📚 Circles: Circumference and Area | 圆：周长与面积
Circles appear throughout the Cambridge KS3 mathematics curriculum, from straightforward circumference calculations to the volume and surface area of cylinders. This article brings together the key formulas, worked examples and common pitfalls you need for confident problem solving.
圆在剑桥 KS3 数学课程中反复出现，从简单的周长计算到圆柱的体积和表面积都会涉及。本文汇总关键公式、例题和常见错误，帮助你自信地解题。
1. Circle Vocabulary and Key Definitions | 圆的基本术语和关键定义
Before using any formula, you must be able to identify the radius, diameter and circumference of a circle. The radius is the distance from the centre to any point on the edge. The diameter is the distance across the circle through the centre. The diameter is exactly twice the radius.
在使用任何公式之前，你必须能辨别圆的半径、直径和周长。半径是从圆心到圆周上任意一点的距离；直径是通过圆心的跨圆距离。直径恰好是半径的两倍。
In symbols, d = 2r and r = d ÷ 2. The circumference is the distance around the outside of the circle, similar to the perimeter of a polygon.
用符号表示为 d = 2r 以及 r = d ÷ 2。周长是围绕圆外部的距离，类似于多边形的周长。
| English term | 中文术语 | Meaning / 含义 |
|---|---|---|
| Centre | 圆心 | The middle point of the circle |
| Radius | 半径 | Distance from centre to the edge |
| Diameter | 直径 | Distance across the circle through the centre |
| Circumference | 周长 | Distance around the circle |
At KS3, you are expected to move quickly between radius and diameter. If a question gives the radius, you can double it to find the diameter. If it gives the diameter, you can halve it to find the radius.
在 KS3 阶段，你需要快速在半径和直径之间转换。如果题目给出半径，你可以把它乘以 2 得到直径。如果给出直径，你可以把它除以 2 得到半径。
2. The Constant π and Its Meaning | 常数 π 及其含义
The number π, written as 'pi', is the constant ratio of the circumference of any circle to its diameter. This ratio is the same for every circle, which is why one simple rule can describe all circles.
数字 π 读作 'pi'，是任何圆的周长与其直径的恒定比值。这个比值对每个圆都相同，因此一个简单的规则就能描述所有圆。
For most KS3 work, you will use the calculator value of π or the approximation 3.14. If a question asks for an answer 'in terms of π', leave the symbol π in your final answer instead of multiplying it out.
在大多数 KS3 题目中，你会使用计算器中的 π 值或近似值 3.14。如果题目要求“用 π 表示”答案，就在最终答案中保留 π 符号，而不要把它乘出来。
C ÷ d = π
This relationship means that once you know either the circumference or the diameter, you can find the other. Rearranging gives C = πd, which is the standard circumference formula.
这个关系意味着，只要你知道周长或直径中的任意一个，就可以求出另一个。变形后得到 C = πd，这就是标准的周长公式。
3. Circumference of a Circle | 圆的周长
The circumference of a circle can be found using C = πd, where d is the diameter. Since d = 2r, the same rule can be written as C = 2πr.
圆的周长可以用 C = πd 求出，其中 d 是直径。因为 d = 2r，同一个规则也可以写成 C = 2πr。
radius 7 cm. First calculate the diameter: d = 2 × 7 = 14 cm. Then C = π × 14 ≈ 43.98 cm.
例题：求半径为 7 cm 的圆的周长。先计算直径：d = 2 × 7 = 14 cm。然后 C = π × 14 ≈ 43.98 cm。
If you are given the diameter directly, do not divide it by 2. For a diameter of 10 m, the circumference is C
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