📚 Circles: Circumference and Area | 圆：周长与面积
Circles are all around us – from wheels and coins to plates and planets. In this article, we will explore the key properties of a circle, learn how to calculate its circumference and area, and practise solving problems that often appear in the Cambridge KS3 mathematics curriculum. Understanding these formulas is not only essential for exams but also helps build a solid foundation for future topics like sectors, arcs, and three-dimensional shapes.
圆在我们身边无处不在——从车轮、硬币到盘子和行星。在本文中，我们将探讨圆的基本性质，学习如何计算圆的周长和面积，并练习解决在剑桥 KS3 数学课程中常见的题目。掌握这些公式不仅对应试至关重要，还能为日后学习扇形、弧长以及立体图形等主题打下坚实的基础。
1. What is a Circle? | 什么是圆？
A circle is a set of points in a plane that are all at the same distance from a fixed point called the centre. The distance from the centre to any point on the circle is the radius. The diameter is the distance across the circle through the centre, and it is exactly twice the radius. These simple relationships are the starting point for all circle calculations.
圆是平面上到一个固定点（称为圆心）距离相等的所有点的集合。从圆心到圆上任意一点的距离叫做半径。直径是通过圆心且两端都在圆上的线段长度，正好是半径的两倍。这些简单的关系是所有圆计算的基础。
- Radius (r): distance from centre to edge
- Diameter (d): distance across the circle through centre, d = 2 × r
- 半径 (r)：从圆心到边的距离
- 直径 (d)：通过圆心的弦长，d = 2 × r
2. Radius and Diameter Practice | 半径与直径练习
Before moving on to circumference and area, you need to be comfortable converting between radius and diameter. If a circle has radius 5 cm, its diameter is 10 cm. If the diameter is 14 m, the radius is 7 m. Many mistakes happen when students forget to halve or double correctly, so always underline which measurement you are given in a question.
在学习周长和面积之前，你需要熟练掌握半径与直径的相互转换。如果一个圆的半径是 5 cm，其直径就是 10 cm。如果直径是 14 m，半径就是 7 m。许多错误都是因为学生忘记正确地除以或乘以 2，因此一定要在读题时圈出所给的测量值。
3. Introducing π (pi) | π (圆周率) 简介
The Greek letter π (pi) represents a special number that links the diameter of any circle to its circumference. Pi is approximately 3.14, but in KS3 you may be asked to use the exact value by leaving π in the answer, or to use 3.14 or 22/7 as an approximation. Always check what the question instructs.
希腊字母 π 代表一个特殊的数，它将任意圆的直径与周长联系起来。π 大约等于 3.14，但在 KS3 阶段你可能会被要求保留 π 表示精确值，或者使用 3.14 或 22/7 作为近似值。务必看清楚题目要求。
| π ≈ 3.1415926535... | π 约等于 3.1415926535... |
| Common approximations | 常用近似值 |
| π ≈ 3.14 (to 3 s.f.) | π ≈ 3.14 (保留三位有效数字) |
| π ≈ 22/7 (fraction) | π ≈ 22/7 (分数形式) |
4. Circumference of a Circle | 圆的周长
with radius 4 cm has circumference 2 × π × 4 = 8π cm. Using π ≈ 3.14 gives about 25.1 cm.
周长是围绕圆一周的长度。公式是 C = π × d，或者等价的 C = 2 × π × r。如果给出的是半径，只需先乘以 2 得到直径。例如，半径为 4 cm 的圆，周长为 2 × π × 4 = 8π cm。用 π ≈ 3.14 可得约 25.1 cm。
C = π × d or C = 2 × π × r
C = π × d 或 C = 2 × π × r
5. Area of a Circle | 圆的面积
quared. For a circle with radius 5 cm, area = π × 5² = 25π cm², which is roughly 78.5 cm².
圆的面积公式是 A = π × r²，其中 r 是半径。注意必须先计算半径的平方，再乘以 π。常见错误是把 π 也一起平方，但公式中只有半径需要平方。对于半径为 5 cm 的圆，面积 = π × 5² = 25π cm²，约等于 78.5 cm²。
A = π × r²
6. Finding Radius from Circumference | 由周长求半径
× 3.14) = 31.4 ÷ 6.28 = 5 cm. Always check by substituting back into the original formula.
有时题目会给出周长，需要你反推半径。将 C = 2πr 变形可得 r = C ÷ (2π)。例如，一个圆的周长是 31.4 cm，取 π = 3.14，则 r = 31.4 ÷ (2 × 3.14) = 31.4 ÷ 6.28 = 5 cm。完成之后一定要代回原公式验算。
7. Finding Radius from Area | 由面积求半径
d then take the square root. For area 78.5 cm² and π = 3.14, r² = 78.5 ÷ 3.14 = 25, so r = √25 tested in reverse problems and requires a solid understanding of squares and square roots.
如果已知面积，将 A = πr² 变形得到 r² = A ÷ π，然后开平方。若面积为 78.5 cm²，π = 3.14，则 r² = 78.5 ÷ 3.14 = 25，因此 r = √25 = 5 cm。逆向求解类题目经常考察这一方法，需要熟练掌握平方与开平方的运算。
8. Perimeter of Semicircles and Quarter Circles | 半圆与四分之一圆的周长
and diameter = 12 cm. Total perimeter = 6π + 12 cm. Using π = 3.14 gives roughly 30.8 cm.
KS3 题目常要求计算半圆的周长（半圆弧长加上直径）。弯曲部分是圆周长的一半，还须加上直线段的直径。半径为 6 cm 的半圆，弧长部分 = ½ × 2 × π × 6 = 6π cm，直径 = 12 cm。总周长 = 6π + 12 cm。用 π = 3.14 计算约为 30.8 cm。
Perimeter of semicircle = πr + 2r
半圆的周长 = πr + 2r
9. Area of Compound Shapes with Circles | 含圆的组合图形面积
You may need to find the area of shapes that combine circles with rectangles or triangles. For example, a running track consists of two straight sections and two semicircular ends. To find the total area, work out the area of the rectangle in the middle and the area of the full circle formed by the two semicircles, then add them together. Always break the shape into simpler parts.
你可能会遇到由圆和矩形、三角形等组合而成的图形的面积。例如，运动场的跑道由两个直道和两个半圆弯道组成。计算总面积时，先计算中间矩形的面积，以及两个半圆拼成的整圆面积，再将它们相加。务必把图形拆分成简单部分分别计算。
10. Common Mistakes to Avoid | 常见错误及避免方法
Many students confuse radius with diameter when substituting into formulas. Always circle the word 'radius' or 'diameter' in the problem. Another common slip is using the area formula for circumference and vice versa. Remember: circumference involves a linear measure (cm, m), while area involves square units (cm², m²). Also, watch out for questions that give the diameter but ask for area – you must halve the diameter first.
许多学生在代入公式时会混淆半径和直径。一定要在题目中圈出“半径”或“直径”字样。另一个常见失误是把面积公式错用在周长上，或者反过来。记住：周长结果带长度单位（cm、m），面积结果带平方单位（cm²、m²）。此外，也要留意给出直径却求面积的情况——必须先算出半径。
11. Real-Life Applications | 实际生活中的应用
give 2 × π × 4² ≈ 100.5 in². So one large pizza often gives more pizza than two small ones!
理解了圆的知识有助于测量实际物体，如自行车轮、披萨大小和圆形池塘。如果你知道一个披萨的半径，就可以比较面积来判断哪种购买方案更划算。一个 12 英寸的披萨，半径 6 英寸，面积 π × 6² ≈ 113 in²；而两个 8 英寸披萨（每个半径 4 英寸）面积总和为 2 × π × 4² ≈ 100.5 in²。因此一个大号披萨实际比两个小号还要多！
12. Practice and Summary | 练习与总结
To master circle calculations, practise a wide range of problems: finding circumference from radius, area from diameter, reverse calculations, and compound shapes. Always write down the formula first, substitute correctly, and include units in your final answer. With regular practice, these formulas will become second nature, and you will be ready to tackle more advanced geometry topics in the future.
要熟练掌握圆的运算，需要练习各种题型：已知半径求周长、已知直径求面积、逆向求解以及组合图形题。解题时务必先写下公式，正确代入数值，并在最后结果中标明单位。通过反复练习，这些公式将成为你的本能反应，未来学习更复杂的几何知识也会游刃有余。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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