jmc

The area formula for a rhombus is $A=\frac{1}{2}\cdot d_1\cdot d_2$, where $d_1$ and $d_2$ are the lengths of its two diagonals. The points $(4,0)$ and $(-4,0)$ are opposing vertices of the rhombus and both lie on the x-axis. Since the third point $(0,K)$ lies on the y-axis, and the diagonals of a rhombus are perpendicular bisectors, it follows that the intersection of the diagonals must be at the origin. Thus, the last vertex is the point $(0,-K)$. It follows that the diagonals have length $8$ and $2K$, and the area is equal to $80=\frac{1}{2}\cdot 8\cdot (2K)=8K$. Thus, $K=\frac{80}{8}=\boxed{10}$.