Question

Find the approximate area of Fort Jefferson in Dry Tortugas National Park, Florida, if each side is 460 feet long and an apothem is about 398 feet long.

Short answer

The area of Fort Jefferson is approximately 549,240 square feet.

Step-by-step solution

Understanding the Problem

To find the area of Fort Jefferson, we need to determine the area of a regular polygon. Based on given measurements, we know each side of the fort is 460 feet long, and it has an apothem measuring 398 feet. The fort has 6 sides, indicating that it is a hexagon.

Formula for Area of a Regular Polygon

The formula for the area of a regular polygon is \( A = \frac{1}{2} \times Perimeter \times Apothem \). We'll use this formula to find the area of Fort Jefferson.

Calculate the Perimeter

The perimeter of a polygon is the sum of the lengths of all its sides. For a hexagon, the perimeter is given by \(6 \times \text{side length}\). Substitute 460 feet for the side length: \[Perimeter = 6 \times 460 = 2760 \text{ feet}\]

Calculate the Area

Substitute the values of the perimeter and apothem into the area formula: \[A = \frac{1}{2} \times 2760 \times 398\]Calculate the area: \[A = \frac{1}{2} \times 2760 \times 398 = 549,240 \text{ square feet}\]

State the Final Answer

The approximate area of Fort Jefferson is 549,240 square feet.

Key concepts

Understanding Hexagons

A hexagon is a six-sided polygon. In a regular hexagon, all sides and angles are equal, making it symmetrical and easy to work with when calculating geometric properties. The hexagon is a common shape found in nature and design due to its efficient use of space and strength. All interior angles in a regular hexagon are 120 degrees, adding up to a total of 720 degrees for the entire shape.

• Symmetrical figure with equal sides and angles
• Each interior angle measures 120 degrees
• Total interior angle sum is 720 degrees

Understanding the basic properties of hexagons is crucial when solving problems related to their area, perimeter, and dimensions.

Calculating the Perimeter

The perimeter of a polygon is the total length around its boundary. For regular polygons like hexagons, where all sides are the same length, calculating the perimeter is straightforward. You simply multiply the length of one side by the number of sides.
For Fort Jefferson, which is shaped like a regular hexagon, each side is 460 feet. Thus, the perimeter can be calculated as follows:

• Number of sides: 6
• Length of one side: 460 feet

Therefore, the perimeter is:\[Perimeter = 6 \times 460 = 2760 \text{ feet}\] This formula applies to any regular hexagon and provides the necessary value to calculate the area using the polygon's apothem.

Role of the Apothem

The apothem of a regular polygon is a line from the center of the polygon perpendicular to one of its sides. Think of it as the shortest radius that can be drawn to the polygon's sides. The apothem is crucial because it simplifies the calculation of the polygon's area.

• Runs from the center to the midpoint of a side
• Used to calculate the area of a regular polygon

In our example with Fort Jefferson, the apothem is 398 feet. Using the formula for the area of a regular polygon, which is \[A = \frac{1}{2} \times \text{Perimeter} \times \text{Apothem}\], we substitute the apothem and the previously calculated perimeter to find the area. The apothem plays a vital role in determining the size of the space contained within a polygon, making it an essential concept in geometry.