Twice the Perimeter

$\begin{aligned} \frac{A_B}{A_A}&=\left(\frac{ℓ_B}{ℓ_A}\right)^2\\ \frac{A_B}{16}&=(2)^2\\ A_B&=16\times 4\\ &=\boxed{64} \end{aligned}$

The side length of square $A$ is $\sqrt{16}=4$ . The perimeter is $4(4)=16$ . So the perimeter of square $B$ is $2(16)=32$ . The side length of square $B$ is $\dfrac{32}{4}=8$ . So the area of square $B$ is $8^2=\boxed{64}$

The side length of A is $\sqrt{16}=4$ ,so the side length of B is $4\times2=8$ ,so the area of B is $8^2=64$

Faster way is that if the perimeter doubles,then the area will times 4,so $16\times4=64$