📚 Sector and Segment Areas | 扇形与弓形面积
A circle can be divided in two useful ways when solving area problems. A sector is the region bounded by two radii and an arc; it looks like a slice of cake. A segment is the region bounded by a chord and an arc; it looks like a cap or crescent. This revision note explains how to calculate these areas accurately and how to avoid common exam mistakes.
在解决圆面积问题时，有两种重要的分割方式。扇形是由两条半径和一条弧围成的区域，形如一片蛋糕；弓形则是由一条弦和一条弧围成的区域，形如帽盖或月牙。本复习笔记将讲解如何准确计算这些面积，并避开常见考试错误。
1. Radian Measure and Angle Conversion | 弧度制与角度换算
π radians = 180°. You must convert degrees to radians before using the formula A = ½ r²θ.
扇形与弓形公式在角以弧度为单位时最为简洁。关键换算为 π 弧度 = 180°。在使用公式 A = ½ r²θ 前，必须先将角度从度数转换为弧度。
| Degrees 度数 | Radians 弧度 |
|---|---|
| 30° | π/6 |
| 45° | π/4 |
| 60° | π/3 |
| 90° | π/2 |
| 120° | 2π/3 |
| 180° | π |
An angle expressed as a multiple of π is already in radians. If an angle is given in degrees, multiply it by π/180 to convert it.
以 π 的倍数表示的角度已经是弧度。若角度以度数给出，则乘以 π/180 即可完成换算。
2. Arc Length | 弧长
ere θ is the central angle in radians. In degrees, the equivalent formula is s = πrθ/180°.
弧长与它所对的圆心角成正比。用弧度表示时，弧长的简洁公式为 s = rθ，其中 θ 是圆心角（弧度）。用度数表示时，等价公式为 s = πrθ/180°。
s = rθ (θ in radians) | s = πrθ/180° (θ in degrees)
弧度制下 s = rθ（θ 为弧度）；度数制下 s = πrθ/180°（θ 为度数）。
ector is therefore P = 2r + s, because a sector has two straight radii and one curved arc.
因此扇形的周长为 P = 2r + s，因为扇形包含两条半径边和一条弧边。
3. Area of a Sector | 扇形面积
A sector is a fraction of the whole circle. Since the full circle has area πr² and corresponds to an angle of 2π radians, the sector area is proportional to its angle. This gives the radian formula Asector = ½ r²θ.
扇形是整圆的一部分。由于整圆面积为 πr²，且对应角度为 2π 弧度，扇形面积与其圆心角成正比，因此得到弧度制公式 Asector = ½ r²θ。
Asector = ½ r²θ (θ in radians) | Asector = πr²θ/360° (θ in degrees)
弧度制公式为 Asector = ½ r²θ（θ 为弧度）；度数制公式为 Asector = πr²θ/360°（θ 为度数）。
d the sector angle is 90° = π/2 radians, the sector area is ½ × 4² × π/2 = 4π ≈ 12.57 cm².
例如，若半径 r = 4 cm，扇形角为 90° = π/2 弧度，则扇形面积为 ½ × 4² × π/2 = 4π ≈ 12.57 cm²。
4. The Triangle Within a Sector | 扇形内的三角形
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