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Approximately what percent of the area of the circle...

by BTGmoderatorLU » Sat Dec 23, 2017 2:22 pm
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Approximately what percent of the area of the circle shown is shaded, if polygon ABCDEF is a regular hexagon?

(A) 24%
(B) 30%
(C) 36%
(D) 42%
(E) 48%

The OA is D.

I'm really confused with this PS question because I know that if the polygon is a regular hexagon then all its sides should be equal and the shaded regions will be isoscels triangles and also the ACE triangle should be equilateral but I don't know how can I determine this percentage. Experts, any suggestion? Thanks in advance.
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by GMATWisdom » Sun Dec 24, 2017 4:09 pm
W
LUANDATO wrote:Image

Approximately what percent of the area of the circle shown is shaded, if polygon ABCDEF is a regular hexagon?

(A) 24%
(B) 30%
(C) 36%
(D) 42%
(E) 48%

The OA is D.

I'm really confused with this PS question because I know that if the polygon is a regular hexagon then all its sides should be equal and the shaded regions will be isoscels triangles and also the ACE triangle should be equilateral but I don't know how can I determine this percentage. Experts, any suggestion? Thanks in advance.
Hexagon is made of 6 equilateral triangles and
the area of each triangle=(√3)/4) r2
hence Area of hexagon = 6(√3)/4) r2 where r is the radius of the circle
And by joining all the radii it will be noticed that
area of shaded portion = half the area of the hexagon
hence area of shaded portion = (½) 6(√3)/4) r2 =3(√3)/4) r2
Area of circle =Ï€r^2
therefore %age of area of shaded portion to that of circle
=[3(√3)/4) r2]/ πr^2 *100 =[3(√3)/4) ]/ π *100 = 41.3%
This is very close to option D
Hence D is the right option.
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