What is the formula for calculating the area of a regular polygon, and how is it solved for the variable ‘a’?

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Understanding the Area of a Regular Polygon

A regular polygon is a polygon with all its sides and angles equal. To find the area of a regular polygon, we typically use the following formula:

A = (1/2) * Perimeter * apothem

In this equation:

  • A represents the area of the polygon.
  • Perimeter is the total length around the polygon, calculated as the number of sides multiplied by the length of one side.
  • Apothem is the distance from the center of the polygon to the midpoint of one of its sides.

Solving for the Apothem (a)

If we want to solve the area formula for the apothem (denoted as ‘a’), we can rearrange the formula. Starting with:

A = (1/2) * Perimeter * a

To isolate ‘a’, we would rearrange the equation as follows:

  1. Multiply both sides by 2 to eliminate the fraction:
    2. 2A = Perimeter * a
  2. Next, divide both sides by the Perimeter:
    3. a = (2A) / Perimeter

Now we have ‘a’ isolated, which represents the formula for the apothem in terms of the area and perimeter of a regular polygon.

Example Calculation

For example, consider a regular hexagon (6-sided polygon) with a perimeter of 18 units and an area of 27 square units:

a = (2 * 27) / 18

Calculating this gives:

a = 54 / 18 = 3

Thus, the apothem of the hexagon is 3 units.

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