Let

x = the width of the garden

y = the length of the garden

2x + 2y = 16

x + y = 16

y = 16 - x

Using the Pythagorean Theorem we find the length of the diagonal 'd':

d^2 = x^2 + y^2

d^2 = x^2 + (16 - x)^2

d^2 = 2x^2 - 32x + 256

We need to find the minimum of the quadratic function: d = 2x^2 - 32x + 256

Since the quotient in front of x^2 is 2>0 the function has minimum in its vertex:

x min = -b/(2a) where a = 2, b = -32

x min = -(-32)/(2*2)

x min = 8

For the width of 8 m the length is 16 - 8 = 8 m.

The area of that the rectangular garden with length of 8 meters and width of 8 meters is:

A = 8*8

A = 64 m^2