Rhombus has perpendicular diagonals that bisects each other. On diagonal has half lenght 12 in (the whole diagonal will have length 24 in). Let second diagonal has length 2x in, then half of this diagonal has length x in.
Consider right triangle, formed by two halves of diagonals (legs) and one rhombus side (hypotenuse). By the Pythagorean theorem,
[tex] \text{hypotenuse}^2=\text{first leg}^2+\text{second leg}^2 [/tex].
Substitude given values:
[tex] 15^2=12^2+x^2,\\ x^2=225-144,\\ x^2=81,\\ x=9 [/tex].
Then the length of whole diagonal is 18 in. The area of the rhombus is
[tex] A_{rhombus}=\dfrac{d_1\cdot d_2}{2} [/tex].
Substitute values:
[tex] A_{rhombus}=\dfrac{18\cdot 24}{2}=216 [/tex] sq. in.
Answer: [tex] A_{rhombus}=216 [/tex] sq. in.
