Abstract and presentation of the lesson "regular polygons". Presentation on regular polygons Presentation on regular polygons

slide 1

slide 2

Definition of a regular polygon. A regular polygon is a convex polygon in which all sides and all (internal) angles are equal.

slide 3

slide 4

A circle circumscribed about a regular polygon. Theorem: around any regular polygon, you can describe a circle, and moreover, only one. A circle is said to be circumscribed about a polygon if all its vertices lie on this circle.

slide 5

A circle inscribed in a regular polygon. A circle is said to be inscribed in a polygon if all sides of the polygon touch the circle. Theorem: In any regular polygon, you can inscribe a circle, and moreover, only one.

slide 6

Let А1 А 2 …А n be a regular polygon, О be the center of the circumscribed circle. When proving Theorem 1, we found out that ∆ OA1A2 = ∆OA2A3= ∆OAnA1 , so the heights of these triangles drawn from the vertex O are also equal. Therefore, a circle with center O and radius OH passes through the points H1, H2, Hn and touches the sides of the polygon at these points, i.e. the circle is inscribed in the given polygon. Given: ABCD…An is a regular polygon. Prove that any regular polygon can be inscribed with a circle, and moreover, only one.

Slide 7

Let us prove that there is only one inscribed circle. Suppose there is another inscribed circle with center O and radius OA. Then its center is equidistant from the sides of the polygon, i.e. the point O1 lies on each of the angle bisectors of the polygon, and therefore coincides with the point O of the intersection of these bisectors.

Slide 8

A D B C O Given: ABCD…An is a regular polygon. Prove that it is possible to draw a circle around any regular polygon, and moreover, only one. Proof: Let's draw the bisectors BO and CO of equal angles ABC and BCD. They will intersect, since the corners of the polygon are convex and each is less than 180⁰. Let the point of their intersection be O. Then, after drawing the segments OA and OD, we obtain ΔBOA, ΔBOC and ΔCOD. ΔBOA \u003d ΔBOC according to the first criterion for the equality of triangles (BO - general, AB \u003d BC, angle 2 \u003d angle 3). Similarly, ΔVOC=ΔCOD. 1 2 3 4 angle2 = angle 3 as halves of equal angles, then ΔBOC is isosceles. This triangle is equal to ΔBOA and ΔCOD => they are also isosceles, so OA=OB=OC=OD, i.e. points A, B, C and D are equidistant from the point O and lie on the circle (O; OB). Similarly, other vertices of the polygon lie on the same circle.

Slide 9

Let us now prove that there is only one circumscribed circle. Consider any three vertices of the polygon, for example, A, B, C. only one circle passes through these points, then only one circle can be circumscribed near the polygon ABC...An. o A B C D

slide 10

Consequences. Corollary #1 A circle inscribed in a regular polygon touches the sides of the polygon at their midpoints. Corollary No. 2 The center of a circle circumscribed near a regular polygon coincides with the center of a circle inscribed in the same polygon.

slide 11

Formula for calculating the area of ​​a regular polygon. Let S be the area of ​​a regular n-gon, a1 its side, P the perimeter, and r and R the radii of the inscribed and circumscribed circles, respectively. Let's prove that

slide 12

To do this, connect the center of the given polygon with its vertices. Then the polygon will be divided into n equal triangles, the area of ​​each of which is equal to Therefore,

slide 13

Formula for calculating the side of a regular polygon. Let's derive the formulas: To derive these formulas, we will use the figure. In a right triangle А1Н1О O А1 А2 А3 Аn H2 H1 Hn H3 Therefore,

slide 14

Assuming in the formula n = 3, 4 and 6, we obtain expressions for the sides of a regular triangle, square and regular hexagon:

slide 15

Task No. 1 Given: circle (O; R) Construct a regular n-gon. the circle is divided into n equal arcs. To do this, draw the radii OA1, OA2, ..., OAn of this circle so that the angle A1OA2 = angle A2OA3 = ... = angle An-1OAn = angle AnOA1 = 360 ° / n (in the figure n = 8). If we now draw the segments A1A2, A2A3, ..., An-1An, AnA1, then we get the n-gon A1A2 ... An. Triangles А1ОА2, А2ОА3,…, АnОА1 are equal to each other, therefore А1А2= А2А3=…= Аn-1Аn= АnА1. It follows that A1A2…An is a regular n-gon. Construction of regular polygons.

slide 16

Task №2 Given: A1, A2...An - regular n-gon Construct a regular 2n-gon Solution. Let's describe a circle around it. To do this, we construct the bisectors of the angles A1 and A2 and denote by the letter O the point of their intersection. Then draw a circle with center O of radius OA1. Divide the arcs A1A2, A2A3..., An A1 in half. Each of the division points B1, B2, ..., Bn will be connected by segments with the ends of the corresponding arc. To construct points B1, B2, ..., Bn, you can use the perpendicular bisectors to the sides of the given n-gon. In the figure, a regular dodecagon A1 B1 A2 B2 ... A6 B6 is constructed in this way.

To use the preview of presentations, create a Google account (account) and sign in: https://accounts.google.com

Slides captions:

REGULAR POLYGONS (geometry grade 9) Volodina n.l.

Lesson objectives: 1. Repeat the concept of a polygon, the formula for the sum of the angles of a convex polygon. 2. Introduce regular polygons, teach how to build regular polygons. 3. To form the skills of solving problems on the topic.

ORAL QUESTIONS: 1. What is the sum of the angles of a convex polygon? (n - 2) ∙ 180 ⁰ 2. How to find one corner of a hexagon if all corners are equal? (6 - 2) ∙ 180 ⁰ / 6 = 120⁰ 3. How to find the angle of an n-gon if all angles are equal? (n - 2) ∙ 180 ⁰ / n

What is the sum of the angles of a triangle? 180⁰

The sum of the angles of a polygon 1. What is the sum of the angles of a convex quadrilateral? 360 ⁰ 2. What is the sum of the angles of a convex hexagon? 720⁰

Divide the polygons into two groups

REGULAR POLYGONS Arbitrary polygons

DEFINITION: A convex polygon is called regular if all sides are equal and all angles are equal.

Right Triangle Equilateral Triangle All sides are equal. All angles are 60.⁰

Regular quadrilateral Square All sides are equal. All angles are 90.⁰

Regular pentagon All sides are equal All angles are 108⁰

Regular hexagon All sides are equal All angles are 120⁰

FINAL QUESTIONS: 1. What polygon is called correct? 2. Does a regular 10-gon exist? 20-gon? 3.How to build a regular polygon?

On the topic: methodological developments, presentations and notes

Non-standard geometry lesson in grade 9. The game "Mathematician - businessman" on the topic "Regular polygons. Circumference and area of ​​a circle...

Development of a lesson in geometry Grade 9 "Formulas for calculating the area of ​​​​a regular polygon, its side and radius of an inscribed circle"

Development of a lesson-study of new material on geometry in grade 9 "Formulas for calculating the area of ​​\u200b\u200ba regular polygon, its side and radius of an inscribed circle" Lesson summary on geomet...

Regular polygons. Order and chaos.

Abstract of a geometry lesson in grade 9 on the topic: "Regular polygons. Order and chaos." One topic is subject, the second is meta-subject ....

Presentation "The area of ​​a regular polygon"

The presentation for the lesson geometry in grade 9 contains the necessary definitions and formulas for calculating the area of ​​​​regular polygons ....

To use the preview of presentations, create a Google account (account) and sign in: https://accounts.google.com

Slides captions:

A polyhedron is a body whose surface consists of a finite number of flat polygons.

Regular polyhedra

How many regular polyhedra are there? - How are they defined, what properties do they have? -Where do they meet, do they have practical application?

A convex polyhedron is called regular if all its faces are equal regular polygons and the same number of edges converge at each of its vertices.

"hedra" - face "tetra" - four hexes "- six "octa" - eight "dodeca" - twelve "icos" - twenty The names of these polyhedra came from ancient Greece and they indicate the number of faces.

Name of a regular polyhedron Type of face Number of vertices of edges of faces of faces converging at one vertex Tetrahedron Regular triangle 4 6 4 3 Octahedron Regular triangle 6 12 8 4 Icosahedron Regular triangle 12 30 20 5 Cube (hexahedron) Square 8 12 6 3 Dodecahedron Regular pentagon 20 30 12 3 Data on regular polyhedra

Question (problem): How many regular polyhedra are there? How to set their number?

α n = (180 °(n -2)) : n Each vertex of the polyhedron has at least three flat angles, and their sum must be less than 360 ° . Shape of faces Number of faces at one vertex Sum of plane angles at the vertex of a polyhedron Conclusion about the existence of a polyhedron α = 3 α = 4 α = 5 α = 6 α = 3 α = 4 α = 3 α = 4 α = 3

L. Carroll

The great mathematicians of antiquity Archimedes Euclid Pythagoras

The ancient Greek scientist Plato described in detail the properties of regular polyhedra. That is why regular polyhedra are called Platonic solids.

tetrahedron - fire cube - earth octahedron - air icosahedron - water dodecahedron - universe

Polyhedra in space and earth sciences

Johannes Kepler (1571-1630) German astronomer and mathematician. One of the founders of modern astronomy - discovered the laws of planetary motion (Kepler's laws)

Kepler Cup Space

"Ecosahedron - dodecahedron structure of the Earth"

Polyhedra in art and architecture

Albrecht Dürer (1471-1528) "Melancholia"

Salvador Dali "The Last Supper"

Modern architectural structures in the form of polyhedrons

Alexandrian lighthouse

Brick polyhedron by a Swiss architect

Modern building in England

Polyhedra in nature

Pyrite (sulphurous pyrites) Monocrystal of potassium alum Crystals of red copper ore NATURAL CRYSTALS

Table salt consists of crystals in the form of a cube. The mineral sylvin also has a crystal lattice in the form of a cube. Water molecules are shaped like a tetrahedron. The mineral cuprite forms crystals in the form of octahedrons. Pyrite crystals are shaped like a dodecahedron

Diamond Diamond, sodium chloride, fluorite, olivine and other substances crystallize in the form of an octahedron.

Historically, the first form of cut that appeared in the XIV century was the octahedron. Diamond Shah Diamond weight 88.7 carats

The task The Queen of England instructed to cut along the edges of the diamond with gold thread. But the cut was not made, because the jeweler was unable to calculate the maximum length of the gold thread, and the diamond itself was not shown to him. The jeweler was given the following data: the number of vertices B=54, the number of faces G=48, the length of the largest edge L=4mm. Find the maximum length of the golden thread.

Regular polyhedron Number of Faces Vertices Edges Tetrahedron 4 4 6 Cube 6 8 12 Octahedron 8 6 12 Dodecahedron 12 20 30 Icosahedron 20 12 30 Research work "Euler's Formula"

Euler's theorem. For any convex polyhedron В + Г - 2 = Р where В is the number of vertices, Г is the number of faces, Р is the number of edges of this polyhedron.

PHYSMINUTE!

Problem Find the angle between two edges of a regular octahedron that have a common vertex but do not belong to the same face.

Problem Find the height of a regular tetrahedron with an edge of 12 cm.

The crystal has the shape of an octahedron, consisting of two regular pyramids with a common base, the edge of the base of the pyramid is 6 cm. The height of the octahedron is 8 cm. Find the lateral surface area of ​​\u200b\u200bthe crystal

Surface area Tetrahedron Icosahedron Dodecahedron Hexahedron Octahedron

Homework: mnogogranniki.ru Using the developments, make models of the 1st regular polyhedron with a side of 15 cm, the 1st semi-regular polyhedron

Thank you for your work!

Lesson on the topic "Regular polygons"

Lesson Objectives:

educational: introduce students to the concept and types of regular polygons, with some of their properties; teach how to use the formula for calculating the angle of a regular polygon

- developing:

- educational:

Course of the lesson:

1. Organizational moment

Lesson motto:

Three paths lead to knowledge:

Chinese philosopher and sage Confucius.

2. Lesson motivation.

Dear Guys!

I hope that this lesson will be interesting, with great benefit for everyone. I really want those who are still indifferent to the queen of all sciences to leave our lesson with a deep conviction that geometry is an interesting and necessary subject.

The French writer of the 19th century, Anatole France, once remarked: “Learning can only be fun ... To digest knowledge, you must absorb it with appetite.”

Let's follow the writer's advice in today's lesson: be active, attentive, absorb with great desire the knowledge that will be useful to you later in life.

3. Actualization of basic knowledge.

Front poll:

What are their elements?

Polygon views

4. Learning new material.

Among the many different geometric shapes on the plane, a large family of POLYGONS stands out.

The names of geometric shapes have a very definite meaning. Look closely at the word "polygon", and say what parts it consists of. The word "polygon" indicates that all the figures of this family have "many corners".

Substitute in the word “polygon” instead of the “many” part a specific number, for example 5. You will get a PENTAGON. Or 6. Then - HEXAGON. Notice how many angles, so many sides, so these figures could well be called multilaterals.

The figure shows geometric shapes. Name these figures using the drawing.

Definition.A regular polygon is a convex polygon in which all angles are equal and all sides are equal.

You are already familiar with some regular polygons - an equilateral triangle (regular triangle), a square (regular quadrilateral).

Let's get acquainted with some properties that all regular polygons have.

The sum of the angles of a polygon
n - number of sides
n-2 - number of triangles
The sum of the angles of one triangle is 180º, multiply by the number of triangles n-2, we get S= (n-2)*180.

S=(n-2)*180
Formula for calculating the angle x of a regular polygon .
We derive a formula for calculating angle x of a regular n-gon.
In a regular polygon, all angles are equal, divide the sum of the angles by the number of angles, we get the formula:
x=(n-2)*180/n

5. Consolidation of new material.

Decide #179, 181, 183(1), 184.

Without turning your head, look around the classroom wall clockwise around the perimeter, the chalkboard around the perimeter counterclockwise, the triangle depicted on the stand clockwise and its equal triangle counterclockwise. Turn your head to the left and look at the horizon line, and now at the tip of your nose. Close your eyes, count to 5, open your eyes and...

We put our hands to our eyes,
Let's set our legs strong.
Turning to the right
Let's look majestic.
And to the left too
Look from under the palms.
And - to the right! And further
Over the left shoulder!
and now we will continue to work.

7. Independent work of students.

Solve #183(2).

8. The results of the lesson. Reflection. D / s.

What do you remember most about the lesson?

What surprised?

What did you like the most?

How would you like to see the next lesson?

D / s. Learn item 6. Solve No. 180, 182 185.

Creative task:

Internet :

View presentation content
"regular polygons"

• - educational: to acquaint students with the concept and types of regular polygons, with some of their properties; teach how to use the formula for calculating the angle of a regular polygon
• - developing: development of cognitive activity, spatial imagination, the ability to choose the right solution, concisely express one's thoughts, analyze and draw conclusions.
• - educational: fostering interest in the subject, the ability to work in a team, a culture of communication.

Lesson motto:

Three paths lead to knowledge:

The way of reflection is the most noble way;

The way of imitation is the easiest way;

The path of experience is the most bitter path.

Chinese philosopher and sage

Confucius.

• What geometric shapes have we already studied?
• What are their elements?
• What shape is called a polygon?
• Polygon views
• What is the perimeter of a polygon?
• What is the sum of the interior angles of the polygon?

Incorrect Correct polygons

• A convex polygon is called regular if all its angles are equal and all sides are equal.

Properties of regular polygons

Sum of angles

polygon

n - number of sides n-2 - number of triangles The sum of the angles of one triangle is 180º, 180º is multiplied by the number of triangles (n -2), we get S= (n-2)*180.

The formula for calculating the right angle P - square

in the right P- in a square, all angles are equal, divide the sum of the angles by the number of angles, we get the formula:

a n =(n-2)*180/n

Test Choose the numbers of the correct statements.

• A convex polygon is regular if all its sides are equal.
• Any regular polygon is convex.
• Any quadrilateral with equal sides is correct.
• A triangle is regular if all its angles are equal.
• Any equilateral triangle is correct.
• Any convex polygon is regular.
• Any quadrilateral with equal angles is regular.

Independent work

a P =(n-2)*180/n

a 3 =(3-2)*180/3= 180/3= 60

Homework

No. 1079 (oral), No. 1081 (b, e), No. 1083 (b)

Creative task:

*Historical information about regular polygons. Possible queries for web search engine Internet :

• Polygons in the school of Pythagoras. Construction of polygons, Euclid. Regular polygons, Claudius Ptolemy.
• Polygons in the school of Pythagoras.
• Construction of polygons, Euclid.
• Regular polygons, Claudius Ptolemy.

slide 3

Regular polygons

slide 4

“Three qualities: extensive knowledge, the habit of thinking and nobility of feelings are necessary for a person to be educated in the full sense of the word.” N.G. Chernyshevsky

slide 5

slide 6

Simonov Monastery

Slide 7

Do you know?

What geometric shapes have we already studied? What are their elements? What shape is called a polygon? What is the smallest number of sides a polygon can have? What is a convex polygon? Show in the figure convex and non-convex polygons. Explain what angles are called corners of a convex polygon, external corners. What is the formula for calculating the sum of the angles of a convex polygon? What is the perimeter of a polygon?

Slide 8

Crossword questions: Sides, angles and vertices of a polygon? What is a polygon with equal sides and angles called? 3. What is the name of a figure that can be divided into a finite number of triangles? 4. Part of a circle? 5.Polygon border? 6. Circle element? 7.Polygon element? 8. Circle border? 9.Polygon with the smallest number of sides? 10. An angle whose vertex is at the center of the circle? 11. Another kind of circle angle? 12. The sum of the lengths of the sides of a polygon? 13. A polygon that is in one half-plane relative to a straight line containing any of its sides?

Slide 9

Slide 10

slide 11

What is each of the corners of a regular a) decagon; b) n-gon.

slide 12

Angle of a regular n-gon

slide 13

Slide 14

Practical work. 1. The seven-headed tower of the White City was a regular hexagon in plan, all sides of which are 14 m. Draw a plan for this tower. 2. Measure the angle AOB. What part of its value is the value of the total angle O? How can you calculate the value of this angle, knowing the number of sides of the polygon? 3.Measure the angle CAK - the outer corner of the polygon. Calculate the sum of the outer angle CAK and the inner angle CAB. Why do these angles always add up to 180°? What is the sum of the exterior angles of a regular hexagon, taken one at each vertex?

slide 15

slide 16

The base diameter of the Dulo tower is 16m. Draw a plan for the base of a 16-sided tower, using the angle at which the side of the polygon is visible from the center of the circle. Calculate the interior and exterior angles of this 16-gon. What is the sum of the exterior angles of a regular 16-gon, taken one at each vertex? What is the sum of the exterior angles of a regular n-gon, taken one at each vertex? No. 1082, 1083.