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Circles: Circumference and Area | 圆：周长与面积

2026-08-21 12:02 · 本文部分内容由AI辅助创作

📚 Circles: Circumference and Area | 圆：周长与面积

e 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。

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。

3. Circumference Formula: C = πd | 周长公式：C = πd

dard 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

r², 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。

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。

FormulaCircumferenceArea
Using radius rC = 2πrA = πr²
Using diameter dC = πdA = π(d/2)²

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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

e 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

g 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

ke 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。

ng 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。

. 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

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。