The regular hexagon can be divided into 6 equilateral triangles of side length a by drawing the six segments from the center of the regular hexagon to each of its 6 vertices. Since the area of the hexagon is
384√3 square inches, the area of each equilateral triangle will be
=64√3 square inches.
Drawing any altitude of an equilateral triangle divides it into two
30°−60°−90° triangles. If the side length of the equilateral triangle is a, then the hypotenuse of each
30°−60°−90° triangle is a, and the altitude of the equilateral triangle will be the side opposite the 60° angle in each of the
30°−60°−90° triangles. Thus, the altitude of the equilateral triangle is
a and the area of the equilateral triangle is
()(a)(a)=a2. Since the area of each equilateral triangle is
64√3 square inches, it follows that
a2=(64√3)=256 square inches. And since the area of the square with side length
a is
a2 , it follows that the square has area 256 square inches.
Choices B, C, and D are incorrect and may result from calculation or conceptual errors.