Download presentation ball. Presentation Creative project on the theme New Year's ball.ppt - Presentation Creative project on the theme "New Year's ball"

“Volume of a ball” - Find the volume of the cut off spherical segment. A ball is inscribed in a cone whose base radius is 1 and its generatrix is ​​2. Find the volume of a sphere inscribed in a cylinder whose base radius is 1. Volume of a torus. Find the volume of a sphere inscribed in a cube with edge equal to one. Exercise 22. Find the volume of a ball whose diameter is 4 cm.

“Circle circle sphere ball” - Ball and sphere. Ball. Circle. Area of ​​a circle. Diameter. Remember how a circle is defined. You are required to be attentive, focused, active, and precise. Geometric pattern. Center of the ball (sphere). Try to define a sphere using the concepts of distance between points. Computer center.

“Sphere and ball” - Three points are given on the surface of the ball. Problem on the theme ball (d/z). Section of a sphere by a plane. Any section of a ball by a plane is a circle. Tangent plane to a sphere. This point is called the center of the sphere, and this distance is called the radius of the sphere. The tale of the emergence of the ball. The section passing through the center of the ball is a large circle. (diametrical section).

“Balloon” - Since ancient times, people have dreamed of the opportunity to fly above the clouds and swim in the ocean of air. Airships are equipped with low-power and economical diesel engines. It is much easier to lift and lower a ball filled with hot air. Speed ​​120-150 km/h. Airships. Aeronautics. It is difficult to imagine the modern world without advertising, and here balloons have been used.

“Cylinder cone ball” - Volume of the spherical sector. Find the volume and surface area of ​​the sphere. Definition of a ball. Problem No. 3. Surface areas of bodies of rotation. Ball sector. The section of a ball by the diametrical plane is called a great circle. Bodies of rotation. The cross section of a cylinder with a plane parallel to the bases is a circle.

“Scientific and practical conference” - M.V. Lomonosov 2003. The focus of Russian education... From the history of the school scientific and practical conference. About how many wonderful discoveries the spirit of enlightenment is preparing for us... The sixth school scientific and practical conference dedicated to Khuzangay 2007. The second school scientific and practical conference dedicated to the 290th anniversary.

Sphere and ball

Creative project name

The many faces of "Round bodies"

Subject, class

Geometry, 11th grade

Brief summary of the project

In life we ​​often use the words sphere, ball. While working on the project, you will become familiar with the scientific concepts of a sphere, a ball and their elements, and in the future you will competently use these terms. Having derived the equation of a sphere, you will learn to write it for a given center and radius and, conversely, to determine from the equation whether the surface is a sphere. It will be quite interesting to consider all possible cases of the arrangement of a sphere and a plane, to get acquainted with the definition of a tangent plane to a sphere and the theorems expressing the properties and sign of a plane tangent to a sphere. Get acquainted with the formula for calculating the area of ​​a sphere. And, of course, you will learn to solve problems on this topic at both compulsory and advanced levels.

Over the centuries, humanity has not ceased to expand its scientific knowledge in one or another field of science. Many scientific geometers, and even ordinary people, were interested in such a figure as a ball and its “shell”, called a sphere. Many real objects in physics, astronomy, biology and other natural sciences are spherical. Therefore, the study of the properties of the ball was given a significant role in various historical eras and is given a significant role in our time.

I wish you success!

Reflective blog

Guys, write your feedback after each stage of the project in a reflective blog

Guiding Questions

Fundamental Question

How to explore the laws and patterns of the Universe?

Problematic issues

• What is the relationship between geometry and other fields of science?
• What are round bodies associated with?
• Why were many scientific geometers interested in such a figure as a ball and its “shell”, called a sphere?

Study questions

1. Give definitions of sphere and ball. What do they have in common and what are their differences?
2. How can a sphere and a sphere be obtained?
3. How to write the equation of a sphere if its center and radius are given?
4. How many possible cases of mutual arrangement of a sphere and a plane? What does it depend on? Sections of a sphere and a ball.
5. What plane is called the plane tangent to the sphere? What is its main property? Is it possible to determine whether a given plane is tangent to a sphere?
6. Formula for the area of ​​a sphere.
7. The relative position of a sphere and a straight line.
8. Ellipse, hyperbola, parabola as sections of a cone.
9. A sphere inscribed in a polyhedron, a sphere circumscribed about a polyhedron.

Project plan

Project business card

Teacher's publication. Booklet for parents

Teacher presentation to identify student ideas and interests

Working groups and research questions

Group “Mathematics” Belyakova Maria, Kobeleva Alena, Morozova Yulia

Summarize the material on the topic “Sphere and Ball” studied in the school geometry course;

Find and compare all definitions of sphere and sphere;

Prepare summary tables and a collection of tasks.

Group “Geographers” Kononykhina Alena, Prokofieva Albina, Samorodov Maxim

Find the first mentions of the Earth as a spherical surface;

Find materials indicating the evolutionary development of planet Earth.

Group “Astronomers” Eremin Vladislav, Kuzmin Evgeniy, Pavlochev Ilya

Find connections between geometry and astronomy;

Find evidence of the sphericity of the Earth from the point of view of astronomy;

Find materials about the structure of the solar system.

Group “Philosophers” Gogoleva Anastasia, Pukosenko Victoria, Chernova Yulia

Find material that connects the geometric body - the sphere with the concepts of philosophy;

Determine the types of spheres from the point of view of philosophy.

Group “Art Critics” Zhaksalikova Nadezhda, Kabanina Yulia, Chemis Valentina

Find paintings and engravings that depict the sphere.

Group “Academic Council” Astanaeva Marina, Balaeva Irina, Rostunova Yulia

Conduct an analysis of Unified State Examination tasks. Select assignments on this topic. Select tasks for final review.

Suggested topics for student projects

"The relative position of the sphere and the plane"

"Ball and Sphere"

“The ball is a symbol of God”

"Harmony of the Ball"

"Music of the Sphere"

"Sphere and ball in architecture"

"Sphere and ball in the world around us"

Email addresses of project participants

I ask all project participants to enter their data into the table after completing registration on the Gmail mail service.

Some materials from the theoretical seminar

Results of student project activities

Formative and summative assessment materials

Materials for support and support of project activities

Useful resources

Theoretical material

Sphere. Dictionaries and encyclopedias on Academician Shar. Dictionaries and encyclopedias on Academician Models of lessons. Sphere and ball. Touches and sections. Parts of a ball and sphere Sphere and sphere. Sections of a sphere and a ball by a plane. Tangent plane to a sphere. Ball and sphere. Abstract. Sphere

Municipal
Municipal
institution
institution
general education
general education
Creative project on the topic:
Creative project on the topic:
"New Year's ball"
"New Year's ball"
Completed by: 8th grade student
Shabalina Alexandra
Head: Vasilyeva Olga Sergeevna
Inshino 2017

Justification for choice
Justification for choice
themesthemes
The people at my school are very wonderful.
teacher - Olga Sergeevna. She teaches us lessons
technologies. Every time we do crafts with her
different topics.
Before the New Year we decided to make a New Year's
ball. We got a very beautiful toy, which
Can be hung on the Christmas tree or given as a gift
friends, relatives.

Goal: to make a New Year's ball that will
decorate the New Year tree.

Product design analysis
Product design analysis
I decided to make such a craft because I
I was attracted by the idea of ​​the composition and the color scheme

Materials
Materials
Satin ribbon (yellow) 4cm x 3m Beads Threads
Green fabric Orange fabric Lace
Styrofoam ball

Tools
Tools
Glue gun Scissors Sewing needle
Pencil
Carpenter's knife
Skewers

Historical reference
Historical reference
The oldest iron compass was discovered in
France during the excavation of an ancient mound. He lay there
in the ground for more than 2 thousand years. In the ashes that fell asleep
Greek city of Pompeii, archaeologists have discovered very
many bronze compasses. There has always been a compass
an indispensable assistant to architects and builders.
It is no coincidence that on the facade of one of the most ancient and
beautiful temples of Georgia depict the hand of the architect, and
behind her is a compass.
Steel compass cutter for applying such a pattern
archaeologists found it during excavations in Novgorod. Is in this
instrument something that makes one relate to
him with respect. This is how I described meeting him in
childhood Yu. Oleina, author of the famous fairy tale “Three
fat man”: “He lies in a velvet bed, legs tightly clenched,
cold sparkling compass. He has a heavy head.
I intend to pick it up. He unexpectedly
opens up and injects into the arm.”
In Ancient Rus' they loved a pattern of small circles.

Practical part
Practical part
Let's prepare in advance: a foam ball,
multi-colored scraps of fabric, cord for
finishing, various satin colored ribbons,
beads, sewing pins, compass, glue
gun, stationery knife, scissors and others
improvised tools for sewing.

Practical part
Practical part
At the very beginning of work we outline the pattern on the ball from
poles of the line. You can use a pen for this,
pencil or disappearing marker.
Now all the lines of the pattern should be cut with stationery
knife to a depth of one centimeter. Work needs to be done
very carefully and carefully.
Let's start working with the first part of the pattern. To the ball
it turned out beautiful, it would be better to use pieces of fabric
different shades or two colors so that you can
alternate. To do this, take pre-cut pieces
selected fabric and apply to this part of the pattern.

Practical part
Practical part
At this stage, you should position the fabric correctly in
depending on the direction of the pile.
Next, using a skewer or knitting needle, we remove the fabric into
slits along the entire contour of the selected part of the pattern. Need to
Make sure that the fabric lies flat and does not
was distorted. It is more convenient to do this work using
skewers, thanks to which the fabric is completely removed
slots.
finished, but you can continue the work if you wish
further. those. give the ball a festive, elegant look.
using a glue gun we will glue the decorative
lace or braid at the borders of the pattern elements.
Basically, at this stage, the ball already looks

Practical part
Practical part
First, coat the joints with glue, and then
apply braid.
In our case, the finished look of the New Year's ball
will add a bow made of narrow braid and gathered lace,
which we will glue together with the bow, but can be used
just a pin. We glue the braid in the center, and then onto it
glue beads at equal distances or
buttons. For hanging we use decorative ribbon.

Safety regulations
Safety regulations
1. Store needles in a cushion or pincushion, wrapped around them
thread. Store pins in a tightly sealed box
closing lid.
box provided for this purpose.
At the end of the work, check for their presence.
pad, do not put it in your mouth, do not stick it into your clothes,
soft objects, walls, curtains. don't leave
needle in the product.
or working box.
when passing, hold them by the closed blades.
scissors.
5. Store scissors in a specific place - in a stand
6. Place the scissors with the blades closed away from you;
7. Work well adjusted and sharpened
8. Do not leave scissors with blades open.
9. Monitor the movement and position of the blades during
10. Use scissors only for their intended purpose.
2. Do not throw a broken needle, but put it in a special
3. Know the number of needles and pins taken for work. IN
4. While working, stick needles and pins into
work.

Economic justification
Economic justification
№
1.
Quantity
in stock
Name
Narrow
ribbon (golden)
Beads
satin tape
(yellow)
Green fabric
Orange fabric
Lace
Foam
ball
2.
3.
4.
5.
6.
7.
TOTAL
8 pcs.
4cm x 3m
4 shreds
4 shreds
in stock
1 PC.
Price
----------
15 rub.
15 rub.
35 rub.
35 rub.
----------
45 rub.
145 rub.

Ecological
Ecological
justification
justification
My work is made from colorful scraps
fabrics using various beads. My product
does not harm the environment, because I purchased
materials in specialized stores, this
guarantees quality. When working with a lighter, I
worked with the teacher and followed all the technical rules
security. My souvenir is environmentally friendly,
because it does not cause allergic reactions and does not
harms your health.

main idea

Over the centuries, humanity has not ceased to expand its scientific knowledge in one or another field of science. Many scientific geometers, and even ordinary people, were interested in such a figure as ball and its “shell”, called sphere. Many real objects in physics, astronomy, biology and other natural sciences are spherical. Therefore, the study of the properties of the ball was given a significant role in various historical eras and is given a significant role in our time.

• Establish connections between geometry and other fields of science.
• To develop the creative activity of students, the ability to independently draw conclusions based on the data obtained as a result of research.
• Develop students' cognitive activity.
• Foster a desire for self-education and improvement.

Working groups and research questions

Group “Mathematics”

1. Summarize the material on the topic “Sphere and Ball” studied in the school geometry course.
2. Find and compare all definitions of sphere and ball.
3. Prepare summary tables and a collection of tasks.

Group “Geographers”

1. Find the first mentions of the Earth as a spherical surface.
2. Find materials indicating the evolutionary development of planet Earth.

Group “Astronomers”

1. Find connections between geometry and astronomy.
2. Find evidence of the sphericity of the Earth from the point of view of astronomy.
3. Find materials about the structure of the solar system.

Group “Philosophers”

1. Find material that connects the geometric body - the sphere with the concepts of philosophy.
2. Determine the types of spheres from the point of view of philosophy.

Group “Art Critics”

Find paintings and engravings that depict the sphere.

Group “Academic Council”

Summarize the lesson and evaluate the work of each group.

Reporting materials

• Summary posters.
• Drawings.
• Messages.
• Collection of problems.
• Presentation (in this article, graphic material from the presentation is used as illustrations).

Lesson type: generalization of knowledge gained in the geometry course about the sphere and ball.

Methods and techniques of work: implementation of design and research technologies.

Equipment:

• Textbook of geometry 10-11, authors L.S. Atanasyan, V.F. Butuzov and others.
• Slides, posters.
• Encyclopedic dictionaries.
• Sphere and ball models.
• Globe, map.

Dear guys! Today’s lesson is a general lesson on the topic “Sphere and Ball”, and it takes place within the framework of design and research technologies. In the lesson we will generalize knowledge about the sphere and ball, and also learn something new about these concepts from other fields of science. Not a single science has ignored these geometric concepts. Many real objects in astronomy, biology, chemistry and other natural sciences have the shape of a sphere and a ball. In various historical eras, the study of these concepts has been and continues to play a significant role.

The epigraph to our lesson will be the words of Wiener: “The highest purpose of geometry is precisely to find hidden order in the chaos that surrounds us.”

Today we will try to streamline the chaos reigning around the sphere and ball.

The following working groups took part in preparing the lesson:

– mathematicians;
– geographers;
– astronomers;
– philosophers;
– art critics.

Each group had its own range of research questions. The general summary of the lesson will be “academic advice”. As usual, in your notebooks you write down the studies that interest you and the conclusions of the groups.

So, let’s write down in the notebooks the date of the lesson, the topic of the lesson (dictate). Today in the lesson we must answer the question “A ball and a sphere – are they ordinary geometric concepts or something more?”

Let's give the floor to a group of mathematicians.

1st student. Our group once again carefully studied the material about the ball and sphere, and then generalized it (a brief summary of the material from the textbook “Geometry 10-11” is considered).

2nd student. We also know what the relative position of the sphere and the plane is. Let R be the radius of the sphere, d be the distance from the center of the sphere to the plane. (Drawings from a textbook about the relative position of a sphere and a plane are considered.)

In addition, when solving problems on the topic “Sphere and ball”, we find its surface area and volume.

3rd student. Our group conducted research on all the definitions of sphere and ball that were found in the mathematical encyclopedic dictionary, in the Great Encyclopedic Dictionary, in the Brockhaus and Efron encyclopedia, in the old geometry textbook by the author Kiselev, published in 1907. And we came to the conclusion that the definitions of a ball and a sphere have undergone virtually no changes over time. For example, in the mathematical encyclopedic dictionary ball is a geometric body obtained by rotating a circle around its diameter; a ball is a set of points whose distance from a fixed point O (center) does not exceed a given R (radius).

The Big Encyclopedic Dictionary gives a similar definition.

In the Brockhaus and Efron encyclopedia ball – a geometric body bounded by a spherical or spherical surface. All points of the sphere are located at equal distances from the center. Distance is the radius of the ball.

In Kiselev’s geometry – a body resulting from the rotation of a semicircle around the diameter limiting it is called. a ball, and the surface formed by a semicircle is called. spherical or spherical surface. This surface is the locus of points equally distant from the same point, called the center of the ball.

Conclusion. So, as a result of the work done by our group, we came to the conclusion that for quite a long time the definitions of a sphere and a ball have not changed. We have prepared a collection of problems on the topic “Sphere and ball”, and we hope that these problems will help to apply theoretical knowledge about the sphere and ball in practice. To support our research, let's put theoretical knowledge into practice (students solve several problems).

Thanks to the group of mathematicians who summarized the material about the sphere and the ball, and also prepared a collection of practical problems. You and I know that the shape of a ball is very common in nature and in the environment around us. The most interesting object with a spherical surface is our planet Earth. Now a group of “geographers” will introduce us to their research. Please.

1st student. The purpose of our work is to study what the Earth was like in the ideas of the ancients, and how the formation of the Earth as a spherical surface took place. While preparing for the lesson, we found a book, or rather, pages from a book, from which we can judge that it was an encyclopedia for children, published before the 1917 revolution; this can be seen from the font.

So, in this book it is written that “a very long time ago people thought that the earth was flat, like a table, and that if you walked straight and straight, you could reach the end of the earth. But then scientists appeared who proved that the earth is a huge ball with no end.”

There is a poem in this book:

I've been standing for hundreds and hundreds of years,
There is no end or edge for me.
I stand like a strong hero,
And cover my chest
Deserts, steppes, mountain ranges,
Forests, fields, meadows,
Villages, villages, cities,
The seas are icy water.
I give shelter here and there,
Animals, people and beasts.
I feed everyone and sing to everyone,
I send my grace to everyone.
I am like a huge round ball!
I am God's work, God's gift!

On the screen we see our land as it is depicted on geographical maps.

2nd student. Continuing our research, we learned that the ancients considered the Earth to be a flat disk surrounded on all sides by the ocean. However, already at that time people began to wonder why water always occupies the lowest places (this applies to seas and oceans); Why is there a gradual appearance or removal of tall objects as you approach or move away from them? While traveling around the world, sailors noticed that when returning to the same place, there was a loss or gain of an entire day, which would be completely impossible if the Earth had the shape of a disk.

So, evidence of the sphericity of the Earth at present is:

1. Always a circular figure of the horizon in the ocean and in open lowlands or plateaus;
2. Gradual approach or removal of objects;
3. Traveling around the world.

3rd student. While studying various geographical maps, we discovered that in geography there are place names associated with the ball. For example, between the Northern and Southern islands of Novaya Zemlya there is a strait that connects the Barents and Kara seas, which is called Matochkin Shar, or a strait between the shores of Vaigach Island and the mainland of Eurasia - Yugorsky Shar. We think that these straits are called balls due to the fact that their size and bottom shape resemble a spherical surface.

Conclusion. Our group studied the Earth as a spherical surface. Of course, what we learned and shared with you is a small fraction of the enormous material about the Earth. We hope you are interested in our research and take the time to read something new.

A student from a group of mathematicians proposes to solve a problem to find the volume of a globe standing on a table.

Thanks to the group of “geographers”.

However, the Earth is not just the surface on which we move, it is also a planet in the solar system. How the study of the sphericity of the Earth took place in the field of astronomy - our “astronomers” will tell us about this.

1st student. Our group studied the Earth from an astronomical point of view. In the course of our research, we learned that in ancient times people believed that the Earth was flat. According to their ideas, the sky was something like an inverted bowl, along which the Sun and stars moved. This is how the Babylonians saw the Earth and the sky (drawing on the screen). However, the movement of people from place to place forced them to look for some signs to choose the right direction. One such sign was the stars.

Thus, from the very beginning of human life, knowledge of the Earth was combined with the study of the sky.

The first impetus for changing views on the shape of the Earth was given by the practice of observing the sky, to which people were forced to turn. They noticed that when moving long distances, the appearance of the sky also changes: some stars cease to be visible, others, on the contrary, appear above the horizon. This speaks in favor of the sphericity of the Earth. Observations of lunar eclipses, during which the round edge of the earth's shadow is invariably visible on the lunar disk, proved that the Earth is spherical.

Lived in the 4th century BC. The greatest Greek scientist Aristotle developed and substantiated the doctrine of the sphericity of the Earth. He believed that all “heavy” bodies tend to approach the center of the world and, gathering around this center, form the globe.

While studying the Earth from an astronomical point of view, our group discovered in an astronomy textbook from the 1939 edition a map of the Earth, which was compiled by the Greek scientist Hecataeus in the 5th century BC. (map on screen). In the same textbook we found a map of the Earth in the Middle Ages - the era of the dominance of the Christian Church. On the map, north is on the left, south is on the right. It depicts the “sacred” Lands, Jerusalem and an imaginary sacred paradise.

2nd student. For the first time, the scientist astronomer Ptolemy tried to unite all the information about the Earth that then existed. According to his teaching, the Earth has the shape of a ball and remains motionless. She is at the center of the world and is the goal of creation. All other celestial bodies exist for the Earth and revolve around it. Ptolemy's theory was geometrically correct and served the practical purpose of pre-calculating the positions of the Sun and planets.

3rd student. Pay attention to the model of the solar system, which is located on the table. You and I see all the planets of our system. The question is: why in this model, as in many others, are all the planets of the solar system represented as spheres? The fact is that, under the influence of the forces of mutual attraction, their entire mass is concentrated in the center and takes the shape of a body whose surface is the smallest. And from geometry we know that of all bodies of rotation, the ball has the smallest surface.

By the way, stars also have the shape of a ball, or, more correctly, a spherical shape.

The volume and surface area of ​​the planets of the solar system cannot be found without information from geometry. This is proven by the independent activity of the Pythagoreans in astronomy. Pythagoras himself taught that the Earth is spherical. The entire universe also has the shape of a ball, in the center of which the Earth freely holds itself. The Earth's axis is also the axis around which the Sun, Moon and planets describe their paths without hindrance. These bodies must have a spherical shape, like the Earth. Because for Pythagoras the ball was perfect. Between the Earth and the sphere of the fixed stars these bodies are located in the following order: Moon, Sun, Mercury, Venus, Mars, Jupiter and Saturn. Their distances from the Earth are in certain harmonic relationships with each other, the consequence of which is the euphony produced by the combined movement of the luminaries, or the so-called music of the spheres.

Conclusion. Our group hopes that you were interested, and you, like us, noticed that none of the sciences can do without geometry. In conclusion, we would like to draw your attention to the screen where you see a photograph of the Earth from space.

Thanks to a group of astronomers. The concept of a sphere, the term “sphere” is used not only in geometry, geography and astronomy. This term is also found in other fields of science. It’s not for nothing that we have a group of philosophers who will now share their research with us.

1st student. Walking in a shady grove, the Greek philosopher talked with his student. “Tell me,” asked the young man, “why are you overcome by doubts? You have lived a long life, are wise by experience and learned from the great Hellenes. How is it that so many unclear questions remain for you?”

In thought, the philosopher drew two circles in front of him with his staff: a small one and a large one. “Your knowledge is a small circle, and mine is a large one. But all that remains outside these circles is the unknown. A small circle has little contact with the unknown. The wider the circle of your knowledge, the greater its border with the unknown. And henceforth, the more you learn new things, the more unclear questions you will have.”

The Greek sage gave a comprehensive answer.

2nd student. Since our class is humanitarian, we decided to study the concept of sphere from a humanitarian point of view, namely, a philosophical one. Sphere is a general scientific concept that denotes the largest part of existence at any level: the universe, physical, chemical, biological, social and individual worlds.

In the social sciences, the concept of sphere has been used very widely and for a very long time. For example, there are 4 spheres of public life - economic, social, political and spiritual. The concept of sphere is one of the central and fundamental concepts of tetrasociology. It distinguishes: 4 spheres of social resources: people, information, organizations, things; 4 spheres of reproduction processes: production, distribution, exchange, consumption; 4 structural spheres of reproduction: social, informational, organizational, material; 4 spheres of states of social development: flourishing, slowing down, decline, death.

3rd student. There is a concept spheral democracy– a new form of democracy that arises in the information (global) society. The structural basis of spheral democracy are 4 spheres of social reproduction:

• sociosphere
– its subject and product are people who are reproduced through humanitarian technologies of education, healthcare, etc.
• infosphere
– its subject and product is information, which is reproduced by information technologies (both areas are directly related to us).
• orgsphere
– its subject and product are social relations (political, legal, financial, managerial)
• technosphere
– its subject and product are things that are reproduced by industrial and agricultural technologies.

4th student. There is also the concept spheral classes – these are 4 large productive groups of people covering the entire population.

• Socioclass –
healthcare, education, social security workers and the non-working population - preschoolers, students, housewives, pensioners and the disabled.
• Infoclass –
workers in the fields of science, culture, art, communications, information services.
• Organizational class –
workers in the fields of management, finance, credit, insurance, defense, state security, customs, Ministry of Internal Affairs, etc.
• Technoclass –
workers and peasants, workers in industries, agriculture and forestry, etc.

Spheral classes are inherent in the population of all countries of the world. Every person lives inside the so-called sphere. This is clearly presented on our table. All factors of the surrounding reality influence a person, and, consequently, the society in which he lives.

Conclusion. Everything we just talked about are the basic concepts of philosophy and sociology. We hope that these concepts will be useful to all of us in social studies lessons.

Thanks philosophers. They introduced us to the concept of sphere from a philosophical point of view. I think this information is very important for all of us. And at the end of the lesson, we will give the floor to art critics.

1st student. Our group also did not stand aside. We explored the work of the Dutch graphic artist Escher. His engravings are beautiful not only from an artistic point of view, but also no less beautiful from the point of view of geometry.

2nd student. Please look at the screen. You see the engravings: “Spirals on a sphere”, “Beech ball”, “Sphere with human figures”, “Three spheres”, “Concentric rings”. Aren't they beautiful? They contain the perfection of geometry, the so-called music of the spheres, which our astronomers spoke about. Escher's engravings contain the principle of symmetry, which can be more clearly seen on the sphere.

Thanks to art critics. Now it's time to give the floor to our academic council.

Thanks to the academic council. I think everyone agrees with him.

So, guys, today in the lesson we summarized the knowledge about the sphere and the ball, we learned a lot of new things. Returning to the epigraph of the lesson (read), we have brought a little order to the chaos that surrounds the sphere and ball.

Thanks to all groups. Your reporting material will be read and studied very carefully.

Homework: repeat everything about the sphere and the ball, prepare for the test work.

Thank you for the lesson. The lesson is over. Goodbye.

Slide 2

A sphere is a surface that consists of all points in space located at a given distance from a given point. This point is called the center, and the given distance is the radius of the sphere, or ball - a body bounded by a sphere. A ball consists of all points in space located at a distance no more than a given point from a given point.

Slide 3

The segment connecting the center of the ball with a point on its surface is called the radius of the ball. A segment connecting two points on the surface of a ball and passing through the center is called the diameter of the ball, and the ends of this segment are called diametrically opposite points of the ball.

Slide 4

What is the distance between diametrically opposite points of the ball if the distance of the point lying on the surface of the ball from the center is known? ? 18

Slide 5

A ball can be considered as a body obtained by rotating a semicircle around a diameter as an axis.

Slide 6

Let the area of ​​the semicircle be known. Find the radius of the ball, which is obtained by rotating this semicircle around the diameter. ? 4

Slide 7

Theorem. Any section of a ball by a plane is a circle. A perpendicular dropped from the center of the ball onto a cutting plane ends up in the center of this circle.

Given: Prove:

Slide 8

Proof:

Consider a right triangle whose vertices are the center of the ball, the base of a perpendicular dropped from the center onto the plane, and an arbitrary section point.

Slide 9

Consequence. If the radius of the ball and the distance from the center of the ball to the section plane are known, then the radius of the section is calculated using the Pythagorean theorem.

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Let the diameter of the ball and the distance from the center of the ball to the cutting plane be known. Find the radius of the circle of the resulting section. ? 10

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The smaller the distance from the center of the ball to the plane, the larger the radius of the section.

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A ball of radius five has a diameter and two sections perpendicular to this diameter. One of the sections is located at a distance of three from the center of the ball, and the second is at the same distance from the nearest end of the diameter. Mark the section whose radius is larger. ?

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Task.

Three points are taken on a sphere of radius R, which are the vertices of a regular triangle with side a. At what distance from the center of the sphere is the plane passing through these three points? Given: Find:

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Consider a pyramid with the top in the center of the ball and the base in this triangle. Solution:

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Let's find the radius of the circumscribed circle, and then consider one of the triangles formed by the radius, the lateral edge of the pyramid and the height. Let's find the height using the Pythagorean theorem. Solution:

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The largest radius of the section is obtained when the plane passes through the center of the ball. The circle obtained in this case is called a great circle. A large circle divides the ball into two hemispheres.

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In a ball whose radius is known, two large circles are drawn. What is the length of their common segment? ? 12

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A plane and a line, tangent to a sphere.

A plane that has only one common point with a sphere is called a tangent plane. The tangent plane is perpendicular to the radius drawn to the point of tangency.

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Let a ball whose radius is known lie on a horizontal plane. In this plane, through the point of contact and point B, a segment is drawn, the length of which is known. What is the distance from the center of the ball to the opposite end of the segment? ? 6

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A straight line is called tangent if it has exactly one common point with the sphere. Such a straight line is perpendicular to the radius drawn to the point of contact. An infinite number of tangent lines can be drawn through any point on the sphere.

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Given a ball whose radius is known. A point is taken outside the ball and a tangent to the ball is drawn through it. The length of the tangent segment from a point outside the ball to the point of contact is also known. How far from the center of the ball is the outer point? ? 4

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The sides of the triangle are 13cm, 14cm and 15cm. Find the distance from the plane of the triangle to the center of the ball touching the sides of the triangle. The radius of the ball is 5 cm. Problem. Given: Find:

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The section of the sphere passing through the points of contact is a circle inscribed in triangle ABC. Solution:

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Let's calculate the radius of a circle inscribed in a triangle. Solution:

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Knowing the radius of the section and the radius of the ball, we will find the required distance. Solution:

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Through a point on a sphere whose radius is given, a great circle and a section are drawn intersecting the plane of the great circle at an angle of sixty degrees. Find the cross-sectional area. ? π

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The relative position of two balls.

If two balls or spheres have only one common point, then they are said to touch. Their common tangent plane is perpendicular to the line of centers (the straight line connecting the centers of both balls).

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The contact of the balls can be internal or external.

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The distance between the centers of two touching balls is five, and the radius of one of the balls is three. Find the values ​​that the radius of the second ball can take. ? 2 8

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Two spheres intersect in a circle. The line of centers is perpendicular to the plane of this circle and passes through its center.

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Two spheres of the same radius, equal to five, intersect, and their centers are at a distance of eight. Find the radius of the circle along which the spheres intersect. To do this, it is necessary to consider the section passing through the centers of the spheres. ? 3

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Inscribed and circumscribed spheres.

A sphere (ball) is said to be circumscribed about a polyhedron if all the vertices of the polyhedron lie on the sphere.

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What quadrilateral can lie at the base of a pyramid inscribed in a sphere? ?

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A sphere is said to be inscribed in a polyhedron, in particular, in a pyramid, if it touches all the faces of this polyhedron (pyramid).

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At the base of a triangular pyramid lies an isosceles triangle, the base and sides are known. All lateral edges of the pyramid are equal to 13. Find the radii of the circumscribed and inscribed spheres. Task. Given: Find:

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Stage I. Finding the radius of the inscribed sphere.

1) The center of the circumscribed ball is removed from all the vertices of the pyramid at the same distance equal to the radius of the ball, and in particular, from the vertices of triangle ABC. Therefore, it lies on the perpendicular to the plane of the base of this triangle, which is reconstructed from the center of the circumscribed circle. In this case, this perpendicular coincides with the height of the pyramid, since its side edges are equal. Solution.