Draw the diagonal of the rhombus.


The diagonals divide the rhombus in four right-angled congruent triangles 

    (since the diagonals of the rhombus bisect each other and are perpendicular).

Each small right-angled triangle has the hypotenuse of 10 cm and one leg of  = 6 cm.

Hence, the second leg of each small right angled triangle is  =  =  = 8 cm.


Then the area of each small right-angled triangle  is  = 24 cm^2.


Thus the area of the rhombus is 4*24 = 96 cm^2.


Then the height of the rhombus =  = 9.6 cm.