Understanding the Area of a Square with Diagonal AC in a Regular Hexagon of Side 1 cm

Given a regular hexagon with each side of length 1 cm, how can we determine the area of a square where one side is the diagonal AC? This article will guide you through the solution, explaining the methodology and detailing each step.

Introduction to the Problem

A regular hexagon is a polygon with six equal sides and angles. This particular problem involves a regular hexagon where each side measures 1 cm. We need to find the area of a square whose side length is the diagonal AC of the hexagon.

Coordinate Placement and Calculations

To solve this problem, we will first place the hexagon in a coordinate system. We assume the center of the hexagon is at the origin (0, 0) and each side has a length of 1 cm. The vertices can be represented as:

A: (0, 0) B: (0.5, 0.866) C: (-0.5, 0.866) D: (-1, 0) E: (-0.5, -0.866) F: (0.5, -0.866)

Next, we need to determine the length of the diagonal AC. Using the distance formula to calculate the distance between points A and C:

Calculating the Diagonal AC

The distance formula for points (x1, y1) and (x2, y2) is:

[d sqrt{(x_2 - x_1)^2 (y_2 - y_1)^2}]

Applying the formula to points A(0, 0) and C(-0.5, 0.866):

[begin{align*} AC sqrt{(-0.5 - 0)^2 (0.866 - 0)^2} sqrt{0.25 0.75} sqrt{1} 1 times sqrt{3} sqrt{3} text{ cm} end{align*}]

Calculating the Area of the Square

The area of a square is given by the formula:

[A s^2]

Where (s) is the side length of the square. In this case, the side length (s sqrt{3}) cm. Thus, the area is:

[A (sqrt{3})^2 3 text{ cm}^2]

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

Simplified Solution

Alternatively, we can solve this problem by leveraging the properties of a regular hexagon and the cosine rule. Since the hexagon is regular, the angle at each vertex is 120 degrees. Therefore, by the cosine rule in triangle ABC, where AC is the diagonal and AB and BC are sides of the hexagon:

[begin{align*} AC^2 AB^2 BC^2 - 2 times AB times BC times cos(120^circ) 1^2 1^2 - 2 times 1 times 1 times -0.5 1 1 1 3 AC sqrt{3} text{ cm} end{align*}]

The area of the square with side length (sqrt{3}) cm is:

[A (sqrt{3})^2 3 text{ cm}^2]

Thus, the area of the square is 3 square centimeters.

Conclusion

The area of the square with diagonal AC as one side in a regular hexagon of side 1 cm is 3 square centimeters. This problem showcases the beauty of geometry and the application of the cosine rule in solving geometric problems.