Problem 63 - Precalculus

Question

A regular hexagon has sides measuring 3 feet. What is its area? Recall that the measure of an angle of a regular \(n\) -gon is given by the formula angle \(=\frac{180^{\circ}(n-2)}{n}\) (IMAGE CANNOT COPY)

Step-by-Step Solution

Answer

The area of the hexagon is approximately 23.38 square feet.

Step 1: Calculate the Central Angle

A hexagon is a polygon with 6 sides, so we have to calculate the measure of a central angle in the hexagon. For a regular hexagon, the formula for the internal angle is \( \frac{180^{\circ}(n-2)}{n} \). Substitute \( n = 6 \): \[ \text{Angle} = \frac{180^{\circ} \times (6-2)}{6} = \frac{180^{\circ} \times 4}{6} = 120^{\circ}. \]

Step 2: Divide the Hexagon into Triangles

A regular hexagon can be divided into 6 equilateral triangles. This is because each central angle at the hexagon's center measures \( \frac{360^{\circ}}{6} = 60^{\circ} \), which is the vertex angle of each triangle. Each equilateral triangle has sides measuring the same as the hexagon sides, 3 feet.

Step 3: Calculate Area of One Equilateral Triangle

The formula for the area of an equilateral triangle with side length \( s \) is \( \frac{\sqrt{3}}{4} s^2 \). Plug in \( s = 3 \) feet: \[ \text{Area of one triangle} = \frac{\sqrt{3}}{4} \times (3^2) = \frac{\sqrt{3}}{4} \times 9 = \frac{9\sqrt{3}}{4}. \]

Step 4: Calculate the Total Area of the Hexagon

Since the hexagon is divided into 6 equilateral triangles, multiply the area of one triangle by 6: \[ \text{Total area} = 6 \times \frac{9\sqrt{3}}{4} = \frac{54\sqrt{3}}{4} = \frac{27\sqrt{3}}{2}. \] This evaluates approximately to 23.38 square feet.

Key Concepts

Regular Hexagon

A regular hexagon is a fascinating geometric shape, notable for its six equal sides and angles. Each side of a regular hexagon is the same length, and each of its interior angles is identical. This consistency makes it one of the most symmetric polygons you'll encounter.

Interestingly, a regular hexagon can be divided into simpler geometric shapes: equilateral triangles. This can be immensely helpful when calculating its area, as breaking it down simplifies the computation process. Inside every regular hexagon, there are six equilateral triangles. Each triangle has a vertex at the center of the hexagon and two vertices on the hexagon's circumference.

By understanding the structure of a regular hexagon, you can uncover the properties of its internal angles and sides, laying the groundwork for more complex calculations involving the hexagon's surface area.

Equilateral Triangle Area

The area of an equilateral triangle is a key concept when dealing with regular hexagons, especially since a hexagon can be divided into six of them. Let's delve into how to find the area of these triangles.

An equilateral triangle, by definition, has three equal sides and three equal angles, each measuring 60 degrees. This perfect symmetry allows us to use a specific formula to find its area:

• The formula is: \( \frac{\sqrt{3}}{4} s^2 \)

where \( s \) is the length of a side.

Substituting the side length of 3 feet, which matches the sides of our hexagon, the area of each triangle becomes:

• \( \frac{\sqrt{3}}{4} \times 9 \) which simplifies to \( \frac{9\sqrt{3}}{4} \)

Knowing the area of one triangle can easily lead us to the total area of the hexagon by simply multiplying by six.

Polygon Angles

Understanding polygon angles is crucial in geometry, particularly when calculating a shape's properties like area and perimeter. For regular polygons, like our hexagon, the angles have specific predictable values.

In a regular hexagon, each internal angle can be calculated using a general formula for any regular \( n \)-gon:

• \( \frac{180^{\circ}(n-2)}{n} \)

Here, \( n \) stands for the number of sides. For a hexagon, \( n = 6 \), meaning each angle measures \( 120^{\circ} \). This internal angle is consistent across all angles in a regular hexagon.

Additionally, the hexagon can be broken down into six equilateral triangles. Each triangle is formed using the central angle derived from dividing the full circle at the hexagon's center into six, giving \( 60^{\circ} \) for each triangle's vertex angle. Understanding these angles helps simplify the process of calculating the hexagon's area by focusing on the constituent triangles.