Example 8 (I) Given two identical equilateral triangular paper pieces, it is required to cut and fold one into a regular triangular pyramid model, and the other into a regular triangular prism model, so that their total surface areas are equal to the area of the original triangle. Please design a cutting and folding method, and mark it with dotted lines in the diagram, with a brief explanation.
(II) Try to compare the volumes of the regular triangular pyramid and the regular triangular prism you have folded.
(2002 National College Entrance Examination Question)
(I) As shown in Fig. $24-3(\mathrm{a})$, folding along the lines connecting the midpoints of the three sides of an equilateral triangle can form a regular tetrahedron.
As shown in Fig. $24-3$ (b), three identical quadrilaterals are cut from the three corners of the equilateral triangle, with the longer pair of adjacent sides being $\frac{1}{4}$ of the triangle's side length, and one pair of opposite angles being right angles. The remaining part is folded along the dotted lines to form a regular triangular prism without the top base, and the three identical quadrilaterals cut out can precisely form the top base of this regular triangular prism.
(II) According to the above cutting and pasting method, we have $V_{\text {III }}>V_{\text {II }}$.
Assuming the side length of the equilateral triangle paper is 2, then the bases of the regular tetrahedron and the regular triangular prism are both equilateral triangles with a side length of 1, and their area is $\frac{\sqrt{3}}{4}$, and their heights are:
$$
\begin{array}{c}
h_{\text {III }}=\sqrt{1-\left(\frac{2}{3} \cdot \frac{\sqrt{3}}{2}\right)^{2}}=\frac{\sqrt{6}}{3}, h_{\text {II }}=\frac{1}{2} \cdot \tan 30^{\circ}=\frac{\sqrt{3}}{6} . \\
V_{\text {III }}-V_{\text {II }}=\left(\frac{1}{3} h_{\text {III }}-h_{\text {II }}\right) \cdot \frac{\sqrt{3}}{4}=\frac{2 \sqrt{2}-3}{24}V_{\text {I }}
\end{array}
$$