Call the side of the square, S
BH = sqrt (2) S - 8 + 4 = sqrt (2)S - 4
BC = 4
EH = sqrt (2)S
EF = S
Using similar triangles
EH/ EF = BH / BC
sqrt (2) S / S = (sqrt (2)S - 4) / 4
sqrt (2) * 4 = sqrt (2)S - 4
4sqrt (2) + 4 = sqrt (2) S
S = 4 ( sqrt (2) + 1) / sqrt (2)
S = 2sqrt(2) ( sqrt (2) + 1) = 4 + 2sqrt (2) = 4 + sqrt (8)
Area of square = (4 + sqrt (8) )^2 = 16 + 8sqrt (8) + 8 = 24 + 8sqrt(8) =
24 + 16sqrt (2) ≈ 46.63