smc

Assume the length of the side of the square is 2, WLOG. This means the side of one t section is 1. As the lines are at clock face positions, each section has a $\frac{360}{12}=30$ degree angle from the center. So each section t is a $30-60-90$ triangle with a long leg of 1. Therefore, the short leg is $\frac{1}{\sqrt{3}}$. This makes the area of each $t=\frac{1}{2}\cdot b\cdot h=\frac{1}{2}\cdot 1\cdot \frac{1}{\sqrt{3}}=\frac{1}{2\sqrt{3}}$ The total area comprises $4q+8t$, so $$4q+(8\cdot \frac{1}{2\sqrt{3}})=2^2=4$$ $$4q+\frac{4}{\sqrt{3}}=4$$ $$4q=4-\frac{4}{\sqrt{3}}$$ $$q=1-\frac{1}{\sqrt{3}}=\frac{\sqrt{3}-1}{\sqrt{3}}$$ $$\frac{q}{t}=\frac{\frac{\sqrt{3}-1}{\sqrt{3}}}{\frac{1}{2\sqrt{3}}}=2\cdot (\sqrt{3}-1)=\boxed{2\sqrt{3}-2}$$ $\boxed{2\sqrt{3}-2}$