KS3 Maths: Circles – Key Revision Points | KS3 数学：圆周运动 考点精讲

📚 KS3 Maths: Circles – Key Revision Points | KS3 数学：圆周运动 考点精讲

Circles appear everywhere in KS3 maths, and mastering them is all about understanding two key formulas and the special number π. This guide breaks down circumference, area, and problem-solving skills you need for tests.

圆在 KS3 数学中无处不在，掌握它的关键在于理解两个核心公式以及特殊的数 π。本指南将分解周长、面积以及考试中所需的解题技巧。

1. What is a Circle? | 什么是圆？

A circle is a set of points that are all the same distance from a fixed centre point. This equal distance is called the radius.

圆是一组到某个固定中心点距离都相等的点。这个相等的距离叫做半径。

2. Key Parts of a Circle | 圆的关键部分

Before using any formula, you must know the basic vocabulary. The radius (r) stretches from the centre to the edge. The diameter (d) goes all the way across the circle through the centre, and it is always twice the radius.

在使用任何公式之前，你必须掌握基本术语。半径 (r) 从圆心延伸到边缘。直径 (d) 穿过圆心横跨整个圆，它始终是半径的两倍。

• Radius = r

  半径 = r

• Diameter = d = 2r

  直径 = d = 2r

• The circumference is the distance around the circle (like the perimeter).

  周长是围绕圆一周的距离（类似于多边形的周长）。

3. Understanding Pi (π) | 理解圆周率 π

Pi, written as the Greek letter π, is a special number roughly equal to 3.14. It represents the ratio of a circle’s circumference to its diameter. This ratio is the same for every circle.

圆周率，写作希腊字母 π，是一个大约等于 3.14 的特殊数字。它表示圆的周长与直径之比。这个比值对任何圆都相同。

In KS3, you will either use π ≈ 3.14 or leave answers in terms of π, depending on the question.

在 KS3 阶段，根据题目要求，你可能使用 π ≈ 3.14，或者将答案保留为 π 的倍数形式。

4. Circumference Formula | 周长公式

The circumference (C) of a circle can be found using two equivalent forms. The first uses diameter, and the second uses radius.

圆的周长 (C) 可以用两种等价形式求得。第一种使用直径，第二种使用半径。

C = πd

（中文：周长 = 圆周率 × 直径）

C = 2πr

（中文：周长 = 2 × 圆周率 × 半径）

Always check whether you are given the radius or the diameter before choosing the easier formula.

在选择使用哪个公式更简单之前，一定要先看清题目给出的是半径还是直径。

5. Example: Calculating Circumference | 例子：计算周长

If a circle has a diameter of 10 cm, the circumference is C = π × 10 ≈ 3.14 × 10 = 31.4 cm. Using the radius of 5 cm, C = 2 × π × 5 ≈ 2 × 3.14 × 5 = 31.4 cm.

如果一个圆的直径是 10 cm，周长 C = π × 10 ≈ 3.14 × 10 = 31.4 cm。如果使用半径 5 cm，C = 2 × π × 5 ≈ 2 × 3.14 × 5 = 31.4 cm。

Both methods give the same result. Remember to include units (cm, m, etc.) in your final answer.

两种方法结果相同。记得在最终答案里带上单位（cm、m 等）。

6. Area of a Circle Formula | 圆面积公式

The area (A) of a circle is found using the radius. The diameter cannot be used directly in the area formula – you must halve it first to get r.

圆的面积 (A) 必须使用半径来计算。面积公式中不能直接使用直径——你必须先将直径除以 2 得到 r。

A = πr²

（中文：面积 = 圆周率 × 半径的平方）

The small ² means the radius is multiplied by itself (r × r) before multiplying by π.

小 ² 表示半径先自乘 (r × r)，再乘以 π。

7. Example: Calculating Area | 例子：计算面积

For a circle with radius 3 cm, first square the radius: 3² = 9. Then A = π × 9 ≈ 3.14 × 9 = 28.26 cm². Notice that the units for area are square units (cm², m²).

对于一个半径 3 cm 的圆，先将半径平方：3² = 9。然后 A = π × 9 ≈ 3.14 × 9 = 28.26 cm²。注意面积的单位是平方单位 (cm², m²)。

If you are given the diameter, say 8 m, first find r = 4 m, then A = π × 4² = π × 16 ≈ 50.24 m².

如果给出的是直径，比如 8 m，先求出 r = 4 m，然后 A = π × 4² = π × 16 ≈ 50.24 m²。

8. Working Backwards: Finding Radius or Diameter | 逆向计算：求半径或直径

Sometimes you know the circumference or area and need to find the radius. For circumference, rearrange C = 2πr to get r = C / (2π). For area, rearrange A = πr² to get r = √(A / π).

有时你已知周长或面积，需要求出半径。对于周长，将公式变形为 r = C / (2π)。对于面积，将 A = πr² 变形为 r = √(A / π)。

After finding r, you can easily double it to obtain the diameter.

求出 r 后，你可以轻松地将其乘以 2 得到直径。

• If C = 44 cm, r = 44 / (2 × 3.14) ≈ 44 / 6.28 ≈ 7.0 cm

  如果 C = 44 cm，r = 44 / (2 × 3.14) ≈ 44 / 6.28 ≈ 7.0 cm

• If A = 154 cm², r² = 154 / 3.14 ≈ 49, so r = √49 = 7 cm

  如果 A = 154 cm²，r² = 154 / 3.14 ≈ 49，因此 r = √49 = 7 cm

9. Compound Shapes Involving Circles | 涉及圆的复合图形

KS3 questions often combine circles with rectangles or triangles. For example, a running track consists of two straight sides and two semicircles at the ends. The total perimeter is the sum of the straight lengths plus the circumference of one whole circle (since two semicircles make one circle).

KS3 题目经常将圆与矩形或三角形组合在一起。例如，一条跑道由两条直道和两端的两个半圆组成。总周长等于直道长度之和加上一个完整圆的周长（因为两个半圆合成一个圆）。

For shaded area problems, find the area of the larger shape and subtract the area of the smaller shape (like a circle cut out of a square).

对于求阴影面积的问题，先求出大图形的面积，再减去小图形的面积（例如从正方形中挖去一个圆）。

10. Common Mistakes to Avoid | 常见错误

One common error is using the diameter in the area formula without halving it. Remember: A = πr², not πd². Always halve the diameter to get the radius first.

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。记住：A = πr²，而不是 πd²。一定要先将直径减半得到半径。

Another mistake is confusing circumference and area units. Circumference is a length (cm, m), while area is in square units (cm², m²).

另一个错误是混淆周长和面积的单位。周长是长度（cm, m），而面积用平方单位（cm², m²）。

Also, when using a calculator, avoid rounding π too early. Use the π button or at least 3.14, and only round the final answer.

另外，使用计算器时，避免过早对 π 取近似值。使用 π 键或至少 3.14，只在最后答案处四舍五入。

11. Exam-style Practice Questions | 考试风格练习题

Try these questions to test your understanding. Answers are provided in brackets, but try to solve them first.

尝试以下问题来检验你的理解。括号中提供了答案，但请先自己尝试解答。

• Q1: The diameter of a circular pond is 14 m. Find its circumference. (C ≈ 43.96 m using π ≈ 3.14)

  问题1：一个圆形池塘的直径是 14 m。求它的周长。（使用 π ≈ 3.14，C ≈ 43.96 m）

• Q2: A coin has radius 1.5 cm. What is its area? (A ≈ 7.065 cm²)

  问题2：一枚硬币的半径是 1.5 cm。它的面积是多少？（A ≈ 7.065 cm²）

• Q3: The circumference of a bicycle wheel is 188.4 cm. Find its radius. Take π = 3.14. (r = 30 cm)

  问题3：自行车轮子的周长是 188.4 cm。求它的半径。π 取 3.14。（r = 30 cm）

• Q4: A semicircle has diameter 10 cm. Calculate its perimeter. (Perimeter = (1/2 × π × 10) + 10 ≈ 15.7 + 10 = 25.7 cm)

  问题4：一个半圆的直径是 10 cm。计算它的周长。（周长 = (1/2 × π × 10) + 10 ≈ 15.7 + 10 = 25.7 cm）

12. Summary and Key Takeaways | 总结与关键要点

To recap, always identify whether a question asks for circumference (distance around) or area (space inside). Memorise the formulas C = πd or 2πr, and A = πr². Practise converting between radius and diameter quickly.

总结一下，始终要明确题目要求的是周长（外围距离）还是面积（内部空间）。熟记公式 C = πd 或 2πr，以及 A = πr²。练习快速在半径和直径之间转换。

With these fundamentals, you can confidently tackle any circle problem in your KS3 exam.

掌握了这些基础，你就能自信地应对 KS3 考试中的任何圆的问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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