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Chapter 7: Problem 88

If the perimeter of a regular hexagon is \(120 \mathrm{ft}\), what is its area?

Short Answer

Expert verified
The area of the hexagon is \ (600 \sqrt{3} \, \text{ft}^2)\.

Step by step solution

01

Determine the Length of One Side

A regular hexagon has six equal sides. To find the length of one side, divide the perimeter by 6. So, the length of one side is \ \[ s = \frac{120 \, \text{ft}}{6} = 20 \, \text{ft} \]
02

Use the Formula for the Area of a Regular Hexagon

The area \ (A) \ of a regular hexagon with side length \ (s) \ is given by: \[ A = \frac{3 \sqrt{3}}{2} s^2 \]
03

Substitute the Side Length into the Formula

Substitute \ (s = 20 \, \text{ft}) \ into the formula: \[ A = \frac{3 \sqrt{3}}{2} (20)^2 \]
04

Simplify the Expression

Calculate \ (A): \ \[ A = \frac{3 \sqrt{3}}{2} \times 400 \implies A = 600 \sqrt{3} \, \text{ft}^2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perimeter
The perimeter of a shape is the total length of all its sides combined. For a regular hexagon, this means summing the lengths of its six equal sides. If you know the perimeter of a regular hexagon, you can easily find the length of one side by dividing the perimeter by 6. For example, if a hexagon's perimeter is 120 feet, each side is \[ s = \frac{120 \, \text{ft}}{6} = 20 \, \text{ft} \]. This step is important because knowing side length is necessary for calculating further properties like area.
Side Length of Hexagon
The side length of a regular hexagon can be found once you know its perimeter. This is because in a regular hexagon, all six sides are of equal length. For example, if the perimeter is 120 feet, the length of one side is achieved with the formula \[ s = \frac{\text{perimeter}}{6} \]. In our exercise, since the perimeter of the hexagon is 120 feet, dividing this by 6 gives us \[ s = 20 \, \text{ft} \]. This uniformity simplifies subsequent calculations, such as determining the area.
Hexagon Area Formula
To find the area of a regular hexagon, we use a specific formula that incorporates the side length. The formula is: \[ A = \frac{3 \sqrt{3}}{2} s^2 \]. Here, \( s \) represents the length of one side of the hexagon. Plugging the side length that we calculated (20 feet) into the formula, we get: \[ A = \frac{3 \sqrt{3}}{2} (20)^2 \]. This formula is derived from dividing the hexagon into six equilateral triangles and calculating the area of those triangles before summing them up.
Simplification of Expressions
After substituting the side length into the hexagon area formula, the expression often needs to be simplified. In our example, we substitute \( s = 20 \) into \[ A = \frac{3 \sqrt{3}}{2} (20)^2 \]. This can be simplified step-by-step: \[ A = \frac{3 \sqrt{3}}{2} \times 400 \], and then \[ A = 600 \sqrt{3} \, \text{ft}^2 \]. Simplifying expressions ensures that your final answer is in its most useful and comprehensible form. Breaking down each step makes it clearer how the numbers interact in the formula.

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