Question

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...

Answer 1

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

Explanation:

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`