2011 AMC 12B Problem 12

Assume the octagon has edge length $1.$ The four corner triangles are right isosceles with legs $\dfrac{\sqrt2}{2}$ and area $\dfrac14$ each. The four rectangles are $1$ by $\dfrac{\sqrt2}{2}$ with area $\dfrac{\sqrt2}{2}$ each, and the center square has area $1.$ The total area is $$4\cdot\dfrac14+4\cdot\dfrac{\sqrt2}{2}+1=2+2\sqrt2.$$ The probability of hitting the center square is $$\dfrac{1}{2+2\sqrt2}=\dfrac{\sqrt2-1}{2}.$$ Thus, the correct answer is A.