Rectangle $ABCD$ has side lengths $AB=8$ and $BC=6$. Equilateral triangles $AKB$, $BLC$, $CMD$, and $DNA$ are constructed outward on its sides. What is the area of $KLMN$?
The quadrilateral consists of rectangle $ABCD$, the four equilateral triangles, and four corner triangles. The corner triangles each have area $\frac12(8)(6)\sin150^\circ=12$, so together they have area $48$.
The equilateral triangles have total area $2\cdot\frac{\sqrt3}{4}(8^2)+2\cdot\frac{\sqrt3}{4}(6^2)=50\sqrt3$. Thus $[KLMN]=48+48+50\sqrt3=\boxed{96+50\sqrt3}$.
