Problem 41 - Precalculus: Functions and Graphs

Question

The Pentagon is the largest office building in the world in terms of ground area. The perimeter of the building has the shape of a regular pentagon with each side of length 921 feet. Find the area enclosed by the perimeter of the building.

Step-by-Step Solution

Answer

The area enclosed by the Pentagon's perimeter is approximately 684,119 square feet.

Step 1: Understand a Regular Pentagon

A regular pentagon is a five-sided polygon where all the sides are of equal length, and all the interior angles are equal.

Step 2: Identify Key Formula

To find the area of a regular pentagon, we can use the formula: \[ A = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2 \] where \( s \) is the side length of the pentagon.

Step 3: Substitute the Given Side Length

Substitute the given side length of 921 feet into the formula: \[ A = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} (921)^2 \]

Step 4: Calculate the Expression Inside the Square Root

Calculate \( 5(5 + 2\sqrt{5}) \). First, find \( 5 + 2\sqrt{5} \), for an approximate value: \( \approx 9.472 \). Then multiply by 5 to get \( \approx 47.36 \).

Step 5: Calculate the Area

Substitute the result from Step 4 and the side length into the formula. Continue calculations: \( A \approx \frac{1}{4} \times 47.36 \times 921^2 \approx 684118.598 \) square feet.

Step 6: Rounding Off

Since the area should be practical and concise, round the result to a reasonable number of decimal places. The area is \( 684,119 \) square feet.

Key Concepts

Perimeter Calculation

In geometry, calculating the perimeter of a regular polygon is relatively straightforward. A regular pentagon, by definition, has five equal sides. Therefore, the perimeter of a regular pentagon can be calculated easily by multiplying the length of one side by five.
Here’s how you do it:

• Identify the length of one side: For our exercise, each side of the pentagon measures 921 feet.
• Multiply the side length by the number of sides: Since a pentagon has five sides, the perimeter (P) is calculated by the formula: \( P = 5 \times s \).
• Substitute the given side length into the formula: \( P = 5 \times 921 \), resulting in a perimeter of 4605 feet.

Calculating the perimeter is crucial, especially when dealing with real-world structures, such as The Pentagon, as it informs on the total boundary length.

Polygon Area Formula

Calculating the area of a regular pentagon is not as straightforward as the perimeter. We use a specific formula derived for regular polygons. The area \( A \) of a regular pentagon can be found using:
\[ A = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2 \]
Where \( s \) represents the side length of the pentagon.

• Key observations: This formula involves using the square root and multiplying by the square of the side length.
• Steps involved: First, compute the expression inside the square root, which is essentially a geometric constant for regular pentagons. Approximate calculations show that this comes to about 47.36.
• Area calculation: For our given side length of 921 feet, substitute into the formula giving: \( A = \frac{1}{4} \times 47.36 \times 921^2 \). This yields an approximate area of 684,119 square feet.

Understanding this formula allows us to compute areas of regular polygons efficiently and is fundamental in geometry applications.

Geometry Applications

Geometry, with its formulas and calculations, extends beyond classroom exercises and into real-life applications. Calculating perimeters and areas of shapes like regular pentagons is crucial in architecture, engineering, and planning.

• Architectural Design: The Pentagon, as an office building, exemplifies the application of geometric calculations. Knowing the perimeter aids in planning logistics, such as security and access control while calculating the area informs on usage and capacity.
• Land and Space Management: Perimeter measurements define boundaries of property, while area calculations help in maximizing usability and functionality of the space. Managing resources effectively is vital in crowded urban spaces.
• Art and Design: Geometry influences designs, from art installations to functional spaces.

These applications highlight why proficiency in geometry and understanding polygon calculations is essential for various professional fields.