The area of the circular enclosure is 50,929.58 ft² and the area of the square 40,000 ft², which indicates that the area of the circular enclosure is larger when the perimeter are the same
The steps used to compare the areas of the circular and square enclosures are as follows;
The area of a circular enclosure made with a 800 ft fence can be found using the formula for the circumference and area of a circle as follows;
The circumference of a circle, C = 2·π·r
Where r is the length of the radius of the circle
The circumference is the length of the fencing, therefore; C = 800 ft
800 = 2·π·r
r = 800/(2·π)
r = 400/π
Area, A, of a circle = π·r²
Therefore, area of the circular enclosure is; A = π × (400/π)²
π × (400/π)² = 160000/π
160000/π ≈ 50929.58
A ≈ 50929.58 ft²
The area of a square enclosure can be found using the formula for the perimeter of a square as follows;
Perimeter of a square, P = 4 × Side length of the square;
Let s represent the side length of the square, we get;
P = 4·s
The length of the enclosure is equivalent to the perimeter of the square, therefore; P = 800 ft
800 ft = 4·s
s = 800 ft/4
s = 200 ft
Area of a square = The square of the side length of the square, s²
Area of the square enclosure = (200 ft)²
(200 ft)² = 40,000 ft²
The area of the circular enclosure, 50,929.58 ft² is much larger than than the area of the of a square enclosure, which is 40,000 ft²