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A regular octagon is formed by cutting four equal isosceles right-angled triangles from the corners of a square of side length $$1$$. The area of the octagon is
Let each removed triangle have equal legs $$x$$. For the remaining octagon to be regular, its horizontal side and sloping side must be equal, so $$1-2x=x\sqrt2$$. Thus $$x=\frac{1}{2+\sqrt2}$$, and the required area is $$1-4\left(\frac{x^2}{2}\right)=1-2x^2=2(\sqrt2-1)$$.
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