let

A = the angle formed by the diagonal of a cube and a diagonal of a face of a cube

x = the length of the side of the cube

x√2 = the length of the face diagonal

x√3 = the length of the space diagonal

cos A = x√2/x√3

cos A = √2/√3

A = 35.26 degrees

the angle formed by the diagonal of a cube and a diagonal of a face of a cube is 35.26 degrees.