请将附件中的题目解答，并配上中文解说和字幕---The image displays a geometry problem involving the net of a polyhedron. **1. Question Stem:** 如图 27, 剪一块硬纸片可以做成一个多面体的纸模型 (沿虚线折, 沿实线粘)。这个多面体的面数、顶点数和棱数的总和是多少？ **English Translation of Question Stem:** As shown in Figure 27, a piece of cardboard can be cut to make a paper model of a polyhedron (fold along dashed lines, glue along solid lines). What is the sum of the number of faces, vertices, and edges of this polyhedron? **2. Options:** No options (A, B, C, D) are provided in the image. **3. Other Relevant Text:** * 图 27 (Figure 27 - label for the diagram) **4. Chart/Diagram Description:** * **Type:** Net of a polyhedron. * **Main Elements:** * The net consists of **four rectangular shapes** arranged horizontally in a row, forming a central strip, and **four triangular shapes** attached to the edges of this strip. * **Rectangles:** Let's denote the vertices of the central strip. Imagine two horizontal rows of points. * Top row of points: P1 (leftmost), P2, P3, P4, P5 (rightmost). * Bottom row of points: P6 (leftmost), P7, P8, P9, P10 (rightmost). * The four rectangles are (P1, P2, P7, P6), (P2, P3, P8, P7), (P3, P4, P9, P8), (P4, P5, P10, P9). * **Triangles:** * One triangle is attached to the edge (P2, P3) of the second rectangle (from the left), pointing upwards. Let its apex be A1. * One triangle is attached to the edge (P4, P5) of the fourth rectangle, pointing upwards. Let its apex be A2. * One triangle is attached to the edge (P6, P7) of the first rectangle, pointing downwards. Let its apex be A3. * One triangle is attached to the edge (P8, P9) of the third rectangle, pointing downwards. Let its apex be A4. * **Lines:** * **Dashed lines:** Indicate fold lines. These include the vertical lines separating the rectangles (e.g., between (P2,P7) and (P3,P8)), and the bases of the four triangles (e.g., (P2,P3) for A1's triangle). * **Solid lines:** Indicate cut lines (outer perimeter of the net) or glue lines (edges that will merge to form the polyhedron's edges). These include the outer vertical edges (P1,P6) and (P5,P10), the outer horizontal edges (P1,P2), (P3,P4), (P7,P8), (P9,P10), and the non-base edges of the triangles (e.g., A1-P2, A1-P3). * **Overall Structure:** The net, when folded, forms a polyhedron. The central strip of four rectangles forms the lateral faces of a quadrilateral prism-like structure. The four triangles attach to specific edges of the top and bottom quadrilaterals of this prism. **5. Extraction and Calculation of F, V, E:** Based on the interpretation that both dashed and solid lines become edges of the polyhedron, and each polygon in the net is a face: * **Number of Faces (F):** * There are 4 rectangular faces. * There are 4 triangular faces. * Total Faces (F) = 4 + 4 = 8. * **Number of Vertices (V):** * Let the vertices of the two main horizontal lines of the rectangular strip be `u_0, u_1, u_2, u_3, u_4` (top row) and `d_0, d_1, d_2, d_3, d_4` (bottom row). The problem's diagram shows this arrangement for the main strip. * When the net is folded and glued: * `u_0` merges with `u_4` to form one vertex (let's call it `U_A`). * `d_0` merges with `d_4` to form one vertex (let's call it `D_A`). * Vertices forming the "body" of the polyhedron: * `U_A`, `u_1`, `u_2`, `u_3` (4 vertices for the upper quadrilateral). * `D_A`, `d_1`, `d_2`, `d_3` (4 vertices for the lower quadrilateral). * These form 8 distinct vertices for the prism-like core. * Vertices from the triangle apexes: Each of the 4 triangles introduces a new, distinct apex vertex. * A1 (apex of triangle on `u_1-u_2`) * A2 (apex of triangle on `u_3-u_4` which becomes `u_3-U_A`) * A3 (apex of triangle on `d_0-d_1` which becomes `D_A-d_1`) * A4 (apex of triangle on `d_2-d_3`) * These are 4 distinct apex vertices. * Total Vertices (V) = 8 (from core) + 4 (from apexes) = 12. * **Number of Edges (E):** * Edges forming the "body" of the polyhedron (after `u_0` merges with `u_4`, and `d_0` merges with `d_4`): * 4 vertical edges: `U_A-D_A` (from `u_0-d_0` and `u_4-d_4`), `u_1-d_1`, `u_2-d_2`, `u_3-d_3`. * 4 horizontal edges on the upper quadrilateral: `U_A-u_1`, `u_1-u_2`, `u_2-u_3`, `u_3-U_A`. * 4 horizontal edges on the lower quadrilateral: `D_A-d_1`, `d_1-d_2`, `d_2-d_3`, `d_3-D_A`. * Total edges for the "body" = 4 + 4 + 4 = 12 edges. * Edges from the triangles (excluding their bases, which are already counted as horizontal edges of the body): * Each of the 4 triangles adds 2 slant edges (connecting its apex to the two vertices of its base). * Total edges from triangles = 2 edges/triangle * 4 triangles = 8 edges. * Total Edges (E) = 12 (from body) + 8 (from triangles) = 20. * **Sum of Faces, Vertices, and Edges:** * F + V + E = 8 + 12 + 20 = 40.