📚 Circles: Circumference and Area | 圆：周长与面积
Circles appear everywhere in maths and the real world, from wheels to coins. In KS3, you need to master two key measurements: the circumference (the distance around a circle) and the area (the space inside it). Both involve the special number π (pi). This article breaks down all you need to know, with clear explanations and worked examples.
圆在数学和现实世界中无处不在，从车轮到硬币。在 KS3 阶段，你需要掌握两个关键度量：周长（围绕圆一圈的距离）和面积（圆内部的空间）。这两者都涉及特殊的数 π（圆周率）。本文将详细分解你需要知道的所有内容，并提供清晰的解释和例题。
1. What Is a Circle? | 什么是圆？
A circle is a set of points in a plane that are all at the same distance from a fixed centre point. That fixed distance is called the radius. The circle is a continuous curved line, and it is not a polygon – it has no straight sides or vertices.
圆是平面内到一定点距离等于定长的所有点的集合。这个定长叫做半径。圆是一条连续的曲线，它不是多边形——没有直边，也没有顶点。
When you draw a circle using a compass, the sharp end marks the centre and the pencil traces all points at the radius length. Every circle is perfectly symmetrical, and any line through the centre splits it into two identical halves.
当你用圆规画圆时，尖的一端固定圆心，铅笔一端画出所有到圆心距离等于半径的点。每个圆都是完全对称的，任何通过圆心的直线都会将它分成两个完全相同的半圆。
2. Key Parts of a Circle | 圆的关键组成部分
Before working with circumference and area, you must know these terms:
在计算周长和面积之前，你必须了解以下术语：
- Radius (r) – the distance from the centre to any point on the circle. 半径（r）——从圆心到圆上任意一点的距离。
- Diameter (d) – the distance across the circle through the centre. It is twice the radius: d = 2r. 直径（d）——穿过圆心连接圆上两点的线段长度。直径是半径的两倍：d = 2r。
- Circumference (C) – the total distance around the circle. 周长（C）——圆一周的总长度。
- Centre – the fixed point inside the circle from which all points on the boundary are equally distant. 圆心——圆内部的定点，边界上所有点到它的距离都相等。
The radius is the most important measurement because all formulas for circles start with r. Always check whether a question gives you the radius or the diameter.
半径是最重要的度量，因为所有关于圆的公式都从 r 开始。在做题时一定要看清楚题目给出的是半径还是直径。
3. Understanding π (Pi) | 理解 π（圆周率）
Pi (π) is a special mathematical constant. It is the ratio of a circle's circumference to its diameter. This ratio is the same for every circle, no matter how large or small. π is approximately 3.14159, but it is an irrational number with infinite decimal places.
π（圆周率）是一个特殊的数学常数。它是圆的周长与直径的比值。无论圆有多大或多小，这个比值始终相同。π 约等于 3.14159，但它是一个无理数，小数部分无限不循环。
In KS3, you will either leave answers in terms of π (for example, 10π cm) or use an approximation such as 3.14 or the π button on a calculator. Your teacher or exam question will tell you which form to use.
在 KS3 中，你可能会把答案保留为带有 π 的形式（例如 10π cm），也可能使用近似值，如 3.14 或计算器上的 π 键。老师或题目会告诉你使用哪一种形式。
π = C ÷ d
4. Formula for Circumference | 周长公式
The circumference of a circle can be calculated using either the radius or the diameter. The two versions of the formula are:
圆的周长可以用半径或直径来计算。公式有两种形式：
C = πd or C = 2πr
double it first and then multiply by π. Both formulas give the same result because d = 2r.
如果你知道直径，将直径乘以 π 就得到周长。如果你知道半径，先将半径乘以 2 再乘以 π。两个公式结果相同，因为 d = 2r。
ius 5 cm has a circumference of 2 × π × 5 = 10π cm. Using π ≈ 3.14, this is about 31.4 cm.
例如，一个半径为 5 cm 的圆，周长为 2 × π × 5 = 10π cm。使用 π ≈ 3.14，大约是 31.4 cm。
5. Formula for Area | 面积公式
The area of a circle is found using the formula:
圆的面积使用以下公式计算：
A = πr²
This means you take the radius, square it (multiply it by itself), and then multiply by π. It is very important that you only square the radius, not the π or any other part.
也就是说，你将半径平方（乘以它自身），然后再乘以 π。非常重要的是，你只对半径进行平方运算，不要对 π 或其他部分平方。
g π ≈ 3.14 gives approximately 28.26 m². Notice the units: area is always in square units.
例如，如果半径是 3 m，面积是 π × 3² = π × 9 = 9π m²。使用 π ≈ 3.14 得到大约 28.26 m²。注意单位：面积始终使用平方单位。
6. Worked Examples – Circumference | 周长例题讲解
Let's go through some examples step by step to see how the circumference formula is used in practice.
让我们逐步看几个例子，了解实际中如何使用周长公式。
Example 1: A circle has a diameter of 8 cm. Find its circumference. Give your answer in terms of π.
C = πd = π × 8 = 8π cm.
例 1：一个圆的直径为 8 cm。求它的周长，答案保留 π。
C = πd = π × 8 = 8π cm。
Example 2: A circular pond has a radius of 12 m. Calculate the circumference to 1 decimal place.
C = 2πr = 2 × π × 12 = 24π m ≈ 24 × 3.14159... = 75.4 m (1 d.p.)
例 2：一个圆形池塘的半径是 12 m。计算周长，精确到一位小数。
C = 2πr = 2 × π × 12 = 24π m ≈ 24 × 3.14159... = 75.4 m（一位小数）
Always write down the formula first, substitute the values, and then calculate. Using the π button on your calculator will give a more accurate answer than using 3.14.
一定要先写出公式，代入数值，然后再计算。使用计算器上的 π 键会比使用 3.14 得到更精确的答案。
7. Worked Examples – Area | 面积例题讲解
Now practise with the area formula. Remember to square the radius before multiplying by π.
现在练习面积公式。记住要先计算半径的平方，再乘以 π。
Example 3: Find the area of a circle with radius 7 cm. Leave your answer in terms of π.
A = πr² = π × 7² = π × 49 = 49π cm².
例 3：求半径为 7 cm 的圆的面积，答案保留 π。
A = πr² = π × 7² = π × 49 = 49π cm²。
Example 4: A circular table top has a diameter of 1.2 m. Calculate its area in m² to 2 decimal places.
First find radius: r = d ÷ 2 = 1.2 ÷ 2 = 0.6 m.
A = πr² = π × (0.6)² = π × 0.36 ≈ 3.14159 × 0.36 = 1.1309... ≈ 1.13 m² (2 d.p.)
例 4：一个圆形桌面的直径为 1.2 m。计算它的面积（m²），精确到两位小数。
先求半径：r = d ÷ 2 = 1.2 ÷ 2 = 0.6 m。
A = πr² = π × (0.6)² = π × 0.36 ≈ 3.14159 × 0.36 = 1.1309... ≈ 1.13 m²（两位小数）
Notice that if you are given the diameter, you must halve it to get the radius before using the area formula. This is a common trap!
注意，如果给出的是直径，你必须先除以 2 得到半径，然后再使用面积公式。这是一个常见的陷阱！
8. Finding Radius or Diameter from Circumference | 由周长求半径或直径
cumference but need to find the radius or diameter. You can rearrange the formula C = 2πr.
有时你知道周长，但需要求半径或直径。你可以将公式 C = 2πr 变形。
To find the diameter: d = C ÷ π. To find the radius: r = C ÷ (2π).
求直径：d = C ÷ π。求半径：r = C ÷ (2π)。
Example 5: A circle has a circumference of 44 cm. Find its diameter, using π = 22/7.
d = C ÷ π = 44 ÷ (22/7) = 44 × (7/22) = 2 × 7 = 14 cm.
例 5：一个圆的周长是 44 cm。使用 π = 22/7 求它的直径。
d = C ÷ π = 44 ÷ (22/7) = 44 × (7/22) = 2 × 7 = 14 cm。
Some problems in KS3 use the fraction 22/7 as an approximation for π because it makes the arithmetic easier. However, it is only an approximate value and should only be used when specifically asked.
KS3 中的一些题目会使用分数 22/7 作为 π 的近似值，因为这样计算更简便。但是，这只是一个近似值，只有在题目明确要求时才使用。
9. Finding Radius or Diameter from Area | 由面积求半径或直径
can find the radius by rearranging A = πr². First divide by π, then take the square root.
如果给出面积，你可以将 A = πr² 变形来求半径。先除以 π，再开平方根。
r = √A ÷ π
Example 6: The area of a circular badge is 154 cm². Find its radius. (Use π = 22/7).
r² = A ÷ π = 154 ÷ (22/7) = 154 × (7/22) = 7 × 7 = 49, so r = √49 = 7 cm.
例 6：一个圆形徽章的面积是 154 cm²。求它的半径。（使用 π = 22/7）。
r² = A ÷ π = 154 ÷ (22/7) = 154 × (7/22) = 7 × 7 = 49，所以 r = √49 = 7 cm。
Once you have the radius, you can easily find the diameter by doubling it. This technique is especially useful in problems involving a circle's dimensions from its area.
得到半径后，只需乘以 2 就能得到直径。这种方法在根据面积求圆的尺寸的问题中非常有用。
10. Semi-circles and Quarter-circles | 半圆和四分之一圆
You will often be asked to find the perimeter or area of a semi-circle or quarter-circle. For these, you combine a fraction of the circle's circumference or area with straight edges.
你经常会被要求求半圆或四分之一圆的周长或面积。这时，你需要将圆周或面积的一部分与直线边组合起来。
For a semi-circle: The area is half of πr². The perimeter is half of the circumference (πr) plus the diameter (2r).
For a quarter-circle: The area is one-quarter of πr². The perimeter is one-quarter of the circumference (πr/2) plus two radii (2r).
对于半圆：面积是 πr² 的一半。周长是周长的一半（πr）加上直径（2r）。
对于四分之一圆：面积是 πr² 的四分之一。周长是周长的四分之一（πr/2）加上两个半径（2r）。
Students often forget to add the straight line(s) when finding the perimeter. Always sketch the shape and mark which edges form the boundary.
学生在求周长时常常忘记加上直线边。始终画出形状并标出哪些边构成边界。
11. Composite Shapes with Circles | 包含圆的组合图形
In KS3, you may combine circles with rectangles, squares or triangles. To find the total area or perimeter, break the shape into parts you know.
在 KS3 阶段，你可能会把圆与长方形、正方形或三角形组合起来。要计算总面积或周长，需将图形分解为你熟悉的各个部分。
Example 7: A running track consists of two semi-circles at the ends of a rectangle of length 100 m and width 50 m. Find the total perimeter of the inner edge.
5 = 50π m. Add the two straight lengths: 100 + 100 = 200 m. Total perimeter = 50π + 200 m.
例 7：一条跑道由两个半圆分别位于一个长 100 m、宽 50 m 的矩形两端构成。求内侧边线的总周长。
两个半圆组成一个直径为 50 m（半径 25 m）的完整圆。该圆的周长 = 2 × π × 25 = 50π m。再加上两条直道长：100 + 100 = 200 m。总周长 = 50π + 200 m。
Always read the question carefully to know whether it wants a numerical approximation or an answer in terms of π. Labelling radii on your diagram helps prevent mistakes.
一定要仔细读题，弄清是需要数值近似解还是保留 π 的答案。在图上标出半径有助于避免出错。
12. Common Mistakes and Tips | 常见错误与提示
- Mistake 1: the diameter in the area formula without halving it first. Always use radius for A = πr². 错误 1：直接将直径代入面积公式，而没有先除以 2。在 A = πr² 中始终使用半径。
- Mistake 2: Forgetting to square the radius. πr² means π × r × r, not π × r × 2. 错误 2：忘记对半径进行平方。πr² 表示 π × r × r，而不是 π × r × 2。
- Mistake 3: Confusing perimeter of a semi-circle with its curved part only. Always add the diameter. 错误 3：将半圆的周长与仅它的曲线部分混淆。一定要加上直径。
- Mistake 4: Mixing up the relationships: diameter = 2 × radius, but area is π × radius². Don't square the 2 by accident. 错误 4：混淆关系式：直径 = 2 × 半径，但面积是 π × 半径²。不要误把 2 也平方了。
To avoid common errors, always start by writing down what you know (r or d, and what formula you need). Show every step clearly – this helps you spot mistakes and earns method marks in tests.
为了避免常见错误，始终从写下已知条件（半径或直径，以及需要什么公式）开始。清晰地展示每一步——这有助于你发现错误，也能在考试中赢得过程分。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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