1956 AMC 12 Problem 35

Every side of the rhombus has length $16,$ so each chord equals the radius

A chord equal to the radius subtends a $60^\circ$ central angle; use $s^2\sin\theta$

All four sides of the rhombus equal the circle’s radius, $16.$ A chord of length equal to the radius forms an equilateral triangle with the two radii to its endpoints, so the included central angle is $60^\circ.$ Therefore the rhombus has area $$\begin{aligned} 16^2\sin60^\circ &=256\cdot\frac{\sqrt3}{2}\\ &=128\sqrt3. \end{aligned}$$

Thus, the correct answer is B.