📚 Arc Length and Sector Area | 弧长与扇形面积
In IGCSE Mathematics, mensuration often includes arc length and sector area. These topics connect circle geometry with proportional reasoning, and they appear in both short and structured questions. Understanding the fraction θ/360 is more important than memorising isolated formulas.
在 IGCSE 数学中，度量衡部分常常包括弧长与扇形面积。这些主题将圆的几何与比例推理联系起来，出现在选择题和结构化问题中。理解 θ/360 这个比例比死记公式更重要。
1. Circles and Proportional Thinking | 圆与比例思想
A sector is the region enclosed by two radii and the arc between them. The central angle θ tells us what fraction of the whole circle we are considering.
扇形是由两条半径和它们之间的弧围成的区域。圆心角 θ 表示我们考虑的是整圆的多少比例。
Since a full circle contains 360°, the fraction is always θ/360. This fraction applies to arc length, sector area, and even sector perimeter if you add the two radii separately.
因为整圆包含 360°，比例始终是 θ/360。这个比例适用于弧长、扇形面积，如果单独加上两条半径，也适用于扇形周长。
This proportional approach means you can solve many questions without memorising extra rules. Just identify the fraction of the circle you are dealing with.
这种比例方法意味着你不需要额外记忆规则就能解决许多问题。只需确定你正在处理的圆的比例即可。
2. Arc Length Formula | 弧长公式
For a circle with radius r, the full circumference is C = 2πr.
对于半径为 r 的圆，整圆周长为 C = 2πr。
l = θ/360 × 2πr
The arc length is simply the fraction θ/360 of the full circumference.
弧长就是整个周长的 θ/360 部分。
°, the arc length is 72/360 × 2π × 10 = 1/5 × 20π = 4π cm, which is approximately 12.6 cm.
例如：若 r = 10 cm，θ = 72°，弧长为 72/360 × 2π × 10 = 1/5 × 20π = 4π cm，约等于 12.6 cm。
Always simplify the fraction θ/360 before multiplying to keep calculations tidy.
在相乘前先化简 θ/360，可以让计算更简洁。
3. Sector Area Formula | 扇形面积公式
The full area of a circle is A = πr².
整圆面积是 A = πr²。
A = θ/360 × πr²
Sector area uses the same fraction θ/360, but multiplied by πr² rather than 2πr.
扇形面积使用相同的比例 θ/360，但乘以 πr² 而不是 2πr。
For r = 10 cm and θ = 72°, the sector area is 72/360 × π × 10² = 1/5 × 100π = 20π cm².
例如 r = 10 cm，θ = 72° 时，扇形面积为 72/360 × π × 10² = 1/5 × 100π = 20π cm²。
ionship: sector area = ½ r × arc length. This connection can be useful in harder problems.
注意一个关系：扇形面积 = ½ r × 弧长。这个联系在较难的问题中很有用。
4. Finding the Angle or Radius | 求角度或半径
Sometimes the question gives arc length or sector area and asks for the radius or angle.
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