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If the area of the regular hexagon is 90, what is the area of the purple region?

 

 May 28, 2020
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Plan:

1) Using the area of the regular hexagon, find the length of one of its sides.

2) Find the number of degrees in each vertex angle of the regular hexagon.

3) Using steps 1) and 2), find the area of one of the white triangles.

4) Subtract the area of both of the white triangles from the area of the regular hexagon to get the area

        of the purple region.

 

1)  Formula for the area of a regular hexagon (A = area; a = length of a side):

          A  =  [ 3·sqrt(3)/2 ] ·a2

         90  =  [ 3·sqrt(3)/2 ] ·a2

       180  =  [ 3·sqrt(3) ] ·a2

         60  =  [ sqrt(3) ] ·a2

        a2  =  60 / sqrt(3)

        a2  =  60·sqrt(3)/3

        a2  =  20·sqrt(3)

         a  =  sqrt[ 20·sqrt(3) ]           <---  length of each side of the hexagon

 

2)  Formula for the number of degrees in each vertex angle of a regular polygon  (n = number of sides):

         degrees  =  (n - 2)·180o / n

         degrees  =  (6 - 2)·180o / 6

         degrees  =  120o      <---  number of degrees in each vertex angle

 

3)  Formula to find the area of a white triangle 

        Area  =  ½·a·b·sin(C)

        Area  =  ½ · sqrt[ 20·sqrt(3) ]  ·sqrt[ 20·sqrt(3) ] · sin(120o)

        Area  =  ½ · 20·sqrt(3) · sqrt(3)/2

        Area  =  15

 

        Area of both white triangles  =  30

 

4)  Area of the purple region:  area(hexagon) - area(both triangle)  =  90 - 30  =  60

 May 28, 2020

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