面积与周长（综合）

考什么 · What it tests

需要先会 · Prerequisites

• 三角形、长方形、圆的面积公式 / Area formulas for triangles, rectangles, and circles
• 相似比与面积比 / Similarity ratios and area ratios
• 扇形面积（角度占比） / Sector area (as a fraction of the circle by angle)

常见套路 · Common moves

1. 整体减空白：阴影算不了就算它旁边的空白，用大图形总面积一减就出——双解里几乎总是「标准第一解」。

   Whole minus blank: if the shaded part resists, compute the unshaded part instead and subtract from the total — almost always the "first solution" in dual-solution write-ups.
2. 平移与翻折归位：把散落的阴影块平移、对称拼成一个整块，高星题里常常一步出答案。

   Shift and fold pieces home: translate or reflect scattered shaded bits into one clean shape — on harder problems this often finishes in a single step.
3. 割补加辅助线：沿着已有顶点、圆心连线切一刀，怪形状就散成三角形和扇形的组合。

   Cut with auxiliary lines: slice through existing vertices or circle centers, and the odd shape falls apart into triangles and sectors.

易错点 · Common pitfalls

• 对着怪形状硬套公式，十分钟耗死在一题上——先想「减法」再想「直接算」。

  Forcing formulas onto an irregular shape and burning ten minutes on one problem — think "subtract" before you think "compute directly."
• 圆弧部分忘了按角度取扇形（$\displaystyle \frac{\theta}{360°} \times \pi r^2$），或半径该平方时没平方。

  Forgetting to scale sectors by angle ($\displaystyle \frac{\theta}{360°} \times \pi r^2$), or failing to square the radius when the formula calls for it.