Example 8 (I) Given two identical equilateral triangular pieces of paper, it is required to cut and assemble one into a regular tetrahedron model and the other into a regular triangular prism model, such that their total surface areas are equal to the area of the original triangle. Please design a cutting and assembly method, and indicate the cuts with dotted lines in the diagram, with a brief explanation.
(II) Compare the volumes of the regular tetrahedron and the regular triangular prism you assembled.
(2002 National College Entrance Examination Question)
(I) As shown in Fig. $24-3(\mathrm{a})$, by folding along the lines connecting the midpoints of the three sides of an equilateral triangle, a regular tetrahedron can be formed.
As shown in Fig. $24-3(b)$, three identical quadrilaterals are cut from the three corners of the equilateral triangle. The longer pair of adjacent sides of these quadrilaterals are $\frac{1}{4}$ of the triangle's side length, and one pair of opposite angles are 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 method of cutting and pasting described above, we have $V_{\text {tetrahedron }}>V_{\text {prism }}$.
Assuming the side length of the equilateral triangle paper is 2, then the bases of both the regular tetrahedron and the regular triangular prism are equilateral triangles with side length 1, and their area is $\frac{\sqrt{3}}{4}$. Moreover, their heights are:
Thus, $V_{\text {tetrahedron }}>V_{\text {prism }}$.