Practice 64.1: Circumference and Area of Circles | 练习 64.1：圆的周长与面积

📚 Practice 64.1: Circumference and Area of Circles | 练习 64.1：圆的周长与面积

Welcome to this complete walkthrough of question 1 from page 64 of your Cambridge Checkpoint Mathematics practice book. We will explore the key skills needed to calculate the circumference and area of a circle, using clear steps and plenty of examples. The aim is to build your confidence so that you can tackle any similar problem with ease.

欢迎来到本次对剑桥 Checkpoint 数学练习册第 64 页第 1 题的详细讲解。我们将探索计算圆的周长和面积所需的关键技能，通过清晰的步骤和大量例子，帮助你建立自信，轻松应对任何类似问题。

1. Understanding π and the Circle | 理解 π 与圆

A circle is a perfectly round shape where every point on its edge is exactly the same distance from the centre. That distance is called the radius, usually written as r. The distance across the circle passing through the centre is the diameter d, which is twice the radius: d = 2r. One of the most fascinating numbers in mathematics, π (pi), links the diameter to the circumference. Regardless of the size of the circle, if you divide the circumference by the diameter you always get π, approximately 3.14159. In KS3, you will often be told to use π as 3.14 or leave your answers in terms of π.

圆是一种完美的封闭曲线，边缘上的每一点到中心的距离完全相等，这个距离称为半径，通常写作 r。穿过圆心的线段长度是直径 d，它是半径的两倍：d = 2r。数学中最神奇的数字之一 π（圆周率）将直径与周长联系起来。无论圆有多大，用周长除以直径总是得到 π，约等于 3.14159。在 KS3 阶段，题目常要求你将 π 取为 3.14 或保留 π 在答案中。

2. Circumference Formula | 周长公式

The circumference is the distance around the circle – its perimeter. There are two versions of the formula you need to remember:

周长是圆周的长度 – 即圆的边界。你需要记住公式的两种写法：

C = π × d

C = 2 × π × r

Because the diameter is twice the radius, both formulas give the same result. Always check whether the question gives you the radius or the diameter so you can pick the most convenient form.

因为直径是半径的两倍，这两个公式结果相同。解题时一定要看清题目给出的是半径还是直径，从而选择最方便的形式。

3. Worked Example: Finding Circumference from Radius | 例题：已知半径求周长

Suppose a circle has a radius of 5 cm. Using C = 2πr, we substitute r = 5:

假设一个圆的半径为 5 cm。使用 C = 2πr，代入 r = 5：

C = 2 × π × 5 = 10π cm

If the problem asks for a numerical answer with π ≈ 3.14, then C ≈ 2 × 3.14 × 5 = 31.4 cm. Leaving the answer as 10π cm is also common and often preferred until the final step.

如果题目要求用 π ≈ 3.14 计算数值，则 C ≈ 2 × 3.14 × 5 = 31.4 cm。将答案保留为 10π cm 也很常见，通常在最终步骤前更受欢迎。

4. Finding Circumference from Diameter | 已知直径求周长

When a question states that the diameter of a circle is 12 cm, you can directly apply C = πd. This avoids the extra step of halving the diameter to obtain the radius. Let us calculate:

当题目给出圆的直径为 12 cm 时，可以直接使用 C = πd，省去将直径除以二得到半径的步骤。计算如下：

C = π × 12 = 12π cm

Using π ≈ 3.14 gives C ≈ 37.68 cm. This approach saves time and reduces the risk of making a careless mistake when halving the diameter.

使用 π ≈ 3.14 得到 C ≈ 37.68 cm。这种方法节省时间，也降低了在将直径除以二时犯粗心错误的风险。

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5. The Area Formula | 面积公式

The area of a circle is the amount of space inside it. The formula relies on the radius:

圆的面积是它内部空间的大小。面积公式依赖于半径：

A = π × r²

Notice that the radius is squared first, and then multiplied by π. This is a very common source of errors – students sometimes multiply the radius by 2 and then square, or square the diameter by mistake. Always take care to apply the correct order of operations: square the radius, then multiply by π.

请注意，必须先对半径进行平方，然后再乘以 π。这是一个常见的错误来源——有些学生先将半径乘以 2 再平方，或者错误地将直径平方。始终要注意正确的运算顺序：先平方半径，再乘以 π。

6. Worked Example: Calculating Area | 例题：计算面积

Let us find the area of a circle with radius 4 cm. Substitute r = 4 into A = πr²:

我们来求半径为 4 cm 的圆的面积。将 r = 4 代入 A = πr²：

A = π × 4² = π × 16 = 16π cm²

If an approximate value is required, multiply 16 by 3.14 to get A ≈ 50.24 cm². Remember that area is always expressed in square units, here cm².

如果需要近似值，将 16 乘以 3.14 得到 A ≈ 50.24 cm²。请记住，面积始终用平方单位表示，这里是 cm²。

7. Solving for Radius Given Circumference or Area | 已知周长或面积求半径

Sometimes the question gives the circumference and asks you to find the radius or diameter. For example, if C = 31.4 cm and you are using π = 3.14, you can rearrange the formula:

有时题目给出周长，让你求半径或直径。例如，已知 C = 31.4 cm，且使用 π = 3.14，你可以将公式变形：

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

Similarly, if you know the area, say A = 78.5 cm² with π = 3.14, you can solve for r by dividing A by π and then taking the square root: r = √(A ÷ π) = √(78.5 ÷ 3.14) = √25 = 5 cm. This inverse thinking is a vital skill for KS3.

类似地，如果知道面积，比如 A = 78.5 cm² 且 π = 3.14，可以先除以 π 再开平方来求半径：r = √(A ÷ π) = √(78.5 ÷ 3.14) = √25 = 5 cm。这种逆向思维是 KS3 的重要技能。

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8. Common Mistakes and Misconceptions | 常见错误与误解

Many students lose marks by confusing radius and diameter. Always read the problem carefully – does it mention ‘radius’ or ‘diameter’? Another classic error is using the wrong area formula, such as A = πd² or A = 2πr. The correct area formula is A = πr². When substituting values, remember to square the radius, not the diameter. Also watch your units: if the radius is in metres, the area is in m². Mixing units like cm and m must be avoided, or you need to convert first.

很多学生因为混淆半径和直径而丢分。务必仔细读题——题目中提到的是“半径”还是“直径”？另一个典型错误是使用错误的面积公式，例如 A = πd² 或 A = 2πr。正确的面积公式是 A = πr²。代入数值时，记住是对半径平方，而不是对直径平方。还要注意单位：如果半径的单位是米，那么面积的单位就是平方米。必须避免 cm 和 m 混用，否则需要先转换单位。

Another misconception involves leaving π in the answer. When a question says ‘Give your answer in terms of π’, you must not substitute 3.14. For instance, the circumference 10π cm is an exact answer, while 31.4 cm is an approximation. Always follow the instruction given.

另一个误解涉及将 π 保留在答案中。当题目要求“用 π 表示你的答案”时，你一定不能代入 3.14。例如，周长 10π cm 是精确答案，而 31.4 cm 是近似值。务必遵循题目给出的指令。

9. Practice Problem Walkthrough (Question 1 from p64) | 练习题详解（p64 第1题）

Now let us apply everything to the specific question from page 64, exercise 1. The problem asks: ‘A circle has a radius of 7 cm. Find (a) the circumference and (b) the area of the circle. Give your answers to 1 decimal place using π = 3.14.’ We will break it down.

现在我们将所有知识应用到第 64 页练习 1 的具体题目中。题目要求：“一个圆的半径为 7 cm。求 (a) 周长和 (b) 圆的面积。使用 π = 3.14，将答案保留一位小数。”我们来逐步分解。

Part (a): Circumference. Using C = 2πr, substitute r = 7 and π = 3.14.

第 (a) 部分：周长。使用 C = 2πr，代入 r = 7 和 π = 3.14。

C = 2 × 3.14 × 7 = 6.28 × 7 = 43.96 cm

Rounded to 1 decimal place, this becomes 44.0 cm. It is important to write 44.0 rather than just 44 to show the required level of precision.

四舍五入到一位小数，结果是 44.0 cm。写成 44.0 而不是 44 很重要，这样才能体现题目要求的精确度。

Part (b): Area. Using A = πr², we calculate:

第 (b) 部分：面积。使用 A = πr²，计算：

A = 3.14 × 7² = 3.14 × 49 = 153.86 cm²

To 1 decimal place, 153.86 rounds to 153.9 cm². Hence the final answers are 44.0 cm and 153.9 cm².

保留一位小数，153.86 四舍五入为 153.9 cm²。因此最终答案为 44.0 cm 和 153.9 cm²。

You should always double‑check your work: quickly redo the multiplication in your head or using a calculator, and confirm that you placed the decimal point correctly.

你应始终检查自己的答案：快速用心算或计算器重新计算乘法，并确认小数点位置正确。

10. Extending the Skill: Circles in Real Life | 拓展技能：生活中的圆

Understanding circumference and area helps you solve problems outside the classroom. For example, if a circular running track has a radius of 35 m, the distance around it is 2 × π × 35 ≈ 220 m (using π ≈ 3.14). If you need to cover the track with a special rubber layer, you would find the area to know how much material to buy.

理解周长和面积能帮助你解决课堂之外的问题。例如，一个圆形跑道半径为 35 m，跑一圈的距离约为 2 × π × 35 ≈ 220 m（使用 π ≈ 3.14）。如果你需要用特殊的橡胶层覆盖跑道，就需要用面积来计算购买多少材料。

Similarly, when designing a round garden or a pizza, the area formula tells you how much space you have for planting or how much surface you get for toppings. This shows that the mathematics you learn now has a direct connection to everyday life.

类似地，在设计圆形花园或披萨时，面积公式可以告诉你种植的空间有多大，或者能铺多少馅料。这说明你现在所学的数学与日常生活直接相关。

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11. Key Terms and Quick Reference | 关键术语与速查表

The table below summarises the essential vocabulary and formulas you must know for this topic.

下表总结了你必须掌握的核心词汇与公式。

Term / 术语	Meaning / 含义	Formula / 公式
Radius (r)	Distance from centre to edge / 从圆心到圆周的距离	r = d ÷ 2
Diameter (d)	Distance across circle through centre / 通过圆心的弦的长度	d = 2r
Circumference (C)	Distance around the circle / 圆的周长	C = πd or C = 2πr
Area (A)	Space inside the circle / 圆内部的面积	A = πr²

12. Summary and Study Tips | 总结与学习建议

Mastering circle calculations in KS3 is all about memorising two simple formulas and knowing when to use them. Always identify whether the problem provides the radius or diameter, and check the required form of your answer – exact (in terms of π) or approximate (using 3.14). Practise with varied numbers, including decimals, to become fluent.

在 KS3 阶段掌握圆的计算，关键在于记住两个简单公式并知道何时使用。始终要识别题目给出的是半径还是直径，并检查答案要求的格式——精确形式（保留 π）还是近似值（使用 3.14）。通过练习各种数字，包括小数，来达到熟练。

Working through past paper problems, marking your own answers, and explaining the steps to a friend are all powerful ways to reinforce your understanding. With careful practice, the question on page 64 will become second nature.

刷历年真题、自己批改答案以及向朋友讲解步骤，都是巩固理解的有力方法。通过细心练习，第 64 页的题目将变得得心应手。

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