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Polygons The word ‘polygon’ is a Greek word. Poly means many and gon means angles. Polygons • The word polygon means “many angles” • A two dimensional object • A closed figure Polygons 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 Examples of Polygons Polygons These are not Polygons Polygons Terminology Side: One of the line segments that make up a polygon. Vertex: Point where two sides meet. Polygons Vertex Side Polygons • 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 Exterior angle Interior angle Polygons Let us recapitulate Exterior angle Vertex Side Diagonal Interior angle Polygons 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 • 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 Examples of Regular Polygons Polygons 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 INTERIOR ANGLES OF A POLYGON Polygons Let us find the connection between the number of sides, number of diagonals and the number of triangles of a polygon. Polygons 180 180o 180o 180 o o 180o 4 sides Quadrilateral 5 sides 2 x 180o = 360o 2 Pentagon o 1 diagonal o 3 x 180 = 540 3 2 diagonals 180o 180o 180o 180o 180o 180o 180o 180o 180o 6 sides 4 Hexagon 4 x 180o = 720o 3 diagonals 7 sides 5 Polygons Heptagon/Septagon 5 x 180o = 900o 4 diagonals Regular Polygon Triangle No. of sides No. of diagonals No. of 3 0 1 Polygons Sum of the interior angles 0 180 Each interior angle 0 180 /3 0 = 60 Regular Polygon No. of sides No. of diagonals No. of Triangle 3 0 1 180 Quadrilateral 4 1 2 2 x180 0 = 360 Polygons Sum of the interior angles 0 Each interior angle 0 180 /3 0 = 60 0 0 360 /4 0 = 90 Regular Polygon No. of sides No. of diagonals No. of Triangle 3 0 1 180 Quadrilateral 4 1 2 2 x180 0 = 360 0 360 /4 0 = 90 Pentagon 5 2 3 3 x180 0 = 540 0 540 /5 0 = 108 Polygons Sum of the interior angles 0 Each interior angle 0 180 /3 0 = 60 0 0 Regular Polygon No. of sides No. of diagonals No. of Triangle 3 0 1 180 Quadrilateral 4 1 2 2 x180 0 = 360 0 360 /4 0 = 90 Pentagon 5 2 3 3 x180 0 = 540 0 540 /5 0 = 108 Hexagon 6 3 4 4 x180 0 = 720 0 720 /6 0 = 120 Polygons Sum of the interior angles 0 Each interior angle 0 180 /3 0 = 60 0 0 0 Regular Polygon No. of sides No. of diagonals No. of Triangle 3 0 1 180 Quadrilateral 4 1 2 2 x180 0 = 360 0 360 /4 0 = 90 Pentagon 5 2 3 3 x180 0 = 540 0 540 /5 0 = 108 Hexagon 6 3 4 4 x180 0 = 720 0 720 /6 0 = 120 Heptagon 7 4 5 5 x180 0 = 900 0 900 /7 0 = 128.3 Polygons Sum of the interior angles 0 Each interior angle 0 180 /3 0 = 60 0 0 0 0 Regular Polygon No. of sides No. of diagonals No. of Triangle 3 0 1 180 Quadrilateral 4 1 2 2 x180 0 = 360 0 360 /4 0 = 90 Pentagon 5 2 3 3 x180 0 = 540 0 540 /5 0 = 108 Hexagon 6 3 4 4 x180 0 = 720 0 720 /6 0 = 120 Heptagon 7 4 5 5 x180 0 = 900 0 900 /7 0 = 128.3 “n” sided polygon n Association with no. of sides Association with no. of sides Polygons Sum of the interior angles 0 Each interior angle 0 180 /3 0 = 60 Association with no. of triangles 0 0 0 0 Association with sum of interior angles Regular Polygon No. of sides No. of diagonals No. of Triangle 3 0 1 180 Quadrilateral 4 1 2 2 x180 0 = 360 0 360 /4 0 = 90 Pentagon 5 2 3 3 x180 0 = 540 0 540 /5 0 = 108 Hexagon 6 3 4 4 x180 0 = 720 0 720 /6 0 = 120 Heptagon 7 4 5 5 x180 0 = 900 0 900 /7 0 = 128.3 “n” sided polygon n n-3 n-2 (n - 2) 0 x180 Polygons Sum of the interior angles 0 Each interior angle 0 180 /3 0 = 60 0 0 0 0 (n - 2) 0 x180 / n 1 Calculate the Sum of Interior Angles and each interior angle of each of these regular polygons. 7 sides Septagon/Heptagon Sum of Int. Angles 900o Interior Angle 128.6o 2 3 4 9 sides 10 sides 11 sides Nonagon Decagon Hendecagon Sum 1260o I.A. 140o Sum 1440o I.A. 144o Sum 1620o I.A. 147.3o Polygons Find the unknown angles below. w 75o 75o 2 x 180o = 360o 360 – 245 = 115o 140o x 100o 70o 125o 100o 115o 3 x 180o = 540o 540 – 395 = 145o 125o z 138o 138o 133o y 4 x 180o = 720o 720 – 603 = 117o 95o 110o 121o 117o Diagrams not drawn accurately. 105o 137o 5 x 180o = 900o Polygons 900 – 776 = 124o EXTERIOR ANGLES OF A POLYGON Polygons An exterior angle of a regular polygon is formed by extending one side of the polygo Angle CDY is an exterior angle to angle CDE B A C F 2 E 1 D Y Exterior Angle + Interior Angle of a regular polygon =180 Polygons 0 1200 600 600 600 1200 Polygons 1200 1200 1200 1200 Polygons 1200 1200 1200 Polygons 3600 Polygons 600 600 600 600 600 600 Polygons 600 600 600 600 600 600 Polygons 3 4 600 600 2 600 600 5 600 600 1 6 Polygons 3 4 600 600 2 600 600 5 600 600 1 6 Polygons 3 4 2 3600 5 1 6 Polygons 900 900 900 900 Polygons 900 900 900 900 Polygons 900 900 900 900 Polygons 2 3 3600 1 4 Polygons 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 In a regular polygon with ‘n’ sides Sum of interior angles = (n -2) x 180 i.e. 2(n – 2) x right angles Exterior Angle + Interior Angle =180 0 Each exterior angle = 360 /n 0 No. of sides = 360 /exterior angle Polygons 0 0 Let us explore few more problems • Find the measure of each interior angle of a polygon with 9 sides. 0 • Ans : 140 • Find the measure of each exterior angle of a regular decagon. 0 • Ans : 36 • How many sides are there in a regular polygon if each interior angle measures 165 ? • Ans : 24 sides • Is it possible to have a regular polygon with an exterior angle equal to 40 ? • Ans : Yes 0 0 Polygons Polygons DG This powerpoint was kindly donated to www.worldofteaching.com Home to well over a thousand free powerpoint presentations submitted by teachers. This a free site. Please visit and I hope it will help in your teaching