Example 1 Given that the three pairs of opposite edges of a tetrahedron are $a$, $b$, and $c$. Find the volume of the tetrahedron and the radius of its circumscribed sphere.
As shown in Figure 1, place the tetrahedron inside a rectangular prism with edge lengths $x$, $y$, and $z$, such that the face diagonals of the rectangular prism are the edges of the tetrahedron.
$$
\text { Then }\left\{\begin{array}{l}
x^{2}+y^{2}=c^{2} \\
y^{2}+z^{2}=a^{2} \\
z^{2}+x^{2}=b^{2}
\end{array}\right. \text {. }
$$

Thus, the volume $(V)$ of the tetrahedron is equal to the volume of the rectangular prism minus the volumes of the four triangular pyramids, i.e.,
$$
\begin{array}{l}
V=x y z-\frac{1}{3} \times \frac{1}{2} x y z \times 4=\frac{1}{3} x y z \\
=\frac{\sqrt{2}}{12} \sqrt{\left(a^{2}+b^{2}-c^{2}\right)\left(c^{2}+b^{2}-a^{2}\right)\left(a^{2}+c^{2}-b^{2}\right)} .
\end{array}
$$

Since the circumsphere of the tetrahedron is the same as the circumsphere of the rectangular prism, the radius $(R)$ is half the length of the space diagonal of the rectangular prism, i.e.,
$$
R=\frac{1}{2} \sqrt{x^{2}+y^{2}+z^{2}}=\frac{\sqrt{2}}{4} \sqrt{a^{2}+b^{2}+c^{2}} .
$$