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If a regular octagon is inscribed in a square

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If a regular octagon is inscribed in a square  [#permalink]

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New post 28 Jul 2017, 12:31
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Question Stats:

16% (03:29) correct 84% (02:41) wrong based on 47 sessions

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If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7
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If a regular octagon is inscribed in a square  [#permalink]

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New post 29 Jul 2017, 03:03
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ammuseeru wrote:
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7



Hi..

look at the att figure..

say each side of Octagon is \(a\sqrt{2}\), then the sides of the square will be \(2a+a\sqrt{2}\)..
this is because each vertices has a isosceles right angled triangle with hypotenuse \(a\sqrt{2}\)..

area of square = \((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)
area of octagon = \((2a+a\sqrt{2})^2-4*\frac{1}{2}*a^2=a^2(2+\sqrt{2})^2-2a^2=a^2(4+2+4\sqrt{2}-2)=a^2*4*(1+\sqrt{2})\)..
ratio of areas = \(2a^2(1+\sqrt{2})^2/a^2*4*(1+\sqrt{2})=1+\sqrt{2}/2\)
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If a regular octagon is inscribed in a square  [#permalink]

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New post 29 Jul 2017, 21:09
ammuseeru wrote:
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7


let area of each small square=1
area of large square=3*3=9
area of octagon=9-(4*1/2)=7
ratio of large square area to octagon area=9 to 7
E
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inscribed octagon.png
inscribed octagon.png [ 8.13 KiB | Viewed 5376 times ]

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Re: If a regular octagon is inscribed in a square  [#permalink]

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New post 29 Jul 2017, 21:31
gracie wrote:
ammuseeru wrote:
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7


let area of each small square=1
area of large square=3*3=9
area of octagon=9-(4*1/2)=7
ratio of large square area to octagon area=9 to 7
E



Hi...

A REGULAR octagon is an octagon with EQUAL sides..

but in your sketch, that is not the case.
4 sides are equal to SIDE of square and other 4 sides are equal to DIAGONAL of square..

that is why the answer is wrong.
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Re: If a regular octagon is inscribed in a square  [#permalink]

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New post 29 Jul 2017, 22:50
chetan2u wrote:
gracie wrote:
ammuseeru wrote:
If a regular octagon is inscribed in a square , what is the ratio of the square’s area to the regular octagon’s area?


(A) \((\sqrt{2}+1)\) to \(2\)
(B) \((\sqrt{2}+2)\) to \(2\)
(C) \((\sqrt{2}-1)\) to \(2\)
(D) \((\sqrt{2}-1)\) to \(1\)
(E) 9 to 7


let area of each small square=1
area of large square=3*3=9
area of octagon=9-(4*1/2)=7
ratio of large square area to octagon area=9 to 7
E



Hi...

A REGULAR octagon is an octagon with EQUAL sides..

but in your sketch, that is not the case.
4 sides are equal to SIDE of square and other 4 sides are equal to DIAGONAL of square..

that is why the answer is wrong.


hi chetan2u,
thank you for pointing out my error.
gracie
GMAT Club Bot
Re: If a regular octagon is inscribed in a square   [#permalink] 29 Jul 2017, 22:50
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