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Polygons—Formulas

Polygons—Formulas. Geometry CP1 (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 2 triangles 3 triangles

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Polygons—Formulas

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  1. Polygons—Formulas Geometry CP1 (Holt 6-1) K.Santos

  2. 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

  3. Why does this formula work? 1 triangle 2 triangles 3 triangles 180 360 540 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

  4. Example—Finding a missing angle Find the missing angle measures. First you have to find out what 121 112 the interior angle sum is. (n-2)180 78 (5-2)180 3(180) x x+3 540 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: 113and 116 (x and x + 3)

  5. 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

  6. 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.

  7. Regular Polygon—One/each interior angle To find one (or each) interior angle in a regular polygon with n, number of sides:

  8. Example—Each interior angle in a regular polygon Find the measure of each interior angle of a regular 18-gon. each angle measures: 160

  9. Regular Polygon—One/each exterior angle To find one (or each) exterior angle in a regular polygon with n, number of sides:

  10. Example—Each exterior angle in a regular polygon Find the measures of each exterior angle in a regular 24-gon. each angle measures: 15

  11. Regular Polygon—Interior and Exterior Angle Relationships Find each interior and exterior angle in a regular 40-gon. Interior Angle Exterior Angle 9 exterior angle 171 interior angle Notice that the interior angle and exterior angle are supplementary (add to 180. Exterior angle interior angle

  12. Example—given the interior angle sum Find the number of sides of a polygon that has an interior angle sum of 3960. Interior angle sum formula: (n-2)180 But we know what the sum is: 3960. So, (n-2)180 = 3960 180n -360=3960 180n = 4320 n = 24 The polygon has 24 sides (24-gon).

  13. Example—given exterior angle sum Find the number of sides of a polygon that has an exterior angle sum of 360. Exterior angle sum of any polygon is 360. So, there is no way to find out how many sides the polygon has.

  14. Example—regular polygon given one exterior angle Find the number of sides of a regular polygon with an exterior angle measurement of 20. The formula to find one exterior angle of a regular polygon is: But we know that the exterior angle is 20. = 20 20n = 360 n = 18 So, the polygon has 18 sides (18-gon).

  15. Example (1st method)—regular polygon given one interior angle Find the number of sides of the regular polygon with one interior angle of 156. The formula to find one interior angle of a regular polygon is: But we know the interior angle is: 156 = 156 156n = (n-2)180 156n = 180n – 360 -24n = -360 n = 15 So, the polygon has 15 sides (15-gon). But there is an easier way to do this problem…

  16. Example (2nd method)—regular polygongiven one interior angle Find the number of sides of the regular polygon with one interior angle of 156. If the interior angle is 156 we can find the exterior angle. 180-156 = 24. So, now we know the exterior angle 24 and we know the formula to find one exterior angle: = 24 24n = 360 n = 15 So, the polygon has 15 sides (15-gon)

  17. Total number of diagonals To find the total number of diagonals: where n is the number of sides Example: find the total number of diagonals in a 20-gon 170 diagonals

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