2011 AMC 10B Problem 22

Let the cube have side length $x$. Take a vertical diagonal cross-section of the pyramid through opposite vertices of the square base. This cross-section is an isosceles right triangle with hypotenuse $\sqrt2$. The cube appears as a rectangle of height $x$ and width $\sqrt2x$, leaving two congruent right isosceles triangles of leg $x$. Thus $\sqrt2=\sqrt2x+2x$, so $x=\sqrt2-1$. The cube volume is $x^3=(\sqrt2-1)^3=5\sqrt2-7$. Thus, A is the correct answer.