Question 761966
It must be a square shape
Let the sides = {{{ W }}} and {{{ L }}}
Let {{{ A }}} = area
Let {{{ P }}} = perimeter
{{{ A = W*L }}}
Suppose you have {{{ 100 }}} units of fencing.
The units can be anything.
Now I can say
{{{ P = 2L + 2W }}}
{{{ 100 = 2L + 2W }}}
{{{ L + W = 50 }}}
{{{ L = 50 - W }}}
By substitution:
{{{ A = W*( 50 - W ) }}}
{{{ A = -W^2 + 50W }}}
If you plot {{{ A }}}  on the vertical axis and
{{{ W }}} on the horizontal, this is a parabola
The maximum is at {{{ W(max) = -b/2a }}}
where the equation has the form {{{ A = a*W^2 + b*W }}}
{{{ W(max) = -50/2*(-1) }}}
{{{ W(max) = 25 }}}
and, since
{{{ L = 50 - W }}}
{{{ L = 50 - 25 }}}
{{{ L = 25 }}}
Both length and width are 25, so {{{ A(max) }}} is a square