A regular decagon $ABCDEFGHIJ$ has area $n$. Let $m$ be the area of pentagon $ACEGI$. What is $\tfrac{m}{n}$?

1. Place the decagon on a circle of radius $1$ centered at the origin, with vertices at angles $36^\circ k$.
2. Pentagon $ACEGI$ is regular, with area $[ACEGI]=5\cdot\frac12\sin72^\circ=\frac52\sin72^\circ$.
3. The full decagon has area $[ABCDEFGHIJ]=10\cdot\frac12\sin36^\circ=5\sin36^\circ$.
4. Thus $\frac{m}{n}=\frac{\sin72^\circ}{2\sin36^\circ}=\cos36^\circ=\frac{1+\sqrt5}{4}$.