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Regular polygons
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“Three qualities: extensive knowledge, habit of thinking and nobility of feelings are necessary for a person to be educated in in every sense words." N.G. Chernyshevsky
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Simonov Monastery
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Do you know?
Which 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.Seven-domed tower White City in plan it 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 internal corner 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 regular n-gon, taken one at each vertex? No. 1082, 1083.
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.
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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 "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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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 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.
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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 right triangleА1Н1О O А1 А2 А3 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 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.
