圆

考什么 · What it tests

需要先会 · Prerequisites

• 圆周率 $\pi$ 的含义 / What $\pi$ means
• 周长 $C=2\pi r$、面积 $S=\pi r^2$ / Circumference $C=2\pi r$ and area $S=\pi r^2$
• 角度与占比（扇形要用） / Angles as fractions (needed for sectors)

常见套路 · Common moves

1. 扇形按角度取占比：弧长和扇形面积都是整圆的 $\displaystyle \frac{\theta}{360°}$ 倍，\text{弧长}=\frac{\theta}{360°}\cdot2\pi r，\text{扇形面积}=\frac{\theta}{360°}\cdot\pi r^2——圆的部分题几乎都靠它。

   Scale a sector by its angle: arc length and sector area are each the whole circle times $\displaystyle \frac{\theta}{360°}$, so $\displaystyle \text{arc}=\frac{\theta}{360°}\cdot2\pi r$ and $\displaystyle \text{sector area}=\frac{\theta}{360°}\cdot\pi r^2$ — nearly every partial-circle problem uses this.
2. 切线连半径出直角：切点连圆心的半径与切线垂直，这条直角是切线题里辅助线的起点。

   A radius to the point of tangency gives a right angle: the radius drawn to the tangent point is perpendicular to the tangent, and that right angle is the starting auxiliary line.
3. 圆周角是圆心角的一半：同一段弧所对的圆周角 = 圆心角的一半，半圆所对的圆周角是 $90°$。

   An inscribed angle is half the central angle: on the same arc, the inscribed angle is half the central one, and an angle inscribed in a semicircle is $90°$.

易错点 · Common pitfalls

• 半径和直径混：公式里全是半径 $r$，题目却常给直径，代进去面积会差 4 倍。

  Confusing radius with diameter: the formulas use radius $r$, but problems often give the diameter — plug that in and the area is off by a factor of 4.
• 求「阴影周长」时漏算或多算弧：只有真正在边界上的弧和线段才计入周长，内部辅助线不算。

  Miscounting arcs in a "shaded perimeter": only arcs and segments actually on the boundary count — interior auxiliary lines don't.