The figure below 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√3 square inches, what is the area, in square inches, of the square?

A regular hexagon is a two-dimensional geometrical shape that is made of six equal sides.

Answer: For a regular hexagon with sides of length a and a square with sides of length a, given area of the hexagon as 384√3 square inches, the area of the square is 256 square inches.

Let's look into the steps below

Explanation:

A regular polygon has equal sides and equal angles.

Thus, the measure of all the sides of the regular hexagon given is 'a'.

Area of the hexagon = 384√3 square inches (Given)

We know that,

Area of a Regular Hexagon = (3√3 a²) / 2, where, a = Side length of the Hexagon

⇒ 384√3 = (3√3 a²) /2 [ Since, Area = 384√3 square inches]

⇒ a² = (384 × 2) / 3

⇒ a² = 256

Taking square root on both sides,

⇒ a = 16

Now, the side length of the square is given as 'a'

Thus,

Area of the square = a² 

⇒ a² = 16²

⇒ a² = 256

Thus, the area of the square is 256 square inches.