Mastering Circle Calculations: Area and Circumference | 掌握圆的计算：面积与周长

📚 Mastering Circle Calculations: Area and Circumference | 掌握圆的计算：面积与周长

In Cambridge KS3 Mathematics, circle calculations appear regularly in both progress tests and checkpoint exams. This article focuses on the two key formulas: circumference and area. You will learn how to identify the radius and diameter, apply the formulas accurately, and handle common exam-style questions, including semicircles and compound shapes.

在剑桥 KS3 数学中，圆的计算经常出现在阶段测试和 checkpoint 考试中。本文重点讲解两个关键公式：周长和面积。你将学会如何识别半径和直径，准确应用公式，并应对常见考试题型，包括半圆和组合图形。

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1. The Circle and Its Key Terms | 圆及其关键术语

rcle through the centre. The diameter is always twice the radius, so d = 2r and r = d ÷ 2.

圆是由与固定中心距离相等的所有点组成的图形。半径 (r) 是从圆心到圆上任意一点的距离，而直径 (d) 是穿过圆心连接圆上两点的距离。直径总是半径的两倍，因此 d = 2r，r = d ÷ 2。

question gives one and asks you to find the other, so quick recall of d = 2r is essential.

在使用任何公式之前，你必须正确识别半径和直径。考试中常见题型是给出其中一个量，让你求另一个量，因此快速掌握 d = 2r 非常关键。

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2. The Number π | 圆周率 π

The number π (pi) is the ratio of the circumference of any circle to its diameter. It is approximately 3.14 or 22/7, but it is an irrational number, so its decimal form never ends or repeats. In Cambridge KS3, you usually leave answers in terms of π unless the question asks for a decimal approximation.

圆周率 π 是任意圆的周长与其直径之比。它的近似值为 3.14 或 22/7，但它是一个无理数，因此它的小数形式永远不会终止或循环。在剑桥 KS3 中，除非题目要求给出小数近似值，否则通常将答案保留为 π 的形式。

When you do approximate, use the π button on your calculator or 3.14 if instructed. Always check whether the question wants an exact answer in terms of π or a rounded decimal.

若需要近似，请使用计算器上的 π 键，或按照题目要求使用 3.14。务必检查题目是要求用 π 表示精确答案，还是要求四舍五入后的小数。

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3. Circumference Formula | 周长公式

The circumference (C) is the distance around the circle. The two equivalent forms are:

周长 (C) 是围绕圆一周的距离。两个等价公式为：

C = πd and C = 2πr

2πr when you know the radius. Since d = 2r, the two formulas always give the same result.

当已知直径时使用 C = πd；当已知半径时使用 C = 2πr。因为 d = 2r，两个公式总是得到相同的结果。

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4. Area Formula | 面积公式

The area (A) of a circle is the amount of space inside it. The formula is:

圆的面积 (A) 是圆内部的空间大小。公式为：

A = πr²

Note that the area formula uses the radius only, not the diameter. If you are given the diameter, first divide by 2 to find the radius, then square it and multiply by π.

请注意，面积公式只使用半径，而不使用直径。如果题目给出的是直径，请先除以 2 求出半径，然后平方并乘以 π。

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5. Choosing Radius or Diameter | 选择半径或直径

, highlight the given measurement and label it as radius or diameter. If the question says 'the radius is 5 cm', use r = 5. If it says 'the diameter is 12 cm', use d = 12 or r = 6.

在计算之前，先标出已知数据，并注明它是半径还是直径。如果题目说“半径为 5 cm”，则使用 r = 5。如果题目说“直径为 12 cm”，则使用 d = 12 或 r = 6。

art of your working helps you avoid the common error of using the diameter in the area formula.

在解题开始时先写出 r = ... 和 d = ...，可以帮助你避免在面积公式中误用直径的常见错误。

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6. Worked Example: Circumference | 周长例题

Question: A circle has radius 7 cm. Work out the circumference. Give your answer in terms of π and as a decimal to 1 decimal place.

题目：一个圆的半径为 7 cm。求它的周长。答案分别用 π 的形式和保留 1 位小数的形式表示。

14π cm. Step 2: Use π ≈ 3.14159: C = 14 × 3.14159 ≈ 43.982 cm, which is 44.0 cm to 1 d.p.

步骤 1：使用 C = 2πr。代入 r = 7：C = 2 × π × 7 = 14π cm。步骤 2：使用 π ≈ 3.14159：C = 14 × 3.14159 ≈ 43.982 cm，保留 1 位小数为 44.0 cm。

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7. Worked Example: Area | 面积例题

Question: A circle has diameter 10 cm. Work out the area. Give your answer to 1 decimal place.

题目：一个圆的直径为 10 cm。求它的面积，答案保留 1 位小数。

= π × 5² = π × 25 = 25π cm². Step 3: 25π ≈ 25 × 3.14159 ≈ 78.54 cm², so 78.5 cm² to 1 d.p.

步骤 1：求半径：r = d ÷ 2 = 10 ÷ 2 = 5 cm。步骤 2：使用 A = πr²。代入 r = 5：A = π × 5² = π × 25 = 25π cm²。步骤 3：25π ≈ 25 × 3.14159 ≈ 78.54 cm²，保留 1 位小数为 78.5 cm²。

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8. Compound Shapes with Circles | 含圆的组合图形

Some exam questions combine circles with rectangles, triangles or semicircles. For a semicircle, find the area of the full circle and divide by 2. For the perimeter of a semicircle, add the curved part (πr) and the straight diameter (2r).

一些考试题目会把圆与矩形、三角形或半圆组合在一起。对于半圆，先求整个圆的面积再除以 2。对于半圆的周长，需要将弯曲部分 (πr) 与直边直径 (2r) 相加。

Example: A shape is made from a rectangle of length 8 cm and width 4 cm with a semicircle of diameter 4 cm added on one side. The total area is area of

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