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Polygons—Formulas Geometry CP2 (Holt 6-1) K.Santos Polygon Interior Angle Sum The sum of the measures of the angles of an n-gon is (n-2)180°. Example: 16-gon (n-2)180° with n=16 (16-2)180 (14)180 2520° Why does this formula work? 1 triangle 180° 2 triangles 360° 3 triangles 540° 1(180) 2(180) 3(180) (3-2)180 (4-2)180 (5-2)180 first number in the formula is number of sides (n-2)180° Example—Finding a missing angle Find the missing angle measures. First you have to find out what the interior angle sum is. (n-2)180° (5-2)180 3(180) 540° Then write an equation to solve for the angles: 121+112+78+x+x+3= 540 314 +2x = 540 2x = 226 x = 113 So the angles measure: 113° and 116° (x and x + 3) 121° 112° 78° x x+3 Polygon—Exterior Angle Sum The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. m<1+m<2+m<3+m<4+m<5 =360° 1 5 2 3 4 Polygon—Exterior Angle Sum For any polygon the exterior angle sum is 360°. So, if it is a 50-gon then it’s exterior angle sum is 360°. So, if it is a dodecagon then it’s exterior angle sum is 360°. Regular Polygon—One/each interior angle To find one (or each) interior angle in a regular polygon with n, number of sides: 𝑡𝑜𝑡𝑎𝑙 𝑖𝑛𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 (n−2)180° 𝑛 Example—Each interior angle in a regular polygon Find the measure of each interior angle of a regular 18-gon. (n−2)180° 𝑛 (18−2)180° 18 (16)180° 2880 18 18 each angle measures: 160° Regular Polygon—One/each exterior angle To find one (or each) exterior angle in a regular polygon with n, number of sides: 𝑡𝑜𝑡𝑎𝑙 𝑒𝑥𝑡𝑒𝑟𝑖𝑜𝑟 𝑎𝑛𝑔𝑙𝑒 𝑠𝑢𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 360° 𝑛 Example—Each exterior angle in a regular polygon Find the measures of each exterior angle in a regular 24-gon. 360° 𝑛 360° 24 each angle measures: 15° Regular Polygon—Interior and Exterior Angle Relationships Find each interior and exterior angle in a regular 40-gon. Interior Angle (n−2)180° 𝑛 (40−2)180° 40 (38)180° 40 6840° 40 Exterior Angle 360° 𝑛 360° 40 9° exterior angle 171° interior angle Notice that the interior angle and exterior angle are supplementary (add to 180°). Exterior angle interior angle