1956 AMC 12 Problem 37

In a $60^\circ$ rhombus, the shorter diagonal equals the side length

The scale is $\frac{800}{3}$ miles per inch, so convert the $\frac{3}{16}$-inch side before finding area

The diagonal opposite the $60^\circ$ angle divides the rhombus into two equilateral triangles, so its length equals the rhombus side. The map scale is $\frac{800}{3}$ miles per inch, hence the actual side length is $$\frac3{16}\cdot\frac{800}{3}=50$$ miles. Therefore the rhombus area is $$\begin{aligned} 50^2\sin60^\circ &=2500\cdot\frac{\sqrt3}{2}\\ &=1250\sqrt3. \end{aligned}$$

Thus, the correct answer is E.