ANGLES IN A POLYGON WHAT IS A REGULAR POLYGON? •A closed figure shape that has more than two sides. •All angles and side are congruent INTERIOR ANGLE SUM • Just like a triangle, every regular polygon has an interior sum when you add up all its angles. • A triangle’s interior sum is 180 degrees. • A square’s interior sum is 360 degrees. • A pentagon’s interior sum is 540 degrees. • Has anyone noticed the pattern? • Every shape increases by 180 degrees from the one before it. FORMULA • What is I told you the sum of a shape that has 54 sides has an interior sum of 9360 degrees? • Do you know how I got that? • Well, there’s a simple formula that we use to calculate the interior sum of any polygon. (n-2)180=Sum *n means the number of sides on your shape. LET’S TRY IT Number of Multiple by Interior Shape Subtract 2 Sides 180 Sum Hexagon 6 6 - 2 = 4 4 x 180 720 Remember, Decagon 10 10 – 2 = 8 8 x 180 1440 you can always count the Pentadecagon 15 15 – 2 = 13 13 x 180 2340 number of sides if given an image 36-gon 36 36 – 2 = 34 34 x 180 6120 EXAMPLE WITH A PICTURE • 1st- count the number of sides • 12 • 2nd- Create your formula • (12-2)180 = Interior Sum • 3rd- Calculate and get your sum • 1800 degrees This Photo by Unknown Author is licensed under CC BY-SA FINDING JUST ONE ANGLE? • What if we need to find out what one angle is? • There is a formula for that! 푛 − 2 180 푛 Just take the interior sum and divide by the number of side. EXAMPLE: FINDING JUST ANGLE ONE (BLUE) Divide Each Interior interior Number of Multiple by Interior Shape Subtract 2 Sum sum by the Sides 180 angle (degrees) number of (degrees) sides Hexagon 6 6 - 2 = 4 4 x 180 720 720 / 6 = 120 120 1440 / 10 = Decagon 10 10 – 2 = 8 8 x 180 1440 144 144 2340 / 15 = Pentadecagon 15 15 – 2 = 13 13 x 180 2340 156 156 6120 / 36 = 36-gon 36 36 – 2 = 34 34 x 180 6120 170 170 EXAMPLE OF FINDING JUST ONE ANGLE WITH A PICTURE • 1st- count the number of sides • 9 • 2nd- Create your formula • (9-2)180 = Interior Sum • 3rd- Calculate and get your sum • 1260 degrees • 4th- Create formula to find individual angle • (1260) / 9 or (9 - 2) 180 / 9 This Photo by Unknown Author is licensed under CC BY-SA • 5th- Calculate and get individual angle • 140 degrees FINDING MEASURES OF INTERIOR ANGLES OF POLYGONS Find the value of x 1st: Find (n-2)180 88 142 (6-2)180=720 136 105 2nd: Set up an equation 136 x 607 + x = 720 x = 113o EXTERIOR ANGLES The sum of the exterior angles of any polygon is 360o. FINDING THE MEASURE OF AN EXTERIOR ANGLE 12x = 360 7x = 360 x = 51.4 x = 30 HELPFUL HINTS What is the relation between an interior 1 angle and an 2 exterior angle? They add to equal 180o. USE THIS FACT!!! FINDING THE NUMBER OF SIDES OR N • A regular polygon has an interior angle with a measure of 140o. How many sides does it have? 1st Find an exterior angle 180 - 140 = 40 2nd Find n IT IS A 360 / 40 = 9 NONAGON!! SUMMARY • Sum of interior (n-2)180o • Sum of exterior always 360o • 1 Interior + 1 Exterior = 180o • To find number of sides when give exterior angle = 360 ¸ ext. FORMULAS TO REMEMBER • Sum of the interior angles of Polygon • (n - 2)180 • Sum of the exterior angles of any polygon • 360° • Each interior angle of a regular polygon • ((n - 2)180)/n • Each exterior angle of a regular polygon • 360/n.