I will examine a 100-sided regular polygon. If you split it into 100 pizza slice triangles, then each one has a base of 1 foot (with central angle of 3.6 degrees), and the other two sides are equal. Split it down the middle to form two right triangles, each with base 0.5 across from an 1.8 degree angle.

To find the area, we need the height of the triangle (and since this is a right triangle, we need the other leg). Since the tangent ratio gives us opposite over adjacent, we have

tan(1.8)=0.5/x ---> x = 0.5/tan(1.8)

Now we have the height of each pizza slice, so we can find the area:

A = 1/2*1*x = x/2

And multiplying this by 100 gives us the polygon's area:

A = 50x = 795.51ft^2... pretty close to the circle!

The ratio of the area of a closed figure to the perimeter of the figure increases as the number of sides is increased. This can be illustrated by starting with a figure with 3 sides, (as it is not possible to have a closed figure with 2 sides)

The area of an equilateral triangle with a perimeter of 100' is given by ((sqrt 3)/4)*(100/3)^2 = 481.12 square feet.

For a square, the area increases to 25^2 = 625 square feet

Similarly, the area of enclosed by the figure formed would increase and reach a maximum when the number of sides is infinity, which is a circle.

The area of the circle is pi*(100^2/4*pi^2) = 100^2/4*pi = 795.77 square feet

A circle gives the largest area.

Circumference of a circle equals 2pi(r)

Area of a circle is pi(r^2).

Radius of circle with 100 ft circumf. is 100/2pi = 50/pi

Area of circle with 100 ft. circumf. = pi (50/pi)^2 = 50^2/pi = 2500/3.14 = 796 square feet.

Now take a square enclosure, sides 25 feet each. Area is 25 x 25 = 625 square feet.

Now take an equilateral right triangle: Area of triangle = 1/2 base x height. For this example, sides 33.3 feet each, area is 1/2 (33.3)(33.3) = 554 square feet.

The circle gives the largest area, compared to square or triangle.

Okay, since the first poster took circles and squares, I will look at hexagons and a pentagon.

A hexagon has 6 sides. Each side will take 16.6 feet of fencing ( 100/6=16.6). A hexagon created by using 100 feet of fencing will have an area of 715.93 square feet.

A pentagon has five sides. Each side of fencing will be 20 20 feet long (100/5=20). Therefore, the area of a pentagon with a 20 foot long side would be 688.19 square feet.

A circle with a 100' circumference will give you an enclosed area of 796 square feet. Circumference is equal to 2 pi radius, so

C = 2(pi)r

100 = 2(pi)r

15.92 = r

the Area of a circle is (pi) r^2, so

A = (pi) r^2

A = (3.1416) (15.92 feet)^2

A = 796 sq feet.

In contrast, consider a square, whose area is length * width. Each side must be 25 feet, so

A = l * w = 25 feet * 25 feet = 625 sq feet.

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