Fraction of shaded region

If the decagon were divided into 10 congruent triangles, each with one of the 10 sides being its base and the third vertex being the centre of the decagon, then each of these triangles would have $\dfrac{1}{10}$ the area of the decagon. The given blue triangle has the same base as one of these triangles but twice the height, and thus has an area $2 \times \dfrac{1}{10} = \boxed{\dfrac{1}{5}}$ that of the decagon.

Let the side length and the apothem of the decagon be $a$ and $h$ respectively. Then the area of the blue triangle $A_{\color{#3D99F6} \text{blue}} = \dfrac 12 a(2h) = ah$ . The area of each of the ten central-angle triangle $A_\triangle = \frac 12 ah$ . The area of the decagon $A_{\text{decagon}} = 10 A_\triangle = 5ah$ . The fraction of the decagon is shaded blue $\dfrac {A_{\color{#3D99F6} \text{blue}}}{A_{\text{decagon}}} = \dfrac {ah}{5ah} = \boxed{\dfrac 15}$ .