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Geometry

Geometry. 11.4 Areas of Regular Polygons. Definitions . New words for the vocab list. Also add median of a trapezoid. Regular polygon-. a polygon that is equiangular and equilateral. In the upper right side of your paper, please draw a regular triangle,

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Geometry

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  1. Geometry 11.4 Areas of Regular Polygons

  2. Definitions New words for the vocab list. Also add median of a trapezoid. • Regular polygon- a polygon that is equiangular and equilateral. In the upper right side of your paper, please draw a regular triangle, a regular quadrilateral, a regular hexagon, and a regular octagon.

  3. Definitions • Center- the center of the circle that circumscribes the polygon. Find the center of each polygon, you may or may not want to draw the circumscribed circle. .center .center .center .center

  4. Definitions • Radius- the segment from the center to a vertex of the polygon. Draw one radii of each regular polygon. . . . . r r r r

  5. Definitions • Central angle- the angle formed by two consecutive radii. Draw one central angle of each regular polygon. Measure of a central angle = 360/n n is the number of sides . . . . 90o 120o 45o 60o 360/4 360/6 360/8 360/3 Many opportunities to use your skills of Pythagorean Theorem, 45-45-90, and 30-60-90 right triangles! Find the measure of each central angle.

  6. Definitions • Apothem- The distance (perpendicular) from the center to a side of the polygon. Draw one apothem of each regular polygon. . . . . a a a a

  7. Area of a Regular Polygon A = ½ a p perimeter apothem WHY? x . x x Area of the green triangle = ½ apothem(x) Area of the regular hexagon = ½ apothem(6x) Area of the regular hexagon = ½ apothem(perimeter) x apothem x This is true for all regular polygons. x The regular hexagon is made up of 6 green triangles.

  8. Fill in the table. 32√2 128 4√2 108 3√6 3√3 A = ½ a p 8√2 64 256 9√2 9 324 6√3 18 1. 2. 3. 4. . . . . 6√3 6√3 18 18 8 90o 3√6 8√2 45o 9√2 8 3√3 9 4√2 45o 45o 3√3 9 8 8√2 6√3 16 18 P = 4(8√2) A = ½ (3√3)(24√3) P = 4(16) A = ½ (9)(72) A = (3√3)(12√3) A = ½ (4√2)(32√2) A = ½ (8)(64) A = (9)(36) A = (2√2)(32√2) A = (4)(64)

  9. Fill in the table. 4 24√3 48√3 A = ½ a p 2 1 3√3 192√3 16 48√3 27√3 3/2 3 4 5. 6. 7. 8. 360o/3 . . . . . 2√3 3√3 2√3 3√3 8 16 2 3 120o 60o 60o 1 60o 8 60o 3/2 4 30o 30o 30o 30o 4√3 √3 8√3 3√3 2√3 16√3 8√3 2 3√3 P = 3(16√3) A = ½ (1)(6√3) P = 3(8√3) A = ½ (3/2)(9√3) A = ½ (8)(48√3) A = ½ (4)(24√3) A = (4)(48√3) A = (2)(24√3)

  10. Fill in the table. Please change some of the numbers and cross off the “Side” column. 5√6 30√2 75√3 2 A = ½ a p 12 6√3 2 5√3 75√3 30 2 2 9. 10. 11. 360o/6 . . . 5√2 2 5 60o √3 5√3 30o 5√6 30o 30o 2 2 1 5√2 5 2 2 5 2 5√2 P = 6(2) P = 6(5) P = 6(5√2) A = ½ (√3)(12) A = ½ (5√3/2)(30) A = ½ (5√6/2)(30√2) A = (5√3/2)(15) A = (5√6/2)(15√2)

  11. Word Problems: Who can write these on the board? Find the area of… 1) An equilateral triangle with radius 6√3. 2) A regular hexagon with perimeter of 48. 81√3 square units 96√3 square units

  12. Word Problems: Who can write these on the board? Find the area of… 3) A square with radius equal to 24. 4) A regular hexagon with apothem equal to 12√3 5) A regular dodecagon(12-sided) with side = r & apothem = s. 1152 square units 864√3 square units 6rs square units

  13. HW • P 443 (1-22 skip 17)

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