Square $ABCD\ :\ A(0,0),B(b,0),C(b,b),D(0,b)$ with area $b^{2}$. Mini-square $A'B'C'D'\ :\ A'(p,q),B'(p+a\cos \alpha,q+a\sin \alpha),C'(p+a\cos \alpha-a\sin \alpha,q+a\sin \alpha+a\cos \alpha),D'(p-a\sin \alpha,q+a\cos \alpha)$ with area $a^{2}$. Area $AA'B'B =\frac{1}{2}(-ap\sin \alpha+aq\cos \alpha+bq+ab\sin \alpha)$. Area $CC'D'D =\frac{1}{2}(-bq-ab\sin \alpha-aq\cos \alpha-a^{2}+ap\sin \alpha+b^{2})$. Sum of these two areas $=\frac{1}{2}(b^{2}-a^{2})$. Area $BB'C'C =\frac{1}{2}(-ap\cos \alpha-a^{2}-aq\sin \alpha-bp+ab\sin \alpha+b^{2})$. Area $DD'A'A =\frac{1}{2}(bp-ab\sin \alpha+aq\sin \alpha+ap\cos \alpha)$. Sum of these two areas $=\frac{1}{2}(b^{2}-a^{2})$.