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# Regular hexagon ABCDEF has a perimeter of 36. O is the

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Manager
Joined: 07 Feb 2007
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Regular hexagon ABCDEF has a perimeter of 36. O is the [#permalink]

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26 Aug 2007, 20:42
This topic is locked. If you want to discuss this question please re-post it in the respective forum.

Regular hexagon ABCDEF has a perimeter of 36. O is the center of the hexagon and of circle O. Circles A, B, C, D, E, and F have centers at A, B, C, D, E, and F, respectively. If each circle is tangent to the two circles adjacent to it and to circle O, what is the area of the shaded region (inside the hexagon but outside the circles)?

108 – 18*pi
54*sqrt3 - 9*pi
54*sqrt3 - 18*pi
108 - 27*pi
54*sqrt3 - 27*pi

Pls see attachment for the figure
Attachments

doc.doc [72.5 KiB]

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Senior Manager
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26 Aug 2007, 21:06
I ger 54 square root 3 minus 27pi

here is how i did it.

Six sides six inches each the area of which will be six equilateral triangles with a hight of 3 sqare root of 3 we have six circles with an area of 9pi each circle takes away an area that is eqaul to 120 degrees of the circle.

{(120/360)54pi} is the area taken away by the outside circles plus the 9pi by the inner circle

subtract the area of the hex from this and you get what is above.

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Manager
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26 Aug 2007, 21:20
I got 54 sqrt 3 - 27*Pi:

1) hexagon consists of 6 equilateral with the side 6, so the area of hexagon:
S1 = 6 * 1/2 * (3 sqrt 3) * 6 = 54 sqrt 3

2) the area of 6 parts of a square =
S2 = 6 * ((Pi * 9) / 3) = 18*Pi

S = S1 - S2 = 54 sqrt 3 - 18*Pi -9*Pi (area of the 7th circle)

Last edited by Whatever on 28 Aug 2007, 01:45, edited 1 time in total.

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Manager
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27 Aug 2007, 07:16
defenestrate wrote:
I ger 54 square root 3 minus 27pi

here is how i did it.

Six sides six inches each the area of which will be six equilateral triangles with a hight of 3 sqare root of 3 we have six circles with an area of 9pi each circle takes away an area that is eqaul to 120 degrees of the circle.

{(120/360)54pi} is the area taken away by the outside circles plus the 9pi by the inner circle

subtract the area of the hex from this and you get what is above.

Thanks! OA is E. I was missing the area included {(120/360)54pi}.

Kudos [?]: 5 [0], given: 0

27 Aug 2007, 07:16
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