AMC 12 Mock Exam 1 Problem 5

The diagonals of a rhombus are perpendicular bisectors of each other. Half of the known diagonal is $6$ and the side is $10$, so half of the other diagonal is $\sqrt{10^2-6^2}=8$, and the other diagonal is $16$. The area is $\frac12\cdot12\cdot16=96$.

The center of the rhombus is equidistant from all four sides, so the inscribed circle is centered there. Splitting the rhombus into four triangles from the center, the area equals $r\cdot\frac{4\cdot10}{2}=20r$. Thus $20r=96$ and $r=\boxed{\tfrac{24}{5}}$.