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

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Re: A regular octagon ABCDEFGH has an area of one square unit. What is the [#permalink]
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Each Interior Angle of a Regular Octagon = 135 degrees

If you draw 2 Straight Parallel Lines from Points A to F and Points B to E you will have the Rectangle in the Middle of the Octagon with 90 degree Interior Angles.

If you draw 2 more Straight Parallel Lines from Points H to C and Points G to D you will have another Rectangle in the Middle of the Regular Octagon with 90 Degree Interior Angles

this will create 4 Right Triangles on the Corners of the Regular Octagon, with the following Hypotenuses -
AH
BC
DE
GF

The Area of the Regular Octagon will now be sub-divided into:

5 Squares with the Side of Each Square = Side of the Regular Octagon

+

4 Isosceles 45/45/90 Right Triangles with the Hypotenuse = Side of the Regular Octagon

Let the Equal Side of the Regular Octagon = A

Area of the Rectangle ABFE = 3 * (Area of our created Square with Side A) = 3 * (A * A) = 3 * (A)^2 = ?

Area of the Regular Octagon =

(5 Squares) * (A)^2

+

(4 Right Isosceles Triangles) * [ (1/2) *(A/sqrt(2) ) * (A/sqrt(2) ) ]

=

1

__________________

5(A)^2 + 4 * (A^2 / 4) = 1

5(A)^2 + (A)^2 = 1

6(A)^2 = 1

the Question is asking for: What is 3(A)^2 = ?

if 6(A)^2 = 1, then HALF of this Area will be 1/2 = 3(A)^2

1/2
(D)
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A regular octagon ABCDEFGH has an area of one square unit. What is the [#permalink]
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A regular octagon ABCDEFGH has an area of one square unit. What is the area of the rectangle ABEF?

A regular octagon can be divided into 8 equal triangles as shown in figure.
Assume that each side of a regular octagon is a and the height of each triangle is h.

Area of regular octagon = 8 * area of one triangle = 8 * ½ * a * h= 4* a * h

We know that Area of regular octagon is given as 1 square unit
4* a * h =1
A* h =1/4

Area of the rectangle ABEF = a* 2h = 2* ¼ = ½

Thanks,
Clifin J Francis,
GMAT Quant SME
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