Presentation on the topic "Round bodies: cylinder, cone, ball." Presentation on the topic "round geometric bodies" Tangent plane to a sphere

Slide 11

Historical information about the cone

CYLINDER.. The word “cylinder” comes from the Greek kylindros, which means “roller”, “roller”. CONE. The Latin word conus is borrowed from the Greek (konos - plug, bushing, pine cone). In Book XI of “Elements” the following definition is given: if a right triangle rotating around one of its legs returns again to the same position from which it began to move, then the described figure will be a cone. Euclid considers only

Slide 12

Sphere A sphere is a surface consisting of all points in space located at a given distance from a given point. Radius is a segment connecting the center with any point on the sphere. Diameter is a segment connecting two points of the sphere and passing through its center. A chord is a segment connecting any two points on a sphere.

Slide 13

Area of ​​a sphere

For the area of ​​a sphere, we take the limit of the sequence of surface areas of polyhedra described around the sphere as the largest size of each face tends to zero. S=4πR^2

Slide 14

Tangent plane to a sphere

A tangent plane to a sphere is a plane that has only one common point with the sphere. The point of contact is their common point. Theorem: The radius of a sphere drawn to the point of contact between the sphere and the plane is perpendicular to the tangent plane. Theorem: If the radius of a sphere is perpendicular to the plane passing through its end lying on the sphere, then this plane is tangent to the sphere

Slide 15

Historical information about the sphere

However, both the words "ball" and "sphere" come from the same Greek word "sphaira" - ball. Moreover, the word “ball” was formed from the transition of the consonants sf to sh. In ancient times, the sphere was held in high esteem. Astronomical observations of the firmament invariably evoked the image of a sphere. The Pythagoreans taught about the existence of ten spheres of the Universe, through which celestial bodies supposedly move. They argued that the distances of these bodies from each other are proportional to the intervals of the musical scale. This was seen as elements of world harmony. Pythagoras’s “music of the spheres” was contained in such semi-mystical reasoning. Aristotle believed that the spherical shape, as the most perfect, is characteristic of the Moon, Sun, Earth and all world bodies. Developing the views of Eudoxus, he believed that the Earth was surrounded by a number of concentric spheres. The sphere has always been widely used in various fields of science and technology. In Book XI of the Elements, Euclid defines a ball as a figure described by a semicircle rotating around a fixed diameter.

Slide 16

Vodovzvodnaya Tower The Vodovzvodnaya Tower was built in 1488. The former name of the tower - Sviblova - is associated with the nearby courtyard of the boyar Sviblova. In 1633, a water pumping machine was installed in the tower to pump water into a reservoir located at the top of the tower. The water spread through the pipes throughout the Kremlin. In 1805-1806, the tower was dismantled and rebuilt according to the design of the architect I.V. Egotov. In 1812, the tower was blown up by the French, and in 1819 it was restored under the leadership of O.I. Bove. The height of the tower to the star is 57.7 meters, with the star - 61.25 meters. The tower is a cylinder. The tower is round in section.

Slide 17

Krivoarbatsky lane, house 10. Two huge white cylinders leaning against each other. Along the perimeter there are sixty small diamond-shaped windows, creating the image of a beehive. On the façade there is a gigantic window, several meters long. Above the window there is an inscription: "Konstantin Melnikov. Architect." The most famous (even iconic) building of the 20s in Moscow. Konstantin Stepanovich Melnikov was born in Moscow in the family of a construction worker, a native of the peasantry, in 1890. After graduating from parish school, he worked as a “boy” in the company “Trading House Zalessky and Chaplin”. Chaplin helped him enroll in 1905. B Moscow School of Painting, Sculpture and Architecture, and then after graduating from Melnikov in 1913. painting department advised him to continue his studies at the Architectural Department, which Konstantin Stepanovich graduated from in 1917. During his senior years at the School and in the first years after graduation, Melnikov worked in the spirit of neoclassicism. However, already in the early 20s, Konstantin Stepanovich sharply broke with various kinds of traditionalist stylizations. The very fact of the widespread implementation of his works forces us to take a different view of those of his works that remained in the projects and which in the 1920s were often declared “fantastic” in the heated polemics of that period. In Melnikov’s projects, one is struck by the degree of uninhibitedness of the master’s creative imagination in matters of form-building. We can say with complete confidence that in the 20th century. There was no other architect who would have created so many fundamentally new projects and such a level of novelty that their originality not only severely separated them from the works of other masters, but also just as strongly distinguished them from the works of their author himself.

According to the definition adopted in 2006 by the International Astronomical Union, a planet is a body orbiting the Sun that is massive enough to have a spherical shape under the influence of its own gravity, in addition, must have space near its orbit that is free from other bodies. If you pay attention to the first part of this formulation, you can ask the question - what is the minimum size of a body so that it has the shape of a ball?

It is believed that this figure is approximately 400 kilometers. At least in our solar system, the 397-kilometer-long Mimas is spherical, making it the smallest known round body.

Mimas

Proteus

Date of: 23.12.2017

Teacher: Kuksenko Natalya Nikolaevna

Item: mathematics

Class: 6

Subject: Round bodies

Formed UUD: the ability to plan ways to achieve goals; the ability to adequately assess the degree of objective and subjective difficulty in completing a learning task

Target: introduce students to geometric bodies: ball, cone, cylinder - and their elements.

Tasks:

be able to operate with the concepts: ball, cone, cylinder, base, height, vertex, sphere, center, radius, diameter, circular sector, section when performing various tasks; be able to recognize studied geometric shapes; be able to give examples of objects that have the shape of the studied bodies of revolution; be able to talk about a ball, cone, cylinder according to plan.

During the classes:

Update

Oral survey.

1. The radii of the circles are 3 cm and 5 cm. What is their relative position if the distance between the centers is

a) 8 cm?; b) 10 cm; c) 6 cm; d) 0

2. Name the equal elements in triangles.

A)

b)

2. Problematization (learning task)

Correctly read the statement written without spaces: Mathematics is the queen of all sciences. Her beloved is truth, her people are simplicity and clarity. The palace of this mistress is surrounded by thorny thickets, and in order to reach it, everyone has to make their way through the thicket. A casual traveler will not find anything attractive in the palace. Its beauty is revealed only to a mind that loves the truth, tempered in the fight against difficulties... (Snyadeck and Jan).

Goal setting

In this lesson you will learn three new geometric shapes. To better understand new material, be attentive, active and smart. The topic of the lesson is encrypted using puzzles. Solve them and you will find out what geometric shapes we will study today.

So, the topic of the lesson is “Round bodies”

- Let's write down the topic of the lesson in a notebook.

What is the purpose of our lesson?

4. Main part

1) Remember what figure was encrypted in the rebus in the form of a hat?

What other objects are cylindrical in shape?

It turns out that the word “cylinder” comes from the Greek word “kylindros”, meaning “roller”, “skating rink”.

At the turn of the 18th - 19th centuries, men in many countries wore hard hats with small brims, which were called top hats because of their great resemblance to the geometric shape of a cylinder.

Let's look carefully at the cylinder (demonstration of the model) and see that the cylinder consists of two bases located in parallel planes and a side surface.

A cylinder is obtained by rotating a rectangle around one of its sides.

What are the bases of the cylinder?

What can you say about the size of these circles?

What is the lateral surface of the cylinder?

Look at the development of the cylinder. What is the lateral surface of the cylinder?

A cylinder has parameters - height and radius.

Let's try to formulate the definition of the height and radius of a cylinder ourselves.

So, height is a segment connecting the centers of the bases, perpendicular to each of them; cylinder radius - the radius of the circle that is the base of the cylinder.

Practical task.

Roll the side surface of the cylinder from a rectangular sheet. What is its height?

Imagine a situation where we need to cut a cylinder.

How can this be done and what will happen in the cross section of the cylinder?

2) - Now let’s move on to considering the cone.

The word "cone" comes from the Greek word "konos", meaning pine cone (showing pine cone). Indeed, there are some similarities.

What objects have the shape of a cone?

The cone consists of a base and a side surface.

A cone is obtained by rotating a right triangle about its right angle side.

What is the base of the cone?

What is the lateral surface?

We will see what the side surface is by unfolding the paper cone onto a plane. The lateral surface of the cone unfolds into a circular sector - a part of the circle limited by two radii.

A cone has a vertex, a height, and a base radius

Let's formulate a definition.

So, the height is a perpendicular drawn from the top of the cone to the center of the base.

If we cut off the vertex and upper part of the cone (I show it on the model), then we get the so-called truncated cone.

- Think and tell me which objects have the shape of a cone or a truncated cone?

How can you cut a cone and what will happen in its cross section?

It turns out that sections of a cone can have the shape of other geometric figures, the names of which we don’t even know yet; we will study them in high school, and therefore we will not talk about them for now

3) - Let's move on to studying the ball.

Give examples of surrounding objects that are spherical in shape.

What do you think a ball has in common with a circle and a circle?

The ball is obtained by rotating a semicircle around a diameter.

The surface of the ball is calledsphere.The word "sphere" comes from the Greek word "sphaira", which is translated into Russian as "ball". There is no need to confuse the concepts of “ball” and “sphere”. A sphere is, one might say, the shell or boundary of a ball.

A ball, a globe are spheres, but a watermelon, an orange, the Sun, the Moon, the Earth and the rest of the planets have the shape of a slightly flattened ball (shows the picture).

Try calling the sections of a ball planes.

Which section will be the largest?

So, we got acquainted with three spatial figures, otherwise they are called geometric bodies. In 5th grade you were introduced to polyhedra. Let's remember their names.

Why were they called polyhedra?

What would you call the new geometric bodies?

Indeed, all geometric bodies are divided into two groups: polyhedra and bodies of rotation.

Working with the textbook

7. Assessment

- Let's generalize our knowledge by completing a test in a notebook.

Problem No. 1. What shape is the tower made of? Call from top to bottom.

(Cone, cube, cylinder)

Task No. 2. The figure shows various geometric bodies. Which of them are polyhedra?

Second (pyramid), third (oblique prism)

Task No. 3. The figure in the first line shows the front view of the figure, and the second line shows the top view of the figure. What figure is this?

1.Cone. 2.Cylinder 3. Quadrangular pyramid. 4. Rectangular parallelepiped. 5.Triangular pyramid. 6.Ball.

Problem No. 4. On a round table there are three cones of different colors - red, blue and green. Children are sitting around the table: Masha, Vanya, Dasha, Kolya, Raya and Petya. Which of the children sees such a picture as shown in the figure under the letter a); b); V)?

a B C)

(Petya) (Vanya) (Masha)

Problem No. 5. The figure shows some geometric bodies. Perhaps the point of view is not very familiar. What bodies, if viewed from the appropriate angle, might look like the one in the picture? Which of the drawings can correspond to the same body?

1. Cube or parallelepiped. 2. Pyramid or cone. 3. Cone, cylinder or ball. 4. Parallelepiped. Figures 2 and 3 can correspond to a cone, and 1 and 4 to a parallelepiped.

8. Reflection

If you think that you understand the topic of the lesson, then draw a green circle.

If you think that you have not learned the material enough, then draw a blue circle.

If you think that you did not understand the topic of the lesson, then draw a red circle.

9. Perspective (homework) № 446, 448