Problem 41 - Algebra and Trigonometry with Analytic Geometry

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 Pentagon encloses an area of approximately 1,457,248 square feet.

Step 1: Understand the Problem

We are asked to find the area enclosed by a regular pentagon, where each side length is 921 feet. A regular pentagon means all sides and angles are equal.

Step 2: Formula for Area of Regular Pentagon

The area of a regular pentagon with side length \( s \) can be found using the formula: \[ \text{Area} = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} \times s^2 \]

Step 3: Substitute the Given Side Length

Substitute \( s = 921 \) feet into the formula:\[ \text{Area} = \frac{1}{4} \sqrt{5(5+2\sqrt{5})} \times 921^2 \]

Step 4: Calculate the Inside of the Square Root

Calculate \( 5 + 2\sqrt{5} \). Approximate \( \sqrt{5} \approx 2.236 \):\[ 5 + 2(2.236) \approx 9.472 \]

Step 5: Calculate the Entire Expression

Now, compute \[ \frac{1}{4} \sqrt{5 \times 9.472} \approx \frac{1}{4} \times \sqrt{47.36} \approx \frac{1}{4} \times 6.88 \approx 1.72 \]

Step 6: Calculate the Area

Now calculate the full area:\[ \text{Area} = 1.72 \times 921^2 \approx 1.72 \times 847,241 \approx 1,457,247.52 \]

Step 7: Final Answer

The area of the Pentagon is therefore approximately 1,457,248 square feet.

Key Concepts

Perimeter of Pentagon

The perimeter of a regular pentagon is quite straightforward to calculate. A regular pentagon is a five-sided polygon where all sides are equal in length. To find the perimeter, you simply multiply the length of one side by the number of sides.

Here's the general formula for a pentagon's perimeter:

• Perimeter = side length \( (s) \times 5 \)

In our example, the given side length of the Pentagon is 921 feet.
So, the calculation would be:

• Perimeter = \( 921 \times 5 = 4605 \) feet.

This value represents the total distance around the pentagon. Understanding this helps in visualizing the entire shape and its enclosed space.

Regular Pentagon Properties

A regular pentagon has some unique properties that distinguish it from other polygons. These characteristics are vital for understanding how to calculate elements like the area.

Some key properties include:

• All five sides are equal in length.
• All five internal angles are equal, each measuring \(108^\circ\).
• Symmetrical nature means it can be divided into five identical triangles, which can be helpful when calculating the area.

These characteristics ensure uniformity across the shape, making mathematical calculations more predictable and easier. The symmetry and equal angles mean there's a balance, which is ideal for applying known geometric formulas.

Pentagon Area Formula

Finding the area of a regular pentagon involves a specific formula that's useful for understanding how much space is enclosed by its sides.

For a regular pentagon, the formula is derived using trigonometry and square roots:

• Area = \(\frac{1}{4} \sqrt{5(5+2\sqrt{5})} \times s^2\)

Here, \(s\) represents the side length of the pentagon. This formula involves:- Understanding of square roots and their approximations (for instance, \(\sqrt{5} \approx 2.236\)).- Plugging the side length into the formula and solving.In the case of the Pentagon building, substituting \(s = 921\) feet gives:

• Area \(\approx 1.72 \times 847,241 = \) approximately 1,457,248 square feet.

This approximation involves multiple calculation steps, where understanding each small mathematical operation, such as squaring the side length and multiplying it by a constant, is crucial.