 256c.

Egorov I.P. Bases of geometry. - M: Education, 1984. - 144С.

Task "Quantum": mathematics. Part 1. / Ed. NB Vasilyeva. M: 1997.

Rosenfeld B.A. The history of non-child geometry. Development of the concept of geometric space. M. Science., 1976. - 408c.

Encyclopedia of elementary mathematics. KN.4 - geometry. M., 1963.

10. Internet resources.

The ball and sphere are primarily geometric shapes, and if the ball is a geometric body, then the sphere is the surface of the ball. These figures were interested in many thousands of years ago BC.

Subsequently, when it was discovered that the Earth is a ball, and the sky is a heavenly sphere, a new fascinating direction in geometry - geometry on the sphere or spherical geometry has developed. In order to argue about the size and volume of the ball, you must first give it a definition.

Ball

The Radius Rag with the center at the point o geometry is called the body, which is created by all points space with a general property. These points are at a distance not exceeding the radius of the ball, that is, fill the entire space less than the ball radius in all directions from its center. If we consider only those points that are equidistant from the center of the ball - we will consider its surface or a bowl.

How can I get a ball? We can cut a circle from paper and start moving it around its diameter. That is, the diameter of the circle will be the axis of rotation. Educated figure - there will be a ball. Therefore, the ball is also called the body of rotation. Because it can be formed by rotating a flat shape - a circle.

Take some plane and cut our ball. Just as we cut the orange knife. A piece that we cut from the ball is called a ball segment.

IN Ancient Greece They could not only work with a ball and a sphere, as with geometric shapes, for example, to use them during construction, and also knew how to calculate the surface area of \u200b\u200bthe ball and the volume of the ball.

The sphere is different called the surface of the ball. The sphere is not a body - this is the surface of the body of rotation. However, since the Earth and many bodies have a spherical shape, such as a drop of water, then the study of geometric ratios within the sphere was greatly distributed.

For example, if we connect two points of the sphere between themselves straight line, then this straight line will call chord, and if this chord is held through the center of the sphere, which coincides with the center of the ball, then the chord will be called the diameter of the sphere.

If we feed a straight line that will affect the sphere just at one point, this line will be called tangent. In addition, this tangent to the sphere at this point will be perpendicular to the sphere radius carried out to the point of touch.

If we continue chord to a straight line in the other side of the sphere, then this chord will be called the Sale. Or can be said otherwise - the sequential to the sphere contains its chord in itself.

Bowl

The formula for calculating the volume of the ball has the form:

where R is a ball radius.

If you need to find the volume of the ball segment - use the formula:

V Seg \u003d πh 2 (R-H / 3), H is the height of the ball segment.

Surface surface of the ball or sphere

To calculate the spheres area or the surface area of \u200b\u200bthe ball (this is the same):

where R is the radius of the sphere.

Archimedy loved the ball and the sphere, he even asked to leave the drawing on his tomb on which a ball entered the cylinder. Archimeda believed that the volume of the ball and its surface is equal to the two thirds of the volume and surface of the cylinder, in which the ball is inscribed.