## Area of a Regular Polygon

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

Area = (1/2) x A x P

Where:

– A represents the apothem of the polygon, which is the perpendicular distance from the center of the polygon to any of its sides

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

Area = (1/2) x A x P

Where:

– A represents the apothem of the polygon, which is the perpendicular distance from the center of the polygon to any of its sides.

– P stands for the perimeter of the polygon, which is the total length of all its sides.

To find the area of a regular polygon, you need to know the length of one side, the number of sides, and the apothem or height. The apothem can be calculated using the formula:

Apothem = s / (2 x tan(180/n))

Where:

– s represents the length of one side of the polygon.

– n is the number of sides.

Once you have the apothem and the perimeter, you can substitute these values into the formula above to find the area of the regular polygon.

Let’s consider an example to better understand the process. Suppose we have a regular hexagon (a polygon with six equal sides) with a side length of 5 units. We need to find its area.

First, let’s find the apothem (A):

Apothem = 5 / (2 x tan(180/6))

Apothem = 5 / (2 x tan(30))

Apothem = 5 / (2 x (sqrt(3)/3))

Apothem = 5 / (2 x sqrt(3)/3)

Apothem = 5 x (3 / (2 x sqrt(3)))

Apothem = 15 / (2 x sqrt(3))

Apothem ≈ 4.33 (rounded to two decimal places)

Next, we calculate the perimeter (P) of the hexagon:

P = 6 x 5 (since there are six sides with each side measuring 5 units)

P = 30 units

Now, we can substitute these values into the formula to find the area (Area):

Area = (1/2) x A x P

Area = (1/2) x 4.33 x 30

Area = 2.165 x 30

Area ≈ 64.95 square units

Therefore, the area of the regular hexagon, with a side length of 5 units, is approximately 64.95 square units.

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