Given an octagon inscribed in a square of side, ss express the shaded area i terms of s? Hint the intersecting circles quarter circles have radius that are related to the diagonal of the square...

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1 Answer
CW
Oct 30, 2016

(1/2pi-1)s^2(12π−1)s2

Explanation:

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Given AB=CD=s, => CB=sqrt2sAB=CD=s,⇒CB=√2s,
=> x=(sqrt2s)/2=s/sqrt2⇒x=√2s2=s√2
Area Delta OCD=1/4s^2

The two green areas in Delta OCD are the same.

One green area A_G=1/4s^2-pi(s/sqrt2)^2*45/360
A_G=(1/4-1/16pi)s^2

Let the black area in Delta OCD be A_B
=> A_B= 1/4s^2-2*A_G
A_B=1/4s^2-2(1/4-1/16pi)s^2
A_B=(1/4-1/2+1/8pi)s^2
A_B=(1/8pi-1/4)s^2

Now, let the shaded area in your diagram be A_S
=> A_S=4*A_B
A_S=4(1/8pi-1/4)s^2
A_S=(1/2pi-1)s^2=0.5708s^2

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