Classic problem of regular polygons presentation. Regular polygons presentation for a geometry lesson (9th grade) on the topic. Updating of reference knowledge

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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. Develop problem solving skills 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 angle of a hexagon if all angles are equal? (6 – 2) ∙ 180⁰ / 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

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 its sides are equal and all angles are equal

Regular 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.Which polygon is called regular? 2.Does a regular 10-gon exist? 20-gon? 3.How to construct a regular polygon?

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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 determined, what properties do they have? -Where are they found, do they have practical applications?

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 “icosas” - twenty The names of these polyhedra come from Ancient Greece and the number of faces is indicated in them.

Name of 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 At each vertex of the polyhedron there are at least three plane 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

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 Cosmic

"Ecosahedron - dodecahedral structure of the Earth"

Polyhedra in art and architecture

Albrecht Durer (1471-1528) "Melancholy"

Salvador Dali "The Last Supper"

Modern architectural structures in the form of polyhedra

Alexandrian lighthouse

Brick polyhedron by a Swiss architect

Modern building in England

Polyhedra in nature FEODARIA

Pyrite (sulfur pyrite) Monocrystal of potassium alum Crystals of red copper ore NATURAL CRYSTALS

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

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

Historically, the first cut form that appeared in the 14th century was the octahedron. Diamond Shah Diamond weight 88.7 carats

Task The Queen of England gave instructions to cut the diamond along the edges with gold thread. But the cutting was not done, since 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 informed of the following data: number of vertices B = 54, number of faces D = 48, length of the largest edge L = 4 mm. 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 B + G - 2 = P where B is the number of vertices, G is the number of faces, P is the number of edges of this polyhedron.

PHYSICAL MINUTE!

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 ​​the crystal

Surface area Tetrahedron Icosahedron Dodecahedron Hexahedron Octahedron

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

Thanks for the work!

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Regular polygons

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“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

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Simonov Monastery

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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? Which polygon is called convex? Show convex and non-convex polygons in the figure. Explain what angles are called the angles of a convex polygon, exterior angles. What formula is used to calculate the sum of the angles of a convex polygon? What is the perimeter of a polygon?

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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 boundary? 6.Element of a circle? 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 type of angle of a circle? 12.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?

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

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Angle of a regular n-gon

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Practical work. 1. The seven-domed tower of the White City in plan was a regular hexagon, all sides of which are equal to 14 m. Draw the plan of 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 size of this angle, knowing the number of sides of the polygon? 3.Measure angle CAK - the outer angle of the polygon. Calculate the sum of the exterior angle CAK and the interior angle CAB. Why do these angles always add up to 180°? What is the sum of the external angles of a regular hexagon, taken one at each vertex?

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The diameter of the base of the Dulo tower is 16m. Draw a plan for the base of a 16-sided tower, using when constructing 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.

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 them to use the formula for calculating the angle of a regular polygon

- developing:

- educational:

Lesson progress:

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

The 19th century French writer Anatole France once remarked: “You can only learn through fun... To digest knowledge, you must absorb it with appetite.”

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

3. Updating basic knowledge.

Frontal survey:

What are their elements?

Polygon views

4. Studying new material.

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

The names of geometric figures have a very specific meaning. Take a close look at the word “polygon” and say what parts it consists of. The word “polygon” indicates that all figures in this family have “many angles.”

Substitute a specific number, for example 5, into the word “polygon” instead of the “many” part. You will get a PENTAGON. Or 6. Then – HEXAGON. Note that there are as many angles as there are sides, so these figures could well be called polylaterals.

The picture shows geometric shapes. Using the drawing, 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 properties that all regular polygons have.

Sum of 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 .
Let us 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.

Solve No. 179, 181, 183(1), 184.

Without turning your head, look around the classroom wall around the perimeter clockwise, the chalkboard around the perimeter counterclockwise, the triangle depicted on the stand clockwise and the 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'll put our palms to our eyes,
Let's spread our strong legs.
Turning to the right
Let's look around majestically.
And you need to go left too
Look from under your palms.
And - to the right! And further
Over your left shoulder!
Now let's continue working.

7. Independent work of students.

Decide No. 183(2).

8. Lesson summary. Reflection. D/z.

What do you remember most about the lesson?

What surprised you?

What did you like the most?

What do you want the next lesson to look like?

D/z. Learn step 6. Solve No. 180, 182 185.

Creative task:

Internet :

View presentation content
"regular polygons"

• - educational: introduce students to the concept and types of regular polygons, and 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 solution, succinctly express one’s thoughts, analyze and draw conclusions.
• - educational: nurturing 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 reflection 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

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 multiplied by the number of triangles (n-2), we get S= (n-2)*180.

Formula for calculating the correct 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 regular.
• A triangle is regular if all its angles are equal.
• Any equilateral triangle is regular.
• 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, 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.

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

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

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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: A circle can be inscribed in any regular polygon, and only one.

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Let A1 A 2 ...A n be a regular polygon, O the center of the circumscribed circle. When proving Theorem 1, we found out that ∆ОА1А2 =∆ОА2А3= ∆ОАnА1, therefore the heights of these triangles drawn from vertex O are also equal. Therefore, a circle with center O and radius OH passes through 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: in any regular polygon you can inscribe a circle, and only one.

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

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A D B C O Given: ABCD…An is a regular polygon. Prove: around any regular polygon you can draw a circle, and only one. Proof: Let us 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, by drawing the segments OA and OD, we obtain ΔBOA, ΔBOC and ΔСOD. ΔBOA = ΔBOS according to the first sign of equality of triangles (VO - general, AB = BC, angle 2 = angle 3). Similar to ΔBOS=ΔCOD. 1 2 3 4 Because angle 2 = angle 3 as halves of equal angles, then ΔВOC is isosceles. This triangle is equal to ΔBOA and ΔCOD => they are also isosceles, which means OA=OB=OC=OD, i.e. points A, B, C and D are equidistant from point O and lie on the circle (O; OB). Similarly, other vertices of the polygon lie on the same circle.

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Let us now prove that there is only one circumscribed circle. Let's consider some three vertices of a polygon, for example A, B, C. Because. Only one circle passes through these points, then only one circle can be described around the polygon ABC...An. o A B C D

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Consequences. Corollary No. 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 about a regular polygon coincides with the center of a circle inscribed in the same polygon.

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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

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

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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 A1H1O O A1 A2 A3 Аn H2 H1 Hn H3 Therefore,

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

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

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