Question

\(A B C D E F\) is a regular hexagon of side \(6 \mathrm{~cm}\). What is the area of triangle \(B D F\) ? (a) \(32 \sqrt{3} \mathrm{~cm}^{2}\) (b) \(27 \sqrt{3} \mathrm{~cm}^{2}\) (c) \(24 \mathrm{~cm}^{2}\) (d) none of these

Short answer

Answer: The area of triangle BDF is 27√3 cm².

Step-by-step solution

Draw the hexagon and the required triangle

Draw a regular hexagon \(ABCDEF\) with side \(6 \mathrm{~cm}\). Mark points \(B, D,\) and \(F\). We need to find the area of triangle \(BDF\).

Divide the hexagon into equilateral triangles

The hexagon can be divided into six equilateral triangles by drawing lines from its center to each of its vertices. By doing so, we can see that we need only the area of three of these equilateral triangles to find the area of triangle \(BDF\).

Apply area formula for equilateral triangle

Let's denote each side of the hexagon, and the equilateral triangles, as \(a\). In this case, \(a = 6 \mathrm{~cm}\). The area of an equilateral triangle with side \(a\) is given by the formula \(A = \frac{\sqrt{3}}{4}a^2\).

Calculate area of one equilateral triangle

Applying the formula, we have: \(A_{1} = \frac{\sqrt{3}}{4}(6 \mathrm{~cm})^2 = \frac{\sqrt{3}}{4}(36 \mathrm{~cm}^{2})=\frac{36\sqrt{3}}{4}\mathrm{~cm}^{2} = 9\sqrt{3}\mathrm{~cm}^{2}\)

Calculate area of triangle \(BDF\)

As previously stated, triangle \(BDF\) consists of three equilateral triangles. Therefore, the area of triangle \(BDF\) is: \(A_{BDF} = 3 \times A_{1} = 3 \times 9\sqrt{3}\mathrm{~cm}^{2} = 27\sqrt{3}\mathrm{~cm}^{2}\). The area of triangle \(BDF\) is \(27\sqrt{3}\mathrm{~cm}^{2}\), which corresponds to option (b).

Key concepts

Regular Hexagon

A regular hexagon is a fascinating shape in geometry. It has six identical sides and angles, making it a symmetrical and balanced figure. Understanding the properties of a regular hexagon can simplify many geometry problems involving them.
Hexagons are two-dimensional shapes, and when all their sides are of equal length, they are termed 'regular'.
Here are some key characteristics to remember about regular hexagons:

• All interior angles are equal, each measuring 120 degrees.
• They can be subdivided into six equilateral triangles, which is quite helpful for calculations involving areas.
• The symmetry of a regular hexagon allows for easy calculations when it comes to finding distances between non-adjacent points.

In problems involving regular hexagons, like the exercise we have, leveraging these properties can lead to simple and efficient solutions.

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are the same length and all three angles are equal. Each angle in an equilateral triangle measures 60 degrees, making it a very balanced shape.
Equilateral triangles are particularly important in the context of regular hexagons, as each regular hexagon can be divided into six identical equilateral triangles.
This division helps simplify calculations related to the hexagon. A few important aspects of equilateral triangles include:

• Each triangle has three lines of symmetry.
• The height can be calculated using the formula: height = \(\frac{a\sqrt{3}}{2}\), where \(a\) is the length of a side.
• Area is given by \(\frac{\sqrt{3}}{4}a^2\), making it straightforward to calculate when the side length is known.

Understanding equilateral triangles is crucial for dividing shapes like hexagons into simpler components, aiding in tasks like area calculation.

Triangle Area Calculation

Calculating the area of a triangle can often seem complex, but when dealing with regular shapes, it becomes much easier. In a regular hexagon divided into equilateral triangles, the area calculation is straightforward thanks to the properties of these triangles.
To calculate the area of an equilateral triangle with side length \(a\), we use the formula:\[A = \frac{\sqrt{3}}{4}a^2\]This formula stems from trigonometric properties and the unique angles in an equilateral triangle.
For example, in our exercise, with each side of the hexagon being 6 cm, the area \[A_{1} = \frac{\sqrt{3}}{4}(6)^2 = 9\sqrt{3} \text{ cm}^2\]is calculated.
Since the triangle of interest, \(\triangle BDF\), consists of three such equilateral triangles, its total area is simply three times this value.Understanding how to apply these formulas efficiently allows for accurate calculation of areas in more complex geometric figures.