6. For a triangle with side lengths $a$, $b$, and $c$, its area is $\frac{1}{4}$, and the radius of the circumscribed circle is 1. If
$$
s=\sqrt{a}+\sqrt{b}+\sqrt{c}, \quad t=\frac{1}{a}+\frac{1}{b}+\frac{1}{c},
$$

then the relationship between $s$ and $t$ is
(A) $s>t$;
(B) $s=t$;
(C) $s<t$;
(D) Uncertain.
6. (C)
\[
\begin{aligned}
\because \quad C & =2 R \sin C=2 \sin C, \quad \text { and } \quad \frac{1}{2} a b \sin C=\frac{1}{4}, \\
\therefore \quad a b c & =1, \text { thus } \\
t & =\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\
& =\frac{1}{2}\left(\frac{1}{a}+\frac{1}{b}\right)+\frac{1}{2}\left(\frac{1}{b}+\frac{1}{c}\right)+\frac{1}{2}\left(\frac{1}{c}+\frac{1}{a}\right) \\
& \geqslant \sqrt{\frac{1}{a b}}+\sqrt{\frac{1}{b c}}+\sqrt{\frac{1}{c a}} \\
& =\frac{\sqrt{c}+\sqrt{a}+\sqrt{b}}{\sqrt{a b c}} \\
& =\sqrt{a}+\sqrt{b}+\sqrt{c}=S .
\end{aligned}
\]

And the equality cannot be achieved, otherwise \(a=b=c=R=1\), which is clearly impossible, hence the answer is (C).