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From history From history Regular polygons were known in ancient times. In Egyptian and Babylonian ancient monuments, there are regular quadrangles, hexagons and octagons in the form of images on the walls and ornaments carved from their stone. Ancient Greek scientists began to show great interest to regular polygons since the time of Pythagoras. The doctrine of regular polygons was systematized and presented in the 4th book of the "Elements" of Euclid.
REGULAR POLYTOPS OF PLATO BODIES: Tetrahedron - "fire" Cube - "earth" Octahedron - "air" Dodecahedron - "whole world" Icosahedron - "water"
REGULAR POLYGONS IN NATURE REGULAR POLYGONS IN NATURE Regular polygons occur in nature. One example is a honeycomb, which is a rectangle covered with regular hexagons. On these hexagons, bees grow from wax cells, which are straight hexagonal prisms. Bees lay honey in them, and then again cover with a solid rectangle of wax.
Sources of information: Children's encyclopedia "I know the world" Mathematics, Moscow, AST, 1998. ru.wikipedia.org/wiki/History of mathematics A.I.Azevich Twenty lessons of harmony: Humanitarian and mathematical course.-M .: School-Press, 1998.
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REGULAR POLYGONS (geometry grade 9) Volodina nl.
Lesson objectives: 1. To review 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 can I find one corner of a hexagon if all angles 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? 180 ⁰
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? 720 ⁰
Divide polygons into two groups
REGULAR POLYGONS Arbitrary polygons
DEFINITION: A convex polygon is called regular if all sides of it are equal and all angles are equal
Regular triangle Equilateral triangle All sides are equal. All corners 60.⁰
Regular quadrangle 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 regular? 2. Does a regular 10-gon exist? 20-sided? 3.How to build a regular polygon?
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Lesson on "Regular Polygons"
Lesson objectives:
educational: to acquaint students with the concept and types of regular polygons, with some of their properties; teach to use the formula for calculating the angle of a regular polygon
- developing:
- educational:
Course lesson:
Lesson motto:
Three paths lead to knowledge:
Chinese philosopher and sage Confucius.
2. Motivation for the lesson.
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.
Anatole France, a French writer of the 19th century, once remarked: "You can only learn fun ... To digest knowledge, you need to absorb it with appetite."
Let's follow the advice of the writer in today's lesson: be active, attentive, absorb with a great desire the knowledge that will be useful to you in later life.
3. Updating basic knowledge.
Frontal 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 tell me what parts it consists of. The word “polygon” indicates that all shapes in this family have “many angles”.
Substitute a specific number in the word “polygon” instead of the part “many”, for example 5. You will get a PENTAGON. Or 6. Then - HEXAGON. Notice how many angles there are as many sides, so these figures could well be called multilaterals.
On the image geometric figures... Using the picture, name these shapes.
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 of the properties that all regular polygons have.
Sum of polygon angles
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
.
Let's derive a formula for calculating angle x of a regular n-gon.
In a regular polygon, all angles are equal, we divide the sum of the angles by the number of angles, we get the formula:
x = (n-2) * 180 / n
5. Securing new material.
Solve No. 179, 181, 183 (1), 184.
Without turning your head, look around the perimeter wall of the classroom clockwise, the chalkboard around the perimeter counterclockwise, the triangle shown on the stand clockwise and its counterclockwise triangle. 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 will put our palm to our eyes,
Let's put our strong legs apart.
Turning to the right
Let's look around majestically.
And you have to go 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 students.
Solve No. 183 (2).
8. Lesson summary. Reflection. D / z.
What did you remember most in the lesson?
What surprised you?
What did you like the most?
How do you want to see the next lesson?
D / z. 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 to calculate the angle of a regular polygon
- - developing: development of cognitive activity, spatial imagination, the ability to choose the right decision, concisely express your 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 path of meditation is the noblest path;
The path of imitation is the easiest path;
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 a polygon?
Incorrect Correct polygons
- A convex polygon is called regular if all its angles are equal and all sides are equal
Regular polygon properties
Sum of angles
polygon
n - number of sides n-2 - number of triangles The sum of the angles of one triangle - 180º, 180º multiply by the number of triangles (n -2), we get S = (n-2) * 180.
Formula for calculating the correct angle NS - square
In the right NS- in a gon, all angles are equal, we 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 of its sides are equal.
- Any regular polygon is convex.
- Any quadrangle with equal sides is regular.
- A triangle is correct if all of its angles are equal.
- Any equilateral triangle is regular.
- Any convex polygon is regular.
- Any quadrangle with equal angles is regular.
Independent work
a NS = (n-2) * 180 / n
a 3 =(3-2)*180/3= 180/3= 60
No. 1079 (orally), No. 1081 (b, d), No. 1083 (b)
Creative task:
* Historical information about regular polygons. Possible queries for a 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.
