Geometry 11.4 Areas of Regular Polygons

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.

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

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

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, , and right triangles!

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

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.

Fill in the table. rapA 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√ P = 4(16) 64 A = ½ (8)(64) A = (4)(64) √2 A = ½ (9)(72) A = (9)(36) 324 A = ½ a p

Fill in the table rapA 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 A = ½ (1)(6√3) 3√3 60 o 30 o √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

Fill in the table. Please change some of the numbers and cross off the “Side” column rapA 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 √ P = 6(2) 12 A = ½ (√3)(12) 6√3 30 o √ P = 6(5) 30 A = ½ (5√3/2)(30) A = (5√3/2)(15) 75√3 2

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 √3 square units 96√3 square units

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 square units 864√3 square units 6rs square units

HW P 443 (1-22 skip 17)