Mastering Circles: Area and Circumference | 掌握圆形：面积与周长

📚 Mastering Circles: Area and Circumference | 掌握圆形：面积与周长

Many KS3 Cambridge questions ask you to work out the circumference or area of a circle, or to use these ideas inside a longer problem. This article reviews the key formulae, worked methods and common exam traps so you can answer page-style questions confidently.

许多 KS3 剑桥题目要求你计算圆的周长或面积，或将这些知识用于更长的应用题。本文复习关键公式、解题步骤和常见考试陷阱，帮助你自信地完成教材风格的练习。

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1. Circle Vocabulary and π | 圆的术语与 π

A circle is the set of points a fixed distance from its centre. The radius r is the distance from the centre to the edge; the diameter d is twice the radius, so d = 2r.

圆是由与圆心距离相等的所有点组成的图形。半径 r 是从圆心到圆周的距离；直径 d 是半径的两倍，因此 d = 2r。

The number π (pi) is a constant about 3.14159. It is the ratio of any circle’s circumference to its diameter. On a calculator you use the π button; if no calculator is allowed, 3.14 or 22/7 is sometimes given in the question.

π（圆周率）是一个约等于 3.14159 的常数，它是任意圆的周长与直径之比。计算器上通常使用 π 键；如果不能用计算器，题目有时给出 3.14 或 22/7。

• Radius r: centre to circle. 半径 r：圆心到圆周。
• Diameter d: edge to edge through centre, d = 2r. 直径 d：经过圆心的两端距离，d = 2r。

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2. Circumference Formula | 周长公式

The circumference C of a circle is given by C = πd or C = 2πr, because d = 2r. Use whichever version matches the value you are given.

圆的周长 C 的公式为 C = πd 或 C = 2πr，因为 d = 2r。题目给出哪个量，就选择相应版本。

C = πd and C = 2πr

For example, if a circle has radius 5 cm, its circumference is C = 2 × π × 5 = 10π cm, which is about 31.4 cm.

例如，一个半径为 5 cm 的圆，其周长为 C = 2 × π × 5 = 10π cm，约等于 31.4 cm。

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3. Area Formula | 面积公式

The area A of a circle is found by A = πr². Note that only the radius is squared, not π. The units of area are square units, such as cm² or m².

圆的面积 A 用公式 A = πr² 计算。注意只有半径被平方，π 不需要平方。面积的单位是平方单位，如 cm² 或 m²。

A = πr²

For a circle with radius 3 m, A = π × 3² = 9π m² ≈ 28.3 m².

对于半径为 3 m 的圆，A = π × 3² = 9π m² ≈ 28.3 m²。

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4. Choosing Radius or Diameter | 区分半径与直径

A very common mistake is to use the diameter in the area formula. If you are given the diameter, halve it first to get the radius, then square that value in A = πr².

一个非常常见的错误是把直径直接代入面积公式。如果题目给的是直径，先除以 2 得到半径，再平方代入 A = πr²。

Example: a circle has diameter 12 cm. Radius r = 12 ÷ 2 = 6 cm. Area A = π × 6² = 36π cm², not π × 12².

示例：一个圆的直径为 12 cm。半径 r = 12 ÷ 2 = 6 cm。面积 A = π × 6² = 36π cm²，而不是 π × 12²。

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5. Worked Example: Circumference | 例题：求周长

Question: Work out the circumference of a circle with radius 7.4 cm. Give your answer to 1 decimal place.

题目：计算半径为 7.4 cm 的圆的周长，答案精确到 1 位小数。

Solution: Use C = 2πr. Substitute r = 7.4 cm. C = 2 × π × 7.4 = 14.8π cm. Press the calculator π key: 14.8 × π ≈ 46.495… cm, so C ≈ 46.5 cm to 1 d.p.

解答：使用 C = 2πr。代入 r = 7.4 cm。C = 2 × π × 7.4 = 14.8π cm。在计算器上按 π 键：14.8 × π ≈ 46.495… cm，所以 C ≈ 46.5 cm（1 位小数）。

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6. Worked Example: Area | 例题：求面积

Question: A circular pond has a diameter of 16 m. Find its area, giving your answer to 2 significant figures.

题目：一个圆形池塘的直径为 16 m。求其面积，答案保留 2 位有效数字。

Solution: Halve the diameter: r = 16 ÷ 2 = 8 m. Area A = π × 8² = 64π m². Using a calculator, A ≈ 201.0619… m², so A ≈ 200 m² to 2 s.f.

解答：先把直径减半：r = 16 ÷ 2 = 8 m。面积 A = π × 8² = 64π m²。用计算器计算，A ≈ 201.0619… m²，所以 A ≈ 200 m²（2 位有效数字）。

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7. Working Backwards from Area or Circumference | 由面积或周长反求半径/直径

Some questions give you the area or circumference and ask for the radius or diameter. Rearrange the formulae:

有些题目给出面积或周长，要求半径或直径。可以变形公式：

r = C ÷ (2π) and r = √(A ÷ π)

Example: if C = 31.4 cm, then r =

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