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Geometry 11.4 Areas of Regular Polygons
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Definitions 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. New words for the vocab list. Also add median of a trapezoid.
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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
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Definitions Radius- the segment from the center to a vertex of the polygon. Draw one radii of each regular polygon..... r r r r
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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 360/3 Find the measure of each central angle. 120 o 360/4 90 o 360/6 60 o 360/8 45 o Many opportunities to use your skills of Pythagorean Theorem, 45-45-90, and 30-60-90 right triangles!
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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
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Area of a Regular Polygon A = ½ a p apothem perimeter WHY?. apothem x Area of the green triangle = ½ apothem(x) x x x x x The regular hexagon is made up of 6 green triangles. Area of the regular hexagon = ½ apothem(6x) Area of the regular hexagon = ½ apothem(perimeter) This is true for all regular polygons.
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Fill in the table. rapA 1. 8 2. 3. 8 4. 72 1. 2. 3. 4.. 8 45 o 90 o... 4√2 8√2 P = 4(8√2) 32√2 A = ½ (4√2)(32√2) 128 6√3 3√3 45 o 3√3 3√6 A = ½ (3√3)(24√3) A = (3√3)(12√3) 108 A = (2√2)(32√2) 8 8√2 8 16 P = 4(16) 64 A = ½ (8)(64) A = (4)(64) 256 18 9 9 9 9√2 A = ½ (9)(72) A = (9)(36) 324 A = ½ a p
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Fill in the table. 5. 6. 7. 8. rapA 5. 8 6. 7. 8 8.. 8... 360 o /3 120 o 60 o 30 o 4 4 4√3 8√3 P = 3(8√3) A = ½ a p A = ½ (4)(24√3) 24√3 A = (2)(24√3) 48√3 2√3 √3 60 o 30 o 1 1 2 2 A = ½ (1)(6√3) 3√3 60 o 30 o 8 16 8√3 16√3 P = 3(16√3) 48√3 A = ½ (8)(48√3) A = (4)(48√3) 192√3. 3√3 60 o 30 o 3/2 3 3√3 2 3/2 3 A = ½ (3/2)(9√3) 27√3 4
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Fill in the table. Please change some of the numbers and cross off the “Side” column. 9. 10. 11. rapA 1. 2. 3. 5... 360 o /6 60 o 30 o 5√2 2 5√6 2 2 A = ½ a p P = 6(5√2) 30√2 A = ½ (5√6/2)(30√2) A = (5√6/2)(15√2) 75√3 30 o √3 1 2 2 2 P = 6(2) 12 A = ½ (√3)(12) 6√3 30 o 5 5 2 5√3 2 2 5 P = 6(5) 30 A = ½ (5√3/2)(30) A = (5√3/2)(15) 75√3 2
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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
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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
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HW P 443 (1-22 skip 17)
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