Find the perimeter and the area of the regular polygon shown. Please show work.

Answer 1

Since we don't have an image provided, let's assume we're looking at a regular polygon with 6 sides (hexagon) for the purpose of this example.

To find the perimeter of a regular polygon, we use the formula:

Perimeter = Number of sides x Length of each side

In a regular polygon, all sides are equal in length, so we just need to find the length of one side. We can do this by using trigonometry.

Let's say the polygon has a radius of 'r'. Using trigonometry, we can determine that the length of each side (s) is given by:

s = 2r * cos(π/n)

where 'n' is the number of sides of the polygon.

For a hexagon (6 sides), we have:

s = 2r * cos(π/6)
s = 2r * cos(30 degrees)
s = 2r * (√3/2)
s = √3r

Now that we have the length of each side, we can find the perimeter:

Perimeter = 6s
Perimeter = 6(√3r)
Perimeter = 6√3r

To find the area of a regular polygon, we use the formula:

Area = 1/2 x Perimeter x Apothem

The apothem is the perpendicular distance from the center of the polygon to the middle of one of its sides. For a regular hexagon, the apothem is equal to the radius of the polygon.

Area = 1/2 x 6√3r x r
Area = 3√3r²

So, the perimeter of the regular hexagon is 6√3r and the area is 3√3r².