📚 Circular Measure | 弧度制与圆弧度量
Circular measure is the branch of trigonometry that measures angles in radians. For Cambridge IGCSE, O-Level and AS-Level mathematics, it provides the essential tools for solving problems involving arc length, sector area and segment area. It also prepares students for calculus, where radian measure is required for standard derivative and integral formulas of trigonometric functions.
圆弧度量是三角学中用弧度来度量角的重要分支。在剑桥 IGCSE、O-Level 和 AS-Level 数学中，它为求解弧长、扇形面积和弓形面积问题提供了核心工具。它也为后续微积分打下基础，因为标准三角函数导数和积分公式都需要使用弧度制。
1. Radian definition | 弧度定义
A radian is defined as the angle at the centre of a circle that is subtended by an arc whose length is equal to the radius of the circle. If the arc length s equals the radius r, then the angle θ is exactly 1 radian. In general, the central angle θ in radians is the ratio of arc length to radius.
1 弧度的定义是：圆心角所对的弧长等于圆的半径，该圆心角就是 1 弧度。如果弧长 s 等于半径 r，则圆心角 θ 恰好为 1 弧度。一般地，用弧度表示的圆心角 θ 等于弧长与半径之比。
θ = s / r
Here θ is the angle in radians, s is the arc length, and r is the radius of the circle. Unlike degrees, radians are a ratio of two lengths, so they are dimensionless. Cambridge papers often write the unit as 'rad' or use the superscript c, for example 1.2 rad or 1.2ᶜ.
其中 θ 为弧度制圆心角，s 为弧长，r 为圆半径。与角度制不同，弧度本质上是两个长度之比，因此是无量纲量。剑桥试卷中常用 rad 或上标 c 表示弧度，例如 1.2 rad 或 1.2ᶜ。
2. Converting degrees and radians | 度与弧度转换
t degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
由于完整的圆周长是 2πr，因此一整圈对应 2π 弧度。于是得到关键恒等式：360° = 2π rad，等价于 180° = π rad。将角度转换为弧度时乘以 π/180；将弧度转换为角度时乘以 180/π。
= 36°. The following table gives common exact values that often appear in Cambridge exams.
例如，120° = 120 × π/180 = 2π/3 rad，π/5 rad = π/5 × 180/π = 36°。下表列出了剑桥考试中常见的精确值。
| Degrees | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
| Radians | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
Knowing these exact conversions is especially useful when answers must be given in terms of π or when using trigonometric functions of standard angles.
掌握这些精确转换非常重要，尤其在答案需要保留 π 或使用标准角的三角函数值时。
3. Arc length formula | 弧长公式
ula is derived directly from the radian definition θ = s/r by multiplying both sides by r.
对于半径为 r、圆心角为 θ（以弧度为单位）的圆，弧长 s 与半径和角度都成正比。该公式直接由弧度定义 θ = s/r 两边同乘 r 得到。
s = rθ
A common mistake is to use θ in degrees in this formula, which gives an incorrect result.
如果角度以度数给出，必须先换算为弧度才能使用该公式。例如，半径为 5 cm 的圆，圆心角为 0.8 rad，则弧长 s = 5 × 0.8 = 4 cm。常见错误是把度数值直接代入该公式，导致结果错误。
4. Sector area formula | 扇形面积公式
² and corresponds to an angle 2π rad, the sector area for angle θ rad is A = (θ/2π) × πr².
扇形是由两条半径和它们之间的弧所围成的区域。它的面积占整个圆面积的比例等于 θ 与 2π 的比。因为整个圆面积为 πr²，对应圆心角 2π rad，所以圆心角为 θ rad 的扇形面积为 A = (θ/2π) × πr²。
A = ½ r²θ
s already known. For example, if r = 4 cm and θ = 1.5 rad, then A = ½ × 4² × 1.5 = 12 cm².
由于弧长 s = rθ，扇形面积也可写成 A = ½rs。当已知弧长时，第二种形式非常实用。例如，若 r = 4 cm，θ = 1.5 rad，则 A = ½ × 4² × 1.5 = 12 cm²。
5. Segment area and chord length | 弓形面积与弦长
A minor segment is the region bounded by a chord and the smaller arc joining its endpoints. Its area is found by subtracting the area of the isosceles triangle formed by the two radii and the chord from the area of the sector. The triangle has sides r, r and included angle θ, so its area is ½r² sin θ.
劣弓形是由一条弦和连接弦两端点的较小弧所围成的区域。它的面积等于扇形面积减去由两条半径和该弦构成的等腰三角形面积。该三角形的两条边都是 r，夹角为 θ，因此面积为 ½r² sin θ。
Segment area = ½ r²θ − ½ r² sin θ = ½ r²(θ − sin θ)
In this formula, θ must be in radians because the sector area term ½r²θ is based on radian measure. The chord length is the base of the same isosceles triangle, obtained by splitting it into two right-angled triangles.
该公式中 θ 必须为弧度，因为扇形面积项 ½r²θ 基于弧度制。弦长是同一个等腰三角形的底边，可通过将其分成两个直角三角形来求出。
Chord length c = 2r sin(θ/2)
This is one of the most useful results for Cambridge questions that connect chords, sectors and circles. Remember that sin(θ/2) is used, not sin θ.
这是剑桥考试中连接弦、扇形和圆的最常用结论之一。注意这里使用的是 sin(θ/2)，而不是 sin θ。
6. Using trigonometry with circular measure | 三角与圆弧综合
Many exam questions require you to combine circular measure with right-angled triangle trigonometry or the cosine rule. If you drop a perpendicular from the centre of a circle to a chord, it bisects the chord. This gives two useful relations: the half-chord is r sin(θ/2), and the distance from the centre to the chord is r cos(θ/2).
许多考试题目要求将圆弧度量与直角三角形三角学或余弦定理结合起来。如果从圆心作弦的垂线，它会平分该弦。由此得到两个重要关系：半弦长为 r sin(θ/2)，圆心到弦的距离为 r cos(θ/2)。
2). These relationships are helpful when you know the radius and chord length and need to find the central angle θ before calculating sector or segment areas.
劣弓形的高度，有时称为弓高，等于半径减去圆心到弦的垂直距离。因此 h = r − r cos(θ/2)。当已知半径和弦长，需要先求圆心角 θ 再计算扇形或弓形面积时，这些关系非常有用。
7. Common pitfalls: radians mode | 常见易错点：弧度模式
The most common error in circular measure is using degree mode on a calculator. In the segment area formula ½r²(θ − sin θ), the first term θ is a radian angle, so sin θ must also be evaluated with the calculator in radian mode. If the calculator is in degree mode, sin θ is interpreted using degrees while θ is treated as radians, producing an incorrect result.
圆弧度量中最常见的错误是计算器处于角度模式。在弓形面积公式 ½r²(θ − sin θ) 中，第一项 θ 是弧度角，因此 sin θ 也必须在弧度模式下计算。如果计算器处于角度模式，sin θ 会按角度解释，而 θ 本身是弧度，结果就会出错。
Another common error is forgetting the factor ½ in the sector
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