One mile is 5280 feet is 1/4 mile is 5280/4 = 1320 feet. Thus a square which is 1/4 mile on each side has an area of 13202 = 1742400 square feet. There are 43,560 square feet in one acre so a 1/4 mile square is 1742400/43560 = 40 acres.
If you have a square with area 40 acres then, from the calculations above, the square will have side length of 1/4 mile and hence a perimeter of one mile. If the 40 acre region is not a square then you need to know the shape to determine the perimeter,