1. Polygons The word ‘polygon’ is a Greek word. Poly gon Poly means many and gon means angles.

2. Polygons The word polygon means “many angles” A two dimensional object A closed figure

3. More about Polygons Made up of three or more straight line segments There are exactly two sides that meet at a vertex The sides do not cross each other Polygons

4. Examples of Polygons Polygons

5. These are not Polygons Polygons

6. Terminology Side: One of the line segments that make up a polygon. Vertex: Point where two sides meet. Polygons

7. Vertex Side Polygons

8. Interior angle: An angle formed by two adjacent sides inside the polygon. Exterior angle: An angle formed by two adjacent sides outside the polygon. Polygons

9. Interior angle Exterior angle Polygons

10. Let us recapitulate Interior angle Diagonal Vertex Side Exterior angle Polygons

11. Types of Polygons Equiangular Polygon: a polygon in which all of the angles are equal Equilateral Polygon: a polygon in which all of the sides are the same length Polygons

12. Regular Polygon: a polygon where all the angles are equal and all of the sides are the same length. They are both equilateral and equiangular Polygons

13. Examples of Regular Polygons Polygons

14. A convex polygon: A polygon whose each of the interior angle measures less than 180°. If one or more than one angle in a polygon measures more than 180° then it is known as concave polygon. (Think: concave has a "cave" in it) Polygons

15. INTERIOR ANGLES OF A POLYGON Polygons

16. Let us find the connection between the number of sides, number of diagonals and the number of triangles of a polygon. Polygons

17. Quadrilateral Pentagon 180 o 2 x 180 o = 360 o 3 4 sides 5 sides 3 x 180 o = 540 o Hexagon 6 sides 180 o 4 x 180 o = 720 o 4 Heptagon/Septagon 7 sides 180 o 5 x 180 o = 900 o 5 2 1 diagonal 2 diagonals 3 diagonals 4 diagonals Polygons

18. Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle301180 0 180 0 /3 = 60 0 Polygons

19. Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle301180 0 180 0 /3 = 60 0 Quadrilateral4122 x180 0 = 360 0 360 0 /4 = 90 0 Polygons

20. Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle301180 0 180 0 /3 = 60 0 Quadrilateral4122 x180 0 = 360 0 360 0 /4 = 90 0 Pentagon5233 x180 0 = 540 0 540 0 /5 = 108 0 Polygons

21. Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle301180 0 180 0 /3 = 60 0 Quadrilateral4122 x180 0 = 360 0 360 0 /4 = 90 0 Pentagon5233 x180 0 = 540 0 540 0 /5 = 108 0 Hexagon6344 x180 0 = 720 0 720 0 /6 = 120 0 Polygons

22. Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle301180 0 180 0 /3 = 60 0 Quadrilateral4122 x180 0 = 360 0 360 0 /4 = 90 0 Pentagon5233 x180 0 = 540 0 540 0 /5 = 108 0 Hexagon6344 x180 0 = 720 0 720 0 /6 = 120 0 Heptagon7455 x180 0 = 900 0 900 0 /7 = 128.3 0 Polygons

23. Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle301180 0 180 0 /3 = 60 0 Quadrilateral4122 x180 0 = 360 0 360 0 /4 = 90 0 Pentagon5233 x180 0 = 540 0 540 0 /5 = 108 0 Hexagon6344 x180 0 = 720 0 720 0 /6 = 120 0 Heptagon7455 x180 0 = 900 0 900 0 /7 = 128.3 0 “n” sided polygon n Association with no. of sides Association with no. of sides Association with no. of triangles Association with sum of interior angles Polygons

24. Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle301180 0 180 0 /3 = 60 0 Quadrilateral4122 x180 0 = 360 0 360 0 /4 = 90 0 Pentagon5233 x180 0 = 540 0 540 0 /5 = 108 0 Hexagon6344 x180 0 = 720 0 720 0 /6 = 120 0 Heptagon7455 x180 0 = 900 0 900 0 /7 = 128.3 0 “n” sided polygon nn - 3n - 2(n - 2) x180 0 (n - 2) x180 0 / n Polygons

25. Septagon/Heptagon DecagonHendecagon 7 sides 10 sides11 sides 9 sides Nonagon Sum of Int. Angles 900 o Interior Angle 128.6 o Sum 1260 o I.A. 140 o Sum 1440 o I.A. 144 o Sum 1620 o I.A. 147.3 o Calculate the Sum of Interior Angles and each interior angle of each of these regular polygons. 1 243 Polygons

26. 2 x 180 o = 360 o 360 – 245 = 115 o 3 x 180 o = 540 o 540 – 395 = 145 o y 117 o 121 o 100 o 125 o 140 o z 133 o 137 o 138 o 125 o 105 o Find the unknown angles below. Diagrams not drawn accurately. 75 o 100 o 70 o w x 115 o 110 o 75 o 95 o 4 x 180 o = 720 o 720 – 603 = 117 o 5 x 180 o = 900 o 900 – 776 = 124 o Polygons

27. EXTERIOR ANGLES OF A POLYGON Polygons

28. An exterior angle of a regular polygon is formed by extending one side of the polygon. Angle CDY is an exterior angle to angle CDE Exterior Angle + Interior Angle of a regular polygon =180 0 D E Y B C A F 1 2 Polygons

29. 120 0 60 0 Polygons

30. 120 0 Polygons

31. 120 0 Polygons

32. 360 0 Polygons

33. 60 0 Polygons

34. 60 0 Polygons

35. 1 2 3 4 5 6 60 0 Polygons

36. 1 2 3 4 5 6 60 0 Polygons

37. 1 2 3 4 5 6 360 0 Polygons

38. 90 0 Polygons

39. 90 0 Polygons

40. 90 0 Polygons

41. 1 2 3 4 360 0 Polygons

42. No matter what type of polygon we have, the sum of the exterior angles is ALWAYS equal to 360º. Sum of exterior angles = 360º Polygons

43. In a regular polygon with ‘n’ sides Sum of interior angles = (n -2) x 180 0 i.e. 2(n – 2) x right angles Exterior Angle + Interior Angle =180 0 Each exterior angle = 360 0 /n No. of sides = 360 0 /exterior angle Polygons

44. Let us explore few more problems Find the measure of each interior angle of a polygon with 9 sides. Ans : 140 0 Find the measure of each exterior angle of a regular decagon. Ans : 36 0 How many sides are there in a regular polygon if each interior angle measures 165 0 ? Ans : 24 sides Is it possible to have a regular polygon with an exterior angle equal to 40 0 ? Ans : Yes Polygons

45. Polygons DG

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