立体几何

考什么 · What it tests

需要先会 · Prerequisites

• 长方体/正方体表面积与体积公式 / Surface-area and volume formulas for boxes and cubes
• 平面图形面积 / Areas of plane figures
• 空间想象（正方体的相对面） / Spatial visualization (opposite faces of a cube)

常见套路 · Common moves

1. 表面积拆成一个个面加起来：别背综合公式，把每个面的面积单独算再求和，最稳当也最通用。

   Build surface area face by face: rather than memorizing one formula, compute each face's area and add — the safest, most general approach.
2. 展开图抓「相对面」：正方体展开图里，隔一个格或错开的两面才是相对面，折回去时相邻的两面绝不可能相对——染色、骰子题都靠它。

   In a net, track opposite faces: on a cube net, faces separated by one square (or offset) are the opposite pair, and two adjacent faces can never be opposite — key for dice and coloring problems.
3. 切块数方块分层数：$n\times n\times n$ 切开后，按「几个面被染色」分类，角块 3 面、棱块 2 面、面块 1 面、内部 0 面。

   Count sliced cubes by category: after cutting $n\times n\times n$, sort by painted faces — corners have 3, edges 2, faces 1, interior 0.

易错点 · Common pitfalls

• 体积和表面积的量纲混：体积是三维（立方），表面积是二维（平方），放大 $k$ 倍时体积变 $k^3$ 倍、表面积变 $k^2$ 倍。

  Confusing the dimensions of volume and area: volume is 3-D (cubic), surface area is 2-D (square), so scaling by $k$ multiplies volume by $k^3$ and surface area by $k^2$.
• 展开图里把相邻面当成相对面——正方体六面里，任何相邻的两面都不可能是相对面。

  Mistaking an adjacent face for the opposite one on a net — among a cube's six faces, no two adjacent faces can ever be opposite.