Question

Let ABCDEF be a regular hexagon with each side of length 1 cm. The area (in sq cm) of a square with AC as one side is

• A. 3√2
• B. 3
• C. 4
• D. √3

Answer 1

The Correct Option is B

To find the area of a square with AC as one side in a regular hexagon ABCDEF where each side is 1 cm, we proceed as follows:

A regular hexagon can be divided into 6 equilateral triangles. Each side of length 1 cm divides into two 30-60-90 right triangles when drawing altitude. The diagonal AC of the hexagon is through the center and two sides away. It spans over 2 equilateral triangles.

In an equilateral triangle with side length 1 cm, the length of each 60 degrees angle is the height: \( \frac{\sqrt{3}}{2} \cdot 1 = \frac{\sqrt{3}}{2} \).

The diagonal AC spans across 2 full equilateral triangles, thus it is double the side length: \( 2 \times 1 = 2 \) cm.

However, we mistakenly considered AC; this is incorrect for the measurement of distance but correct structurally.

The correct distance for the square's side (AC) using trigonometry involves finding twice the cosine component of adjacent axes forming this diagonal:

\(AC=2 \cos(30^{\circ})\)

\(= 2 \cdot \frac{\sqrt{3}}{2} = \sqrt{3}\)

The area \(A\) of the square with side length \(\sqrt{3}\) is given by:

\(A = (\sqrt{3})^2 = 3\) sq cm.

Hence, the area of the square with AC as one side is 3 square centimeters.