Question

A regular hexagon is made up of six congruent equilateral triangles. Find the area of a regular hexagon whose perimeter is 24 feet.

Short answer

The area is \(24\sqrt{3}\) square feet.

Step-by-step solution

Find the Side Length of the Hexagon

The perimeter of the regular hexagon is given as 24 feet. Since the hexagon has 6 sides of equal length, the side length can be found by dividing the perimeter by the number of sides. Therefore, the side length of the hexagon is \( \frac{24}{6} = 4 \) feet.

Calculate the Area of One Equilateral Triangle

The hexagon can be divided into 6 equilateral triangles. To find the area of one triangle, use the formula \( \text{Area} = \frac{\sqrt{3}}{4}a^2 \), where \( a = 4 \) feet is the side length.\[ \text{Area of one triangle} = \frac{\sqrt{3}}{4} \times 4^2 = 4\sqrt{3} \text{ square feet}\]

Calculate the Total Area of the Hexagon

Since the hexagon is made up of 6 congruent equilateral triangles, the area of the hexagon is 6 times the area of one triangle. Therefore, the total area is:\[ \text{Area of hexagon} = 6 \times 4\sqrt{3} = 24\sqrt{3} \text{ square feet}\]

Key concepts

Equilateral Triangle

Understanding the basics of an equilateral triangle is key to exploring a regular hexagon since each regular hexagon can be divided into six congruent equilateral triangles. An equilateral triangle has three sides of equal length and three angles that are each 60 degrees.
This uniformity makes calculations involving equilateral triangles consistent and straightforward. The side length in our example of a regular hexagon is 4 feet, and knowing this applies to each of the six equilateral triangles. Recognizing these properties allows for seamless area and perimeter calculations, as everything within an equilateral triangle is balanced and predictable.

Perimeter

The perimeter of a shape is simply the total length of its outer boundary.
For a regular hexagon, calculating the perimeter is straightforward since all sides are of equal length. Given in the problem, the perimeter is 24 feet. To find how long each side of the hexagon is, you divide this total by the number of sides.

• Perimeter formula for a hexagon: Divide the total perimeter by 6: \( \text{side length} = \frac{\text{perimeter}}{6} \).
• In our example, \( \frac{24}{6} = 4 \) feet per side.

This step makes it clear how congruency simplifies understanding shapes like regular hexagons. Knowing the length of each side helps in further operations like calculating area.

Area Calculation

Finding the area of a geometric shape often involves understanding its composition.
In this case, a regular hexagon's area is found by first calculating the area of one of its equilateral triangles. The formula to use for an equilateral triangle's area when you know its side length is \( \frac{\sqrt{3}}{4}a^2 \), where \( a \) is the side length.

• Plug in the side length of 4 feet to get the area of a single triangle: \( \text{Area} = \frac{\sqrt{3}}{4} \times 4^2 = 4\sqrt{3} \text{ square feet} \).
• Since the hexagon is composed of six of these triangles, multiply the area of one triangle by 6.
• This results in the total area of the hexagon: \( 6 \times 4\sqrt{3} = 24\sqrt{3} \text{ square feet} \).

Multiplying by the number of triangles leverages their symmetry, allowing for an efficient calculation of the hexagon's full area.