Given an octagon inscribed in a square of side, #s# 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
Oct 30, 2016

#(1/2pi-1)s^2#

Explanation:

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Given #AB=CD=s, => CB=sqrt2s#,
# => x=(sqrt2s)/2=s/sqrt2#
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#