About Regular Polygon Area Calculation

What is a Regular Polygon?

A regular polygon is a closed shape with all sides equal in length and all interior angles equal in measure. Examples include equilateral triangles, squares, regular pentagons, and so on.

Formula for Regular Polygon Area

The formula to calculate the area of a regular polygon is:

\[A = \frac{1}{2} n s a\]

Where:

• \(A\) is the area of the regular polygon
• \(n\) is the number of sides
• \(s\) is the length of each side
• \(a\) is the apothem (the distance from the center to the middle of any side)

Calculation Steps

1. Determine the number of sides (n) and the side length (s).
2. Calculate the apothem using the formula: \(a = \frac{s}{2 \tan(\frac{\pi}{n})}\)
3. Apply the area formula: \(A = \frac{1}{2} n s a\)
4. Simplify and calculate the final result.

Example

Let's calculate the area of a regular hexagon with side length 5 units:

1. Given:
   • Number of sides (n) = 6
   • Side length (s) = 5 units
2. Calculate the apothem:
   \(a = \frac{5}{2 \tan(\frac{\pi}{6})} \approx 4.33\) units
3. Apply the area formula:
   \(A = \frac{1}{2} \times 6 \times 5 \times 4.33\)
   \(A = 64.95\) square units

Visual representation:

Therefore, the area of the regular hexagon is approximately 64.95 square units.