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# A regular octagon ABCDEFGH has an area of one square unit. What is the

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Math Expert
Joined: 02 Sep 2009
Posts: 54366
A regular octagon ABCDEFGH has an area of one square unit. What is the  [#permalink]

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20 Mar 2019, 23:15
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55% (hard)

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36% (02:53) correct 64% (01:39) wrong based on 11 sessions

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A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF?

(A) $$1 - \frac{\sqrt{2}}{2}$$

(B) $$\frac{\sqrt{2}}{4}$$

(C) $$\sqrt{2} - 1$$

(D) 1/2

(E) $$\frac{1 + \sqrt{2}}{4}$$

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Re: A regular octagon ABCDEFGH has an area of one square unit. What is the  [#permalink]

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21 Mar 2019, 00:42
Let each side of octagon be x

Area of octagon=2 (1+rt2)* x^2
As area given is 1 sq unit
We will get , x^2=1/(2(1+rt2)) {i}

Also to find area of rectangle ABFE, we need to find BF
For this extend AB and DC to meet at M,
Triangle CMB is a isosceles rt angle triangle, where CM=MB and BC=x
So CM =x/rt2
Similarly by extending CD and FE to meet at N we can find DN=x /rt2

Now BE=CM+CD+DN=x*(1+rt2)
And area ABEF= BE×AB=x^2(1+rt2)
Pitting value of x^2 from {i}

We get area ABEF=1/2

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A regular octagon ABCDEFGH has an area of one square unit. What is the  [#permalink]

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21 Mar 2019, 00:58
Join AF & BE we see a rectangle been formed whose area would be equal to the 8 ∆ which are formed by joining opposite sides of the octagon
so area of the rectangle = sum of 8 ∆
each side of length and breadth has 4 ∆ so ratio ; 4/8 = 1/2
IMO D

replying to PM ; NamGup ;

Bunuel wrote:

A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF?

(A) $$1 - \frac{\sqrt{2}}{2}$$

(B) $$\frac{\sqrt{2}}{4}$$

(C) $$\sqrt{2} - 1$$

(D) 1/2

(E) $$\frac{1 + \sqrt{2}}{4}$$

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A regular octagon ABCDEFGH has an area of one square unit. What is the   [#permalink] 21 Mar 2019, 00:58
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# A regular octagon ABCDEFGH has an area of one square unit. What is the

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