Calculating the Area of Rectangle ACDF in a Regular Hexagon

Introduction

The goal of this article is to explore the geometric properties of a regular hexagon and calculate the area of a specific rectangle, ACDF, formed by specific segments of the hexagon. We will use Geometry and Trigonometry to solve the problem in multiple ways.

Step-by-Step Solution

First Method: Using Area Formulas

Given a regular hexagon ABCDEF of side length ( sqrt{10} ) cm, we are to find the area of rectangle ACDF.

To begin, the area of the regular hexagon can be calculated using the formula:

( Area_{hexagon} 6 times frac{sqrt{3}}{4} times side^2 )

Plugging in the side length ( sqrt{10} ) cm, we get:

( 6 times frac{sqrt{3}}{4} times (sqrt{10})^2 15sqrt{3} ) cm2 )

The area of the rectangle ACDF can be determined by subtracting the areas of triangles ABC and DEF from the area of the hexagon:

( Area_{ACDF} 15sqrt{3} - 2 times frac{1}{2} times sqrt{10} times sqrt{10} times sin 120° )

( 15sqrt{3} - 10 times frac{sqrt{3}}{2} 10sqrt{3} ) cm2 )

Therefore, the area of rectangle ACDF is 10sqrt{3} cm2 or approximately 17.32 cm2.

Second Method: Using Cosine Rule

Another way to calculate the area is by using the Cosine Rule in triangle ABC:

( AC^2 AB^2 BC^2 - 2 times AB times BC times cos(120°) )

Multiplying through by ( sqrt{10} ) and simplifying with trigonometric values, we get:

( AC^2 20 - 20 times left( -frac{1}{2} right) 30 )

( AC sqrt{30} ) cm

The area of rectangle ACDF is then ( AC times CD sqrt{30} times sqrt{10} 10sqrt{3} ) cm2 ), or approximately 17.32 cm2.

Third Method: Using Apothem and Trigonometry

For a regular hexagon with side ( sqrt{10} ) cm, the length of the apothem can be calculated as:

( a frac{sqrt{10}}{2} times tan(frac{pi}{3}) frac{sqrt{30}}{2} )

The diagonal ( AC ) is twice the apothem:

( AC 2 times frac{sqrt{30}}{2} sqrt{30} ) cm

The area of the rectangle is then:

( Area_{ACDF} AC times CD sqrt{30} times sqrt{10} 5sqrt{12} 10sqrt{3} ) cm2 )

Thus, we find that the area of rectangle ACDF is 10sqrt{3} cm2 or approximately 17.32 cm2.

Conclusion

In this article, we explored three different methods to calculate the area of rectangle ACDF in a regular hexagon. Each approach provided a consistent result, demonstrating the power of geometric and trigonometric principles.

Understanding these concepts is crucial for solving problems related to regular polygons and their various segments.