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Slide 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 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 "icos" - twenty The names of these polyhedra came from Ancient Greece and they indicate the number of faces.
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
Alexandria 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 crystal lattice in the shape of a cube. 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 maximum length gold thread, but 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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Slide 2
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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Slide 4
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.
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: A circle can be inscribed in any regular polygon, and only one.
Slide 6
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.
Slide 7
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.
Slide 8
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 СО 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.
Slide 9
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
Slide 10
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.
Slide 11
Formula for calculating the area of a regular polygon. Let S be the area regular n-gon, a1 is its side, P is the perimeter, and r and R are the radii of the inscribed and circumscribed circles, respectively. Let's prove that
Slide 12
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,
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 right triangleА1Н1О O А1 А2 А3 Аn H2 H1 Hn H3 Therefore,
Slide 14
Putting n = 3, 4 and 6 in the formula, we obtain expressions for the sides of a regular triangle, square and regular hexagon:
Slide 15
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.
Slide 16
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.
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Slide 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. 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?
On the topic: methodological developments, presentations and notes
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Presentation for a geometry lesson in 9th grade, contains the necessary definitions and formulas for calculating the area of regular polygons....
From history From history Regular polygons were known back in ancient times. In Egyptian and Babylonian ancient monuments, regular quadrangles, hexagons and octagons are found in the form of images on the walls and decorations carved from stone. Ancient Greek scientists began to show great interest in regular polygons since the time of Pythagoras. The doctrine of regular polygons was systematized and presented in book 4 of Euclid’s Elements.
REGULAR POLYHEDRON PLATONIAN solids: Tetrahedron – “fire” Cube – “earth” Octahedron – “air” Dodecahedron – “the whole world” Icosahedron – “water”
REGULAR POLYGONS IN NATURE REGULAR POLYGONS IN NATURE Regular polygons are found in nature. One example is the honeycomb, which is a rectangle covered with regular hexagons. On these hexagons, bees grow cells from wax that are straight hexagonal prisms. The bees deposit honey in them, and then cover them again with a solid rectangle of wax.
Sources of information: Children's encyclopedia "I explore the world" Mathematics, Moscow, AST, 1998. ru.wikipedia.org/wiki/History of Mathematics A.I.Azevich Twenty Lessons of Harmony: Humanities and Mathematics Course. - M.: Shkola-Press, 1998.
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:
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.
French writer 19th century 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 of 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 sides as there are angles, so these figures could well be called polylaterals.
In the picture 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 - equilateral triangle (regular triangle), 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 one more thing
Over your left shoulder!
Now let's continue working.
7. Independent work 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 cognitive activity, spatial imagination, ability to choose the right decision, concisely express your 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 amount? internal corners 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 n - square
In the right n- 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 regular.
- Any convex polygon is regular.
- Any quadrilateral with equal angles is regular.
Independent work
A n =(n-2)*180/n
A 3 =(3-2)*180/3= 180/3= 60
No. 1079 (oral), No. 1081 (b, d), No. 1083 (b)
Creative task:
*Historical information about regular polygons. Possible requests for search engine networks 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.
