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

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

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

Interior angle Exterior angle Polygons

Let us recapitulate Interior angle Diagonal Vertex Side Exterior 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

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 diagonal 2 diagonals 3 diagonals 4 diagonals Polygons

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

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

Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle /3 = 60 0 Quadrilateral4122 x180 0 = /4 = 90 0 Pentagon5233 x180 0 = /5 = Polygons

Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle /3 = 60 0 Quadrilateral4122 x180 0 = /4 = 90 0 Pentagon5233 x180 0 = /5 = Hexagon6344 x180 0 = /6 = Polygons

Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle /3 = 60 0 Quadrilateral4122 x180 0 = /4 = 90 0 Pentagon5233 x180 0 = /5 = Hexagon6344 x180 0 = /6 = Heptagon7455 x180 0 = /7 = Polygons

Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle /3 = 60 0 Quadrilateral4122 x180 0 = /4 = 90 0 Pentagon5233 x180 0 = /5 = Hexagon6344 x180 0 = /6 = Heptagon7455 x180 0 = /7 = “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

Regular Polygon No. of sides No. of diagonals No. ofSum of the interior angles Each interior angle Triangle /3 = 60 0 Quadrilateral4122 x180 0 = /4 = 90 0 Pentagon5233 x180 0 = /5 = Hexagon6344 x180 0 = /6 = Heptagon7455 x180 0 = /7 = “n” sided polygon nn - 3n - 2(n - 2) x180 0 (n - 2) x180 0 / n Polygons

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

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

EXTERIOR ANGLES OF A POLYGON Polygons

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

Polygons

120 0 Polygons

120 0 Polygons

360 0 Polygons

60 0 Polygons

60 0 Polygons

Polygons

Polygons

Polygons

90 0 Polygons

90 0 Polygons

90 0 Polygons

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 i.e. 2(n – 2) x right angles Exterior Angle + Interior Angle =180 0 Each exterior angle = /n No. of sides = /exterior angle Polygons

Let us explore few more problems Find the measure of each interior angle of a polygon with 9 sides. Ans : 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 ? Ans : 24 sides Is it possible to have a regular polygon with an exterior angle equal to 40 0 ? Ans : Yes Polygons

Polygons DG

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