Let  x, y and z be three perpendicular dimensions of the box: the length, the width and the height respectively.

Then you are given

4x + 4y + 4z = 140 cm    (the sum of lengths of all 12 edges),  or, dividing by 4 both sides,

x + y + z = 35 cm.        (1)


Also, you are given that 

 = 21 cm.      (2)    ("the distance from one corner of the box to the farthest corner is 21 cm.")


I will not go to explanation why (2) is that distance: it is easy consequence of the Pythagorean theorem in 3D.


Now square both sides of (2) to get

 = 21^2 = 441.    (3)


Thus we have these two equations simultaneously:

x + y + z = 35,            (1)
 = 441.           (3)


Square both sides of equation (1). You will get

 =  = 1225.


In the last equation, replace  by 441, based on (3). You will get

441 + 2xy + 2xz + 2 yz = 1225,  ====>

2xy + 2xz + 2 yz = 1225 - 441,  ====>

2*xy + 2*xz + 2*yz = 784.


Now notice that the left part of the last equation is nothing else as the total surface are of the box.


So, we got this 


Answer.  The total surface area of the box is equal to 784 cm^2.