Circle Measurements: Circumference and Area | 圆的测量：周长与面积

📚 Circle Measurements: Circumference and Area | 圆的测量：周长与面积

Welcome to Lesson 16-3 of your IGCSE Mathematics revision series. In this lesson, we focus on the two fundamental measurements of a circle: circumference and area. These concepts appear in almost every IGCSE paper, either as direct questions or as parts of composite figures. We will explore the formulas, their origins, and how to apply them confidently under exam conditions.

欢迎来到 IGCSE 数学复习系列第 16-3 课。本节课我们聚焦圆的两个基本测量：周长与面积。这两个概念几乎出现在每份 IGCSE 试卷中，既可能是直接考题，也可能是组合图形的一部分。我们将深入探讨公式、公式的由来，以及如何在考试条件下自信地运用它们。

1. The Circle and Its Key Parts | 圆及其关键要素

Before we calculate anything, we must identify the essential parts of a circle. The centre is the point exactly in the middle. The radius, denoted as r, is the distance from the centre to any point on the circle. The diameter, denoted as d, is the distance across the circle passing through the centre, so d = 2r.

在计算之前，我们必须识别圆的基本要素。圆心是正中间的点。半径，用 r 表示，是从圆心到圆上任意一点的距离。直径，用 d 表示，是穿过圆心的圆上两点间的距离，因此 d = 2r。

A chord is a line segment joining two points on the circle. The longest chord is the diameter. An arc is part of the circumference, and a sector is the region enclosed by two radii and an arc. Understanding these terms helps us solve more advanced questions later.

弦是连接圆上两点的线段。最长的弦就是直径。弧是圆周的一部分，扇形是由两条半径和一段弧围成的区域。理解这些术语有助于我们解决后面更复杂的问题。

Radius (半径) r: centre to circumference
Diameter (直径) d: across the circle through the centre
Circumference (周长) C: the distance around the circle
Sector (扇形): a “pizza slice” of the circle

2. The Magic of Pi (π) | π 的神奇之处

The ratio of the circumference of any circle to its diameter is always the same constant, approximately 3.14159. This constant is called pi and is written using the Greek symbol π. This means that for every circle, C ÷ d = π, regardless of the circle’s size.

任何圆的周长与其直径的比值总是一个相同的常数，约为 3.14159。这个常数被称为圆周率，用希腊符号 π 表示。这意味着对于每一个圆，C ÷ d = π，无论圆的大小如何。

In IGCSE examinations, you are usually expected to use π ≈ 3.14 or the fraction 22/7 when a numerical answer is required. However, if the question asks for an “exact value” or says “in terms of π”, you must leave π in your answer.

在 IGCSE 考试中，通常要求你使用 π ≈ 3.14 或分数 22/7 来计算数值答案。但是，如果题目要求”精确值”或”用 π 表示”，你就必须在答案中保留 π。

π ≈ 3.14159… ≈ 3.14 ≈ 22/7

3. Circumference Formula | 周长公式

From the definition of π, we obtain the circumference formula. Since C ÷ d = π, we can rearrange to get C = πd. Because d = 2r, we also write C = 2πr. Both formulas are equally valid; choose whichever matches the information given in the question.

根据 π 的定义，我们得到周长公式。因为 C ÷ d = π，可以变形为 C = πd。又因为 d = 2r，所以也可以写成 C = 2πr。两个公式同样有效；根据题目给出的信息选择使用哪一个。

C = πd = 2πr

Example: A circle has a radius of 7 cm. Find its circumference, using π = 22/7.

例题：一个圆的半径为 7 cm，取 π = 22/7，求其周长。

C = 2 × (22/7) × 7 = 44 cm

Notice that the 7 in the denominator cancels with the radius 7, leaving 2 × 22 = 44. This cancellation is why 22/7 is convenient when the radius is a multiple of 7.

注意分母中的 7 与半径 7 相消，剩下 2 × 22 = 44。这就是为什么当半径是 7 的倍数时，22/7 很方便。

4. Area of a Circle | 圆的面积

The area of a circle is the amount of space enclosed within its circumference. The formula is derived from a limiting process: if you cut a circle into many thin sectors and rearrange them, they form an approximate rectangle whose width is r and length is πr. Hence the area is length × width = πr × r = πr².

圆的面积是圆周所围成的空间大小。这个公式来源于一个极限过程：如果将圆切成许多细小的扇形并重新排列，它们会形成一个近似矩形，其宽度为 r，长度为 πr。因此面积为长 × 宽 = πr × r = πr²。

A = πr²

Example: Find the area of a circle with diameter 10 cm. Give your answer to 1 decimal place.

例题：求直径为 10 cm 的圆的面积，答案保留一位小数。

First find the radius: r = d/2 = 10/2 = 5 cm. Then A = π × 5² = 25π ≈ 25 × 3.14 = 78.5 cm².

先求半径：r = d/2 = 10/2 = 5 cm。然后 A = π × 5² = 25π ≈ 25 × 3.14 = 78.5 cm²。

5. Choosing Between Exact and Approximate Values | 精确值与近似值的选择

Many students lose marks because they do not read the question carefully. If a question says “leave your answer in terms of π”, you must not substitute a decimal. For example, the area above would be written as 25π cm². If the question says “take π = 3.14”, use that value. If it says “give your answer correct to 3 significant figures”, use the π button on your calculator and round at the end.

许多学生因为没有仔细读题而失分。如果题目说”答案用 π 表示”，你就不能代入小数。例如，上面的面积应写成 25π cm²。如果题目说”取 π = 3.14″，就使用该值。如果题目说”答案精确到 3 位有效数字”，就使用计算器上的 π 键并在最后四舍五入。

Instruction (指令)	Required output (所需输出)
In terms of π (用 π 表示)	25π cm²
Take π = 3.14 (取 π = 3.14)	78.5 cm²
Correct to 3 s.f. (精确到3位有效数字)	78.5 cm²
Calculator value (计算器值)	78.5398… cm²

6. Working Backwards: Finding the Radius | 逆向思维：求半径

Sometimes the circumference or area is given, and you must find the radius or diameter. For circumference, divide by 2π: r = C ÷ (2π). For area, take the square root: r = √(A/π). These inverse operations are common in non-calculator papers.

有时题目给出周长或面积，要求半径或直径。对于周长，除以 2π：r = C ÷ (2π)。对于面积，开平方：r = √(A/π)。这些逆运算在不能使用计算器的试卷中很常见。

Example: The circumference of a circle is 31.4 cm. Using π = 3.14, find the radius.

例题：一个圆的周长为 31.4 cm，取 π = 3.14，求半径。

r = 31.4 ÷ (2 × 3.14) = 31.4 ÷ 6.28 = 5 cm

Always check your answer for reasonableness. A circle with radius 5 cm has diameter 10 cm, and the circumference should be slightly more than three times the diameter, which it is.

始终检查答案是否合理。半径为 5 cm 的圆，直径为 10 cm，周长应略大于直径的三倍，而确实如此。

7. Arc Length and Sector Area | 弧长与扇形面积

A sector is a fraction of the whole circle. If the sector has central angle θ degrees, then it represents the fraction θ/360 of the full circle. Therefore the arc length is θ/360 × 2πr, and the sector area is θ/360 × πr².

扇形是整个圆的一部分。如果扇形的圆心角为 θ 度，那么它占整个圆的 θ/360。因此弧长为 θ/360 × 2πr，扇形面积为 θ/360 × πr²。

Arc length (弧长) = (θ/360) × 2πr

Sector area (扇形面积) = (θ/360) × πr²

Example: A sector has radius 6 cm and angle 60°. Find the sector area in terms of π.

例题：一个扇形半径为 6 cm，圆心角为 60°。用 π 表示扇形面积。

A = (60/360) × π × 6² = (1/6) × 36π = 6π cm²

8. Composite Shapes Involving Circles | 含圆的组合图形

IGCSE questions often combine circles with rectangles, triangles, or semicircles. For a composite shape, calculate the area of each component separately and then add or subtract as required. Always draw a clear diagram and label known lengths.

IGCSE 题目经常将圆与矩形、三角形或半圆结合。对于组合图形，分别计算每个组成部分的面积，然后按要求相加或相减。务必画清晰图示并标注已知长度。

Example: A rectangle of length 12 cm and width 8 cm has a semicircle attached to one of its longer sides. Find the total area, using π = 3.14.

例题：一个长 12 cm、宽 8 cm 的矩形，在一条长边上接一个半圆。取 π = 3.14，求总面积。

Rectangle area (矩形面积): 12 × 8 = 96 cm²
Semicircle radius (半圆半径): 6 cm (half of 12)
Semicircle area (半圆面积): (1/2) × π × 6² = 18 × 3.14 = 56.52 cm²
Total area (总面积): 96 + 56.52 = 152.52 cm²

9. Perimeter of Composite Shapes | 组合图形的周长

Be careful with perimeter! The perimeter of a composite shape includes only the outer boundary. For the example above, the perimeter is the sum of the three rectangle sides plus the semicircular arc. Do not include the shared side between the rectangle and the semicircle because it is inside the shape.

注意周长！组合图形的周长只包含外部边界。对于上面的例子，周长是矩形三条边加上半圆弧长之和。不要包括矩形与半圆共用的那条边，因为它在图形内部。

P = 12 + 8 + 8 + (1/2) × 2π × 6 = 28 + 6π ≈ 46.84 cm

Notice that we excluded the 12 cm side that is inside the figure. This is a classic trap in IGCSE exams.

注意我们排除了图形内部那条 12 cm 的边。这是 IGCSE 考试中的经典陷阱。

10. Common Mistakes and How to Avoid Them | 常见错误及避坑指南

The most frequent errors are using the wrong formula, confusing radius with diameter, and forgetting to halve the diameter. Another common mistake is using circumference when the question asks for area. Write down the formula first, substitute carefully, and always check units.

最常见的错误包括：用错公式、混淆半径与直径、忘记将直径除以 2。另一个常见错误是题目求面积时却用了周长公式。先写出公式，仔细代入，并始终检查单位。

Mistake (错误): Using d instead of r in A = πr²
Fix (纠正): Always write r = d/2 first
Mistake (错误): Adding π to answers numerically
Fix (纠正): Keep π until the final step
Mistake (错误): Forgetting squared units for area
Fix (纠正): Area is always in cm², m², etc.

11. Practice Problems | 练习巩固

Now attempt these five questions. Cover the solutions below and test yourself first.

现在尝试以下五个问题。先遮住下面的解答，自我测验。

Find the circumference of a circle with radius 10 cm, in terms of π.
Find the area of a circle with diameter 14 cm, using π = 22/7.
The area of a circle is 154 cm². Taking π = 22/7, find the radius.
A sector has radius 9 cm and angle 40°. Find the arc length in terms of π.
A square of side 10 cm has a circle inscribed inside it. Find the shaded area outside the circle, using π = 3.14.

求半径为 10 cm 的圆的周长，用 π 表示。
求直径为 14 cm 的圆的面积，取 π = 22/7。
圆的面积为 154 cm²，取 π = 22/7，求半径。
扇形半径为 9 cm，圆心角为 40°，用 π 表示弧长。
边长为 10 cm 的正方形内切一个圆，求圆外阴影部分面积，取 π = 3.14。

12. Solutions and Exam Strategy | 解答与应试策略

Solutions / 解答

1. C = 2πr = 2 × π × 10 = 20π cm.

2. r = 7 cm, so A = (22/7) × 7² = (22/7) × 49 = 154 cm².

3. πr² = 154, so r² = 154 × (7/22) = 7 × 7 = 49, hence r = 7 cm.

4. Arc length = (40/360) × 2π × 9 = (1/9) × 18π = 2π cm.

5. Square area = 100 cm². Circle radius = 5 cm, so circle area = 3.14 × 25 = 78.5 cm². Shaded area = 100 − 78.5 = 21.5 cm².

1. C = 2πr = 2 × π × 10 = 20π cm。

2. r = 7 cm，所以 A = (22/7) × 7² = (22/7) × 49 = 154 cm²。

3. πr² = 154，所以 r² = 154 × (7/22) = 7 × 7 = 49，因此 r = 7 cm。

4. 弧长 = (40/360) × 2π × 9 = (1/9) × 18π = 2π cm。

5. 正方形面积 = 100 cm²。圆半径 = 5 cm，所以圆面积 = 3.14 × 25 = 78.5 cm²。阴影部分面积 = 100 − 78.5 = 21.5 cm²。

Exam strategy: In the exam, read every question twice. Underline whether you need circumference or area, and note whether π is specified. Show all working steps because partial credit is awarded in IGCSE. Finally, always include the correct units.

应试策略：考试时，每题读两遍。标出你需要的是周长还是面积，并注意 π 是否被指定。写出所有步骤，因为 IGCSE 考试会给步骤分。最后，始终写上正确的单位。

With these tools, you can master all circle measurement questions in Lesson 16-3 and beyond. Practise regularly, and you will see these problems become automatic.

掌握了这些工具，你就能攻克第 16-3 课及以后所有圆测量问题。定期练习，你会发现这些问题变得得心应手。

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Circle Measurements: Circumference and Area | 圆的测量：周长与面积