The figure above shows a regular hexagon with sides of length a and a square with sides of length a. If the area of the hexagon is 384 `sqrt(3)` square inches, what is the area, in square inches, of the square?

Options

A. 256
B. 192
C. `64sqrt(3)`
D. `16sqrt(3)`

Explanation

Choice A is correct. 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 `384sqrt(3)` square inches, the area of each equilateral triangle will be `(384sqrt(3))/(6)=64sqrt(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 `(sqrt(3))/(2)`a, and the area of the equilateral triangle is `(1)/(2)(a) ((sqrt(3))/(2)a)=(sqrt(3))/(4)a^(2)`. Since the area of each equilateral triangle is `64 sqrt(3)` square inches, it follows that `a^(2)= (4)/(sqrt(3))(64sqrt(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