Area of a Circle | 圆的面积

📚 Area of a Circle | 圆的面积

Understanding how to calculate the area of a circle is a fundamental skill in KS3 mathematics. This topic builds on your knowledge of shapes, π (pi), and formulas. Whether you are working out the space inside a coin, a pizza, or a circular garden, the area formula A = πr² is your key tool. In this article, we will explore every part of that formula, learn how to use it step by step, and tackle common problems you may meet in the Cambridge syllabus.

理解如何计算圆的面积是KS3数学中的基本技能。这个主题建立在你对形状、π（圆周率）和公式的认识之上。无论你是在计算硬币、披萨还是圆形花园的内部空间大小，面积公式 A = πr² 都是你的关键工具。在本文中，我们将深入探讨这个公式的每一部分，逐步学习如何使用它，并解决你在剑桥课程中可能遇到的常见问题。

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1. What is a Circle? | 什么是圆？

A circle is a perfectly round 2D shape where every point on its boundary is the same distance from the centre. That distance is called the radius. Unlike polygons, circles have no straight sides or corners, so we need a special method to find their area.

圆是一种完美的二维圆形形状，其边界上的每一个点到中心的距离都相等。这个距离被称为半径。与多边形不同，圆没有直边或角，因此我们需要一种特殊的方法来计算它的面积。

2. Key Parts of a Circle | 圆的关键部分

Before using any formula, you must be able to identify three important parts: the radius (r) – the distance from the centre to the edge; the diameter (d) – the distance across the circle through the centre, which is twice the radius; and the circumference (C) – the distance all the way around the circle. The diameter is always 2 × r, so if you are given the diameter, you can halve it to find the radius.

在使用任何公式之前，你必须能够识别三个重要部分：半径 (r)——从圆心到边缘的距离；直径 (d)——穿过圆心横跨圆的距离，它是半径的两倍；以及圆的周长 (C)——围绕圆一圈的距离。直径总是 2 × r，所以如果给出直径，你可以将它除以二得到半径。

3. Introducing Pi (π) | 圆周率 π 简介

Pi (π) is a special mathematical constant that comes from the relationship between a circle’s circumference and its diameter. No matter the size of the circle, C ÷ d always equals about 3.14159… We write this number as π. For KS3 work, you will often use π as 3.14 or press the π button on your calculator for more accurate answers.

圆周率 π 是一个特殊的数学常数，它来源于圆的周长与直径之间的关系。无论圆有多大，C ÷ d 始终约等于 3.14159…… 我们把这个数写作 π。在 KS3 的学习中，你通常会使用 π = 3.14，或者在计算器上按下 π 键以获得更精确的答案。

4. The Circumference Formula | 周长公式

Although our focus is area, knowing the circumference formula helps you understand π better. The circumference of a circle is given by C = πd or C = 2πr. This formula reminds you that the circumference is just over three times the diameter. Practising with this formula makes working with π feel more natural.

虽然我们的重点是面积，但了解周长公式有助于你更好地理解 π。圆的周长公式为 C = πd 或 C = 2πr。这个公式提醒你，周长刚好是直径的三倍多一点。练习使用这个公式会让你对 π 的运用更加得心应手。

5. The Area Formula: A = πr² | 面积公式：A = πr²

The area of a circle is calculated by multiplying π by the square of the radius: A = π × r². Note that r² means r × r, not r × 2. It is crucial to square the radius before multiplying by π. For example, if r = 5 cm, then r² = 25 cm², and A = π × 25 ≈ 78.5 cm² (using π ≈ 3.14).

圆的面积通过 π 乘以半径的平方来计算：A = π × r²。注意，r² 表示 r × r，而不是 r × 2。关键在于先求半径的平方，再乘以 π。例如，如果 r = 5 cm，那么 r² = 25 cm²，A = π × 25 ≈ 78.5 cm²（使用 π ≈ 3.14）。

6. Worked Example 1: Radius Given | 例题1：已知半径

A circle has a radius of 7 cm. Find its area. Use π = 3.14.
Step 1: Write the formula A = πr².
Step 2: Substitute r = 7 → A = π × 7².
Step 3: Square the radius: 7² = 49.
Step 4: Multiply: 3.14 × 49 = 153.86.
Answer: The area is approximately 153.9 cm² (to 1 decimal place).

一个圆的半径为 7 cm，求它的面积。使用 π = 3.14。
步骤1：写出公式 A = πr²。
步骤2：代入 r = 7 → A = π × 7²。
步骤3：半径平方：7² = 49。
步骤4：相乘：3.14 × 49 = 153.86。
答案：面积约为 153.9 cm²（保留一位小数）。

7. Worked Example 2: Diameter Given | 例题2：已知直径

A circular pond has a diameter of 10 m. Calculate its area. Use π = 3.14.
First, find the radius: r = d ÷ 2 = 10 ÷ 2 = 5 m.
Then use A = πr²: A = 3.14 × 5² = 3.14 × 25 = 78.5 m².
The pond covers 78.5 square metres. Always check whether you are given the radius or diameter before substituting into the formula.

一个圆形池塘的直径为 10 m，请计算它的面积。使用 π = 3.14。
首先，求半径：r = d ÷ 2 = 10 ÷ 2 = 5 m。
然后使用 A = πr²：A = 3.14 × 5² = 3.14 × 25 = 78.5 m²。
池塘占地 78.5 平方米。在代入公式之前，务必确认给出的条件是半径还是直径。

8. Finding the Radius from the Area | 通过面积求半径

Sometimes you know the area and need to work backwards. Rearrange the formula: r² = A ÷ π, then take the square root. For example, if A = 154 cm² and π ≈ 3.14, then r² = 154 ÷ 3.14 ≈ 49. r = √49 = 7 cm. This is a valuable problem‑solving skill in KS3.

有时你知道面积，需要反向求解。重新整理公式：r² = A ÷ π，然后开平方根。例如，如果 A = 154 cm²，π ≈ 3.14，那么 r² = 154 ÷ 3.14 ≈ 49。r = √49 = 7 cm。这是 KS3 中一项很有价值的问题解决技能。

9. Area of a Semicircle | 半圆的面积

A semicircle is exactly half of a full circle. To find its area, first calculate the area of the whole circle, then divide by 2: Aₛₑₘᵢ = (πr²) ÷ 2. Don’t forget to include the straight edge if the question asks for the perimeter, but area only involves the curved region.

半圆恰好是整个圆的一半。要计算它的面积，先求出整个圆的面积，然后除以 2：Aₛₑₘᵢ = (πr²) ÷ 2。如果题目要求周长，别忘了加上直边，但面积只涉及曲线区域。

10. Area of a Quarter Circle | 四分之一圆的面积

Similarly, a quarter circle is one‑fourth of a full circle. The formula becomes A = (πr²) ÷ 4. This often appears in composite shapes, where you combine quarter circles with squares or rectangles. Break the shape down into parts you know, and add or subtract areas.

同理，四分之一圆是整个圆的四分之一。公式变为 A = (πr²) ÷ 4。这种形状常出现在组合图形中，你需要将四分之一圆与正方形或长方形结合起来。将图形分解成你熟悉的部分，然后对面积进行加减运算。

11. Real‑life Applications | 实际生活中的应用

Area of a circle is used everywhere: calculating the amount of paint for a round ceiling, fabric for a circular tablecloth, or the surface of a watch face. Architects and engineers rely on this formula to design roundabouts, tunnels, and pipes. Understanding A = πr² connects classroom maths to the world around you.

圆的面积应用广泛：计算圆形天花板所需的油漆量、圆形桌布所需的布料，或是手表表面的面积。建筑师和工程师依靠这个公式来设计环形交叉口、隧道和管道。理解 A = πr² 可以将课堂数学与你周围的世界联系起来。

12. Common Mistakes and Tips | 常见错误与提示

Never confuse radius with diameter. Do not multiply r by 2 before squaring – r² and 2r are different. Also, remember to include the correct units: area is always in square units (cm², m², etc.). When using a calculator, use the π button rather than 3.14 for greater accuracy, unless the question tells you otherwise. Practise with a mix of simple and worded problems to build confidence.

切勿混淆半径与直径。不要在平方之前将 r 乘以 2 —— r² 与 2r 是不同的。还要记住带上正确的单位：面积总是以平方单位（cm²、m² 等）表示。使用计算器时，尽量使用 π 键而不是 3.14 以提高精度，除非题目另有要求。通过练习简单的和文字题来建立信心。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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