📚 Circles: Circumference and Area | 圆：周长与面积
In the Cambridge KS3 Mathematics curriculum, questions on circles often ask you to move between radius, diameter, circumference, and area. A typical page-191 exercise, such as the one in p191_2.pdf, usually provides one measurement and expects you to find the others. This article explains the core formulas, worked examples, and common mistakes so you can answer these questions confidently.
在剑桥 KS3 数学课程中，圆的题目通常要求你在半径、直径、周长和面积之间进行转换。像 p191_2.pdf 这样的第 191 页练习，通常会给出一个已知量，让你求出其他量。本文讲解核心公式、例题和常见错误，帮助你自信作答。
1. The Circle and Its Parts | 圆的构成要素
A circle is the set of all points that are the same distance from a fixed centre. The radius is the distance from the centre to any point on the circle, and the diameter is the distance across the circle through the centre.
圆是由到固定中心距离相等的所有点组成的图形。半径是从圆心到圆上任意一点的距离，直径是经过圆心贯穿圆两端的距离。
The diameter is always twice the radius. In symbols, d = 2r and r = d ÷ 2. Many errors in circle questions come from confusing these two measurements.
直径总是半径的两倍。用符号表示为 d = 2r，r = d ÷ 2。圆题中的许多错误都来自混淆这两个量。
2. From Diameter to Radius | 直径与半径的关系
If a question gives the diameter, divide by 2 to get the radius. If it gives the radius, multiply by 2 to get the diameter. Always check which measurement you have before putting it into a formula.
如果题目给出直径，除以 2 得到半径；如果给出半径，乘以 2 得到直径。在代入公式之前，一定要先确认你拿到的是哪个量。
- Diameter 12 cm → radius 6 cm | 直径 12 cm → 半径 6 cm
- Radius 3.5 m → diameter 7 m | 半径 3.5 m → 直径 7 m
3. Circumference Formula: C = πd | 周长公式：C = πd
The circumference is the distance around the circle. The standard formula is C = πd, where d is the diameter. Since d = 2r, you can also write C = 2πr.
周长是圆一圈的长度。标准公式为 C = πd，其中 d 是直径。因为 d = 2r，所以也可以写成 C = 2πr。
Pi (π) is the special number that connects a circle’s circumference to its diameter. It is approximately 3.14, but it is an irrational number with unending decimals.
圆周率 π 是连接圆周长与直径的特殊数。它约等于 3.14，但它是无理数，小数部分无限不循环。
4. Area Formula: A = πr² | 面积公式：A = πr²
The area of a circle is found using A = πr², where r is the radius. Notice that only the radius is squared, not π. This is a very common area mistake.
圆的面积用公式 A = πr² 计算，其中 r 是半径。注意只有半径被平方，π 不被平方。这是面积计算中最常见的错误之一。
If you are given the diameter, you must first divide it by 2 before using the area formula. For example, a diameter of 8 cm gives a radius of 4 cm, so A = π × 4² = 16π cm².
如果题目给出直径，你必须先除以 2，再代入面积公式。例如，直径 8 cm，半径为 4 cm，所以 A = π × 4² = 16π cm²。
5. Using π: Approximation and Exact Form | π 的使用：近似值与精确形式
Cambridge KS3 questions often accept answers in two ways: as an exact multiple of π, or as a decimal approximation. Exact form means leaving π in your answer, such as 10π cm or 25π cm².
剑桥 KS3 题目通常接受两种答案：用 π 的倍数表示精确值，或用小数表示近似值。精确形式就是在答案中保留 π，例如 10π cm 或 25π cm²。
For a decimal answer, use the π button on your calculator if possible; otherwise use 3.14. Always round to the required degree of accuracy, usually 3 significant figures or 2 decimal places unless the question says otherwise.
如果需要小数答案，尽量使用计算器上的 π 键；否则使用 3.14。一定要按照题目要求的精度四舍五入，通常是 3 位有效数字或 2 位小数，除非题目另有说明。
| Formula | Circumference | Area |
| Using radius r | C = 2πr | A = πr² |
| Using diameter d | C = πd | A = π(d/2)² |
6. Worked Example: Finding Circumference | 例题：求圆的周长
A circle has a diameter of 12 cm. Find the circumference, giving your answer in terms of π and as a decimal correct to 3 significant figures.
一个圆的直径为 12 cm。求它的周长，答案分别用 π 表示和保留 3 位有效数字的小数表示。
C = πd = π × 12 = 12π cm
Using π ≈ 3.14, the circumference is approximately 12 × 3.14 = 37.68 cm, which rounds to 37.7 cm to 3 significant figures.
使用 π ≈ 3.14，周长约为 12 × 3.14 = 37.68 cm，保留 3 位有效数字为 37.7 cm。
7. Worked Example: Finding Area | 例题：求圆的面积
A circle has a radius of 5 m. Find its area, leaving your answer in terms of π, and then find the decimal approximation to 2 decimal places.
一个圆的半径为 5 m。求它的面积，答案先用 π 表示，再求保留 2 位小数的近似值。
A = πr² = π × 5² = 25π m²
Using a calculator value of π, A ≈ 78.5398… m², so to 2 decimal places the area is 78.54 m².
使用计算器中的 π 值，A ≈ 78.5398… m²，保留 2 位小数为 78.54 m²。
8. Composite Shapes: Semicircles and Quarter Circles | 组合图形：半圆与四分之一圆
Many KS3 questions ask you to find the perimeter or area of a semicircle or quarter circle. For a semicircle, the curved length is half the circumference: πd ÷ 2 or πr. The total perimeter also includes the straight diameter.
许多 KS3 题目会要求你求半圆或四分之一圆的周长或面积。对于半圆，弯曲部分的长度是圆周长的一半：πd ÷ 2 或 πr。总周长还要加上直线部分的直径。
For a quarter circle, the curved length is one quarter of the circumference: πd ÷ 4 or πr ÷ 2. The total perimeter includes two radii. The area of a semicircle is half the full circle area: πr² ÷ 2.
对于四分之一圆，弯曲部分的长度是圆周长的四分之一：πd ÷ 4 或 πr ÷ 2。总周长包括两条半径。半圆的面积是整个圆面积的一半：πr² ÷ 2。
9. Common Errors and How to Avoid Them | 常见错误与避免方法
One common mistake is using the diameter in the area formula without dividing by 2. Always remember: area needs radius, not diameter. Another mistake is forgetting to square the entire radius, for example writing π × 5 instead of π × 5².
一个常见错误是在面积公式中直接使用直径而没有除以 2。请始终记住：面积需要半径，而不是直径。另一个错误是忘记将半径整体平方，例如写成 π × 5 而不是 π × 5²。
Students also sometimes confuse perimeter and area units. Perimeter is a length and has units such as cm or m; area has square units such as cm² or m². Finally, if the question asks for an exact answer, do not round π prematurely.
学生有时也会混淆周长和面积的单位。周长是长度，单位是 cm 或 m；面积是平方单位，如 cm² 或 m²。最后，如果题目要求精确答案，不要过早将 π 取近似值。
10. Exam-Style Practice Question | 考试型练习题
A circular garden has a diameter of 14 m. Find: (a) the circumference, (b) the area, giving exact answers in terms of π and decimal answers correct to 3 significant figures.
一个圆形花园的直径为 14 m。求：(a) 周长，(b) 面积。答案分别用 π 的精确形式和保留 3 位有效数字的小数形式表示。
For part (a), radius r = 14 ÷ 2 = 7 m. Circumference = 2πr = 2 × π × 7 = 14π m. Using π ≈ 3.14, the circumference is about 43.96 m, which is 44.0 m to 3 significant figures.
对于 (a)，半径 r = 14 ÷ 2 = 7 m。周长 = 2πr = 2 × π × 7 = 14π m。使用 π ≈ 3.14，周长约为 43.96 m，保留 3 位有效数字为 44.0 m。
For part (b), area = πr² = π × 7² = 49π m². Using a calculator, 49π ≈ 153.938… m², so the area is 154 m² to 3 significant figures.
对于 (b)，面积 = πr² = π × 7² = 49π m²。使用计算器，49π ≈ 153.938… m²，保留 3 位有效数字为 154 m²。
11. Summary and Key Points | 总结与要点
For any circle question, identify whether you are given the radius or the diameter. Use C = πd or 2πr for circumference, and A = πr² for area. Keep π in your answer when exact form is required, and round only at the final step.
对于任何圆的题目，先判断已知量是半径还是直径。周长用 C = πd 或 2πr，面积用 A = πr²。当题目要求精确形式时保留 π，并且只在最后一步进行四舍五入。
- Diameter = 2 ×
Published by TutorHao | KS3 Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导