A regular polygon is a polygon with all sides equal in length and all internal angles equal. There are currently two methods used in the industry to calculate the area of a regular polygon, which are as follows:

1.Side Length Calculation Method

The area formula for a regular polygon is given by:

Area = (Side Length^2 × Number of Sides) / (4 × tan(π/Number of Sides))

Where “l” represents the side length of the regular polygon, and “n” represents the number of sides. In other words, the area of a regular polygon is equal to the value obtained by multiplying the square of the side length by the number of sides, and then dividing the result by (4 times the tangent of π divided by the number of sides).

To calculate the area of a regular polygon with a given number of sides and side length, we can use the formula provided earlier. Let’s take an example of a regular polygon with 4 sides (i.e., a square) and a side length of 5.

Using the area formula for a regular polygon:

Area = (Side Length^2 × Number of Sides) / (4 × tan(π/Number of Sides))

For a square (4-sided regular polygon) with a side length of 5:

Area = (5^2 × 4) / (4 × tan(π/4))

Since tan(π/4) is equal to 1 (because the tangent of 45 degrees is 1), the formula simplifies to:

Area = (25 × 4) / (4 × 1)
Area = 100 / 4
Area = 25

Therefore, the area of a square with a side length of 5 is 25 square units.

2.Inscribed Circle Radius Calculation Method

The Inscribed Circle Radius Method is another way to calculate the area of a regular polygon. This method is based on the radius of the inscribed circle of the regular polygon (i.e., the perpendicular distance from the center of the polygon to any one of its sides)

The area of a regular polygon can be calculated using the formula:

Area = (n × l × r) / 2

Where:
n is the number of sides of the regular polygon
l is the length of each side
r is the radius of the inscribed circle

This formula provides an alternative method to determine the area of a regular polygon by considering its number of sides, side length, and the radius of the circle that can be inscribed within it.

The formula for the area of a regular polygon is derived by dividing the polygon into n (where n is the number of sides) triangles and then summing up the areas of these triangles.

Let’s illustrate this with an example: Consider a regular pentagon with a side length of 4cm and an inscribed circle radius of 2.38cm. To calculate its area using the formula:

Area = (n × l × r) / 2

We substitute the given values:
n = 5 (because it’s a pentagon)
l = 4cm (the side length)
r = 2.38cm (the radius of the inscribed circle)

Area = (5 × 4cm × 2.38cm) / 2
= 23.8cm^2

So, the area of the regular pentagon is approximately 47.6 square centimeters.

For irregular polygons, the area calculation is more complex and may require different methods or approximations. You can refer to formulas or techniques specifically designed for calculating the area of irregular polygons.

Regarding your teacher’s reminder, equilateral triangles and squares are indeed special cases of regular polygons, and their area can be calculated using the same formula for regular polygons.