Answer:
B. regular hexagon
Explanation:
Let's say he gives you x length of fencing.
Rectangle
The rectangle has length L and width 2L.
The perimeter is L + 2L + L + 2L
L + 2L + L + 2L = x
6L = x
L = x/6
2L = 2x/6 = x/3
The area of the rectangle is
A = length * width
A = x/6 * x/3 = x^2/18
A = 0.0556x^2
For the rectangle, the area is x^2/18.
Square
The length of each side is L.
The perimeter is 4L.
4L = x
L = x/4
The area of the square is
A = side^2
A = (x/4)^2 = x^2/16
A = 0.0625x^2
For the square, the area is x^2/16.
Regular hexagon
The length of fencing is x.
The regular hexagon has 6 congruent sides.
The length of each side is L.
The perimeter is 6L
6L = x
L = x/6
The length of each side is x/6.
A hexagon can be divided into 6 congruent equilateral triangles.
Each triangle can be divided into 2 congruent right triangles by drawing an altitude from the vertex at the center of the triangle to the midpoint of the side of the hexagon. Each right triangle is a 30-60-90 triangle. Half of the side of the hexagon measures x/12. By the ratios of the lengths of sides of a 30-60-90 triangle, the other leg has length x*sqrt(3)/12. The area of the hexagon is 6 times the area of one equilateral triangle.
area = 6 * area of equilateral triangle
area = 6 * bh/2
area = [6 * x/6 * x * sqrt(3)/12]/2
area = x^2*sqrt(3)/24
A = 0.0722x^2
The approximate areas are:
rectangle: 0.0556x^2
square: 0.0625x^2
hexagon: 0.0722x^2
The largest area is that of the regular hexagon.