Each corner is an isosceles right triangle, so has leg dimension equal to x/√2. There are two of them in the 18-inch measure, so the value of x will satisfy ...
x/√2 + x + x/√2 = 18 in
x(1 + √2) = 18 in . . . . . . . . simplify
x = (18 in)/(1 +√2) . . . . . . . divide by the coefficient of x
x ≈ 7.46 in