Arc Length and Sector Area | 弧长与扇形面积

📚 Arc Length and Sector Area | 弧长与扇形面积

Welcome to Lesson 16-2 of our IGCSE Mathematics revision series! In this lesson, we will explore the fascinating geometry of circles — specifically, how to calculate the length of an arc and the area of a sector. These concepts appear frequently in IGCSE examinations and form an essential foundation for more advanced topics in trigonometry and calculus.

欢迎来到本IGCSE数学复习系列的第16-2课！在本课中，我们将探索圆的有趣几何性质——具体来说，是如何计算弧长和扇形面积。这些概念在IGCSE考试中频繁出现，是学习更高级的三角学和微积分主题的重要基础。

1. What Are Arcs and Sectors? | 什么是弧和扇形？

An arc is part of the circumference of a circle. When we cut a circle along two radii, the curved edge between the two radii is the arc. A sector is the region bounded by two radii and the arc between them — it looks like a slice of pizza or a slice of cake.

弧是圆周的一部分。当我们沿两条半径切割一个圆时，两条半径之间的曲边就是弧。扇形是由两条半径及其之间的弧所围成的区域——它看起来就像一片披萨或一块蛋糕。

Every sector is defined by its central angle, θ, which is the angle formed at the centre of the circle between the two radii. The central angle is measured in degrees and can range from 0° to 360°.

每个扇形都由其圆心角θ来定义，圆心角是在圆心处两条半径之间形成的角。圆心角以度为单位，范围从0°到360°。

2. Key Terms and Notation | 关键术语与符号

Before we dive into the formulas, let us establish the notation we will use throughout this lesson:

在我们深入研究公式之前，让我们先确定本课中使用的符号：

r — the radius of the circle | 圆的半径
d — the diameter of the circle (d = 2r) | 圆的直径（d = 2r）
π — the mathematical constant pi, approximately equal to 3.14159 | 数学常数圆周率，约等于3.14159
θ — the central angle of the sector in degrees | 扇形的圆心角（度）
l — the length of the arc | 弧长
A — the area of the sector | 扇形面积

Remember that the circumference of a full circle is C = 2πr, and the area of a full circle is A = πr². These two formulas are the building blocks for everything we will learn today.

请记住，整个圆的周长是C = 2πr，整个圆的面积是A = πr²。这两个公式是我们今天所学一切的基础。

3. Deriving the Arc Length Formula | 推导弧长公式

A full circle corresponds to a central angle of 360°. If we take only a fraction of the circle, the arc length is the same fraction of the circumference. The fraction is simply θ divided by 360°.

整个圆对应360°的圆心角。如果我们只取圆的一部分，弧长就是圆周长的相应比例。这个比例就是θ除以360°。

Therefore, the arc length formula is:

因此，弧长公式为：

l = (θ/360) × 2πr

For example, if θ = 90° and r = 10 cm, then the arc length is (90/360) × 2π × 10 = ¼ × 20π = 5π ≈ 15.7 cm. Notice that a 90° sector gives us exactly one quarter of the full circumference.

例如，如果θ = 90°且r = 10 cm，则弧长为(90/360) × 2π × 10 = ¼ × 20π = 5π ≈ 15.7 cm。注意90°的扇形正好给出整个圆周的四分之一。

4. Deriving the Sector Area Formula | 推导扇形面积公式

By the same logic, the area of a sector is the same fraction of the area of the full circle. Since the full circle has area πr², we multiply this by the fraction θ/360° to obtain the sector area.

同样的逻辑，扇形的面积是整个圆面积的相应比例。由于整个圆的面积是πr²，我们将其乘以θ/360°的比例，得到扇形面积。

Therefore, the sector area formula is:

因此，扇形面积公式为：

A = (θ/360) × πr²

For the example above, where θ = 90° and r = 10 cm, the sector area is (90/360) × π × 10² = ¼ × 100π = 25π ≈ 78.5 cm².

对于上面的例子，θ = 90°且r = 10 cm时，扇形面积为(90/360) × π × 10² = ¼ × 100π = 25π ≈ 78.5 cm²。

5. Worked Example 1: Finding Arc Length | 例题1：求弧长

Problem: A circle has a radius of 8 cm. A sector of this circle has a central angle of 135°. Find the arc length, giving your answer correct to 3 significant figures.

题目：一个圆的半径为8 cm。该圆的一个扇形圆心角为135°。求弧长，答案保留3位有效数字。

Solution: We substitute r = 8 and θ = 135 into the arc length formula:

解答：我们将r = 8和θ = 135代入弧长公式：

l = (135/360) × 2π × 8 = (3/8) × 16π = 6π ≈ 18.8 cm

Step by step

Published by TutorHao | IGCSE Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Arc Length and Sector Area | 弧长与扇形面积