I cant draw the figure here. Lets let the sides of the square be "x"

But there are important points we need to understand about a diagonal of a square..A diagonal of a square bisects the square into to equal triangles. Hence the 90° angle is also bisected into two 45° angles..

You then can use the formula of area of a triangle A = ab x sinФ..a and b are sides of the triangle.

For triangle ABP, the area is:

          A = ab x sinФ
             = (x)(x/4) x sin45
          A = (x²/4) x (0.707)
But since area of quadrilateral ABPD is 1 square unit, it follows that the area of triangles ABP and ABD are each 1/2 square units..

So we will have:

          1/2 = (x²/4) x (0.707)
           2 = 0.707x²
      1.68 = x

Now we got the length of each side of the square ABCD which is 1.68.

Area of the square ABCD = x²
                                     = 1.68²
                             Area = 2.82 square units   ANSWER

i apologize for the formula A = ab x sin(phi) should be A = 1/2 ab x sin(phi)..
please correct the formula..and you will get x = 2.38 units...and Area = 5.66 square units.