Arc Length Calculation | 弧长的计算

📚 Arc Length Calculation | 弧长的计算

In A-Level Mathematics, the arc length of a circle is a fundamental concept in trigonometry and geometry. It measures the distance along the circumference between two points on a circle, often within a sector. Mastering this calculation is essential for solving problems involving circles, sectors, and radians.

在 A-Level 数学中，圆的弧长是三角学与几何学中的基础概念。它衡量圆周上两点之间沿圆周的距离，通常出现在扇形中。掌握这一计算对于解决涉及圆、扇形和弧度的问题至关重要。

1. Understanding Arc Length | 弧长简介

An arc is simply a portion of the circumference of a circle. The length of this portion depends on the radius of the circle and the angle subtended at the centre. This angle can be measured in degrees or radians, but in A-Level mathematics, radians are the preferred unit because they simplify many formulas.

弧就是圆周长的一部分。这一部分的长度取决于圆的半径以及圆心角的大小。该角可以用角度或弧度来度量，但在 A-Level 数学中，弧度是更常用的单位，因为它能简化许多公式。

If you think of a circle as a complete turn of 360° or 2π radians, then an arc is a fraction of that full turn. For a given angle θ, the arc length is directly proportional to θ when the radius is fixed.

如果把一个圆看作完整的 360° 或 2π 弧度旋转，那么弧就是整个旋转的一部分。在半径固定的情况下，对于给定的角度 θ，弧长与 θ 成正比。

2. Radians vs Degrees | 弧度与角度

Radian measure is defined as the ratio of arc length to radius. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius. Therefore, a full circle of 360° is equal to 2π radians.

弧度的定义是弧长与半径之比。当弧长等于半径时，所对应的圆心角大小为一弧度。因此，完整的圆 360° 等于 2π 弧度。

Key conversions to remember:

需要记住的关键换算：

180° = π radians

180° = π 弧度
90° = π/2 radians

90° = π/2 弧度
360° = 2π radians

360° = 2π 弧度
1 radian ≈ 57.2958°

1 弧度 ≈ 57.2958°

When solving arc length problems, always check whether the angle is given in radians or degrees. Using the wrong unit is one of the most common errors in exams.

在解决弧长问题时，务必检查角度是以弧度还是以角度为单位。使用错误的单位是考试中最常见的错误之一。

3. Formula for Arc Length in Radians | 弧度制下的弧长公式

For a circle of radius r and an angle θ measured in radians, the arc length s is given by:

对于半径为 r、圆心角 θ 以弧度表示的圆，弧长 s 的公式为：

s = rθ

This elegant formula works only when θ is in radians. If θ is in degrees, you must first convert it to radians or use a different version of the formula.

这个简洁的公式仅在 θ 以弧度为单位时成立。如果 θ 以角度为单位，则必须先将其转换为弧度，或使用另一形式的公式。

Notice that when θ = 2π, the arc length becomes s = 2πr, which is exactly the circumference of the circle. This confirms that the formula is consistent with the known perimeter of a full circle.

注意当 θ = 2π 时，弧长变为 s = 2πr，这正是圆的周长。这证实了该公式与圆的周长公式是一致的。

4. Derivation of the Formula | 公式推导

The definition of a radian gives the derivation directly. By definition, an angle of 1 radian subtends an arc of length r. Therefore, if the angle is θ radians, the arc length must be θ times larger:

弧度的定义可直接用于推导。根据定义，1 弧度的角对应长度为 r 的弧。因此，若角度为 θ 弧度，弧长必然是 θ 倍：

s = r × θ

Alternatively, consider the proportion of the arc to the full circumference. The fraction of the complete circle is θ / (2π), so:

或者，考虑弧长占整个圆周的比例。占完整圆的比例为 θ / (2π)，因此：

s = (θ / 2π) × 2πr = rθ

This proportional reasoning is useful because it also leads to the degree-based formula when θ is in degrees.

这种比例推理非常有用，因为当 θ 以角度为单位时，它也能引导出基于角度的公式。

5. Example 1: Basic Calculation | 例题1：基础计算

Find the arc length of a sector with radius 5 cm and central angle 1.2 radians.

已知扇形的半径为 5 cm，圆心角为 1.2 弧度，求弧长。

Using s = rθ directly:

直接使用 s = rθ：

s = 5 × 1.2 = 6 cm

Therefore, the arc length is 6 cm. This is a straightforward application of the formula.

因此，弧长为 6 cm。这是公式的直接应用。

6. Example 2: Finding the Angle | 例题2：求圆心角

An arc of length 14 cm is drawn in a circle of radius 8 cm. Find the central angle in radians.

在一个半径为 8 cm 的圆中，一段弧长为 14 cm。求圆心角（以弧度表示）。

Rearrange s = rθ to solve for θ:

由 s = rθ 变形，解出 θ：

θ = s / r = 14 / 8 = 1.75 radians

Hence the central angle is 1.75 radians. This type of problem tests your ability to rearrange formulas accurately.

因此圆心角为 1.75 弧度。这类问题考查你准确变形公式的能力。

7. Example 3: Finding the Radius | 例题3：求半径

A sector has an arc length of 22 cm and a central angle of 2 radians. Calculate the radius.

一个扇形的弧长为 22 cm，圆心角为 2 弧度。求半径。

Using s = rθ, we have r = s / θ:

使用 s = rθ，得 r = s / θ：

r = 22 / 2 = 11 cm

So the radius of the circle is 11 cm. Always ensure that the angle is in radians before applying this rearrangement.

因此圆的半径为 11 cm。在应用这个变形之前，一定要确保角度以弧度为单位。

8. Arc Length in Degrees | 角度制下的弧长

Sometimes the angle is given in degrees, especially in problems that do not specify radians. In that case, the formula becomes:

有时角度以度数给出，尤其是在没有指定弧度的题目中。此时公式变为：

s = (θ / 360°) × 2πr

This formula represents the fraction of the full circle that the angle covers, multiplied by the full circumference.

该公式表示角度所覆盖的完整圆的比例，再乘以整个圆的周长。

For example, find the arc length of a sector with radius 9 cm and angle 60°.

例如，求半径为 9 cm、圆心角为 60° 的扇形的弧长。

s = (60° / 360°) × 2π × 9 = (1/6) × 18π = 3π cm ≈ 9.42 cm

Alternatively, convert 60° to π/3 radians and use s = rθ:

或者将 60° 转换为 π/3 弧度，然后使用 s = rθ：

s = 9 × (π/3) = 3π cm

Both methods produce the same result. The degree version is often safer when the angle is a familiar degree value such as 30°, 45°, or 60°.

两种方法得到相同的答案。当角度是常见的度数如 30°、45° 或 60° 时，使用角度制公式往往更安全。

9. Perimeter of a Sector | 扇形的周长

The perimeter of a sector includes the arc length plus the two straight radii. If the sector has radius r, arc length s, then the perimeter P is:

扇形的周长包括弧长加上两条半径。如果扇形半径为 r，弧长为 s，则周长 P 为：

P = 2r + s = 2r + rθ

This is a common exam question that combines arc length with perimeter. Many students forget to add the two radii, so read the question carefully.

这是常见的考试题型，将弧长与周长结合。许多学生会忘记加上两条半径，因此要仔细审题。

For example, a sector has radius 6 cm and angle 0.8 radians. Find its perimeter.

例如，扇形半径为 6 cm，圆心角为 0.8 弧度，求其周长。

s = 6 × 0.8 = 4.8 cm, then P = 2 × 6 + 4.8 = 16.8 cm

Thus the perimeter is 16.8 cm.

因此周长为 16.8 cm。

10. Area of a Sector and Relation to Arc Length | 扇形面积及其与弧长的关系

The area of a sector in radians is A = ½ r²θ. Because s = rθ, we also have θ = s / r, which allows the area to be written as:

弧度制下扇形面积为 A = ½ r²θ。由于 s = rθ，可得 θ = s / r，于是面积可以写成：

A = ½ r s

This elegant relation mirrors the formula for the area of a triangle: half the base times the height. It can be useful when the arc length is given but the angle is not.

这个优美的关系类似于三角形面积公式：底乘以高的一半。当已知弧长而未知角度时，这个公式非常有用。

For instance, a sector has radius 10 cm and arc length 6 cm. Its area is:

例如，扇形半径为 10 cm，弧长为 6 cm，其面积为：

A = ½ × 10 × 6 = 30 cm²

This connection between arc length and sector area shows how interrelated the circle formulas are.

弧长与扇形面积之间的联系显示了圆的各种公式之间的紧密关系。

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

Here are some common pitfalls and helpful tips when working with arc length:

以下是在处理弧长问题时常见的陷阱和实用提示：

Always check the unit of the angle. If using s = rθ, the angle must be in radians. If it is in degrees, convert or use the degree formula.

始终检查角度的单位。如果使用 s = rθ，角度必须为弧度。如果是角度制，请转换或使用角度制公式。
Do not confuse arc length with sector area. Arc length is a length, so its unit is cm, m, etc. Area is measured in square units.

不要将弧长与扇形面积混淆。弧长是长度，单位是 cm、m 等。面积使用平方单位。
When finding the perimeter of a sector, remember to include the two radii.

在求扇形周长时，记得包含两条半径。
If a question gives the angle in radians as a multiple of π, keep π in your answer unless a decimal is requested.

如果题目给出的弧度角是 π 的倍数，除非要求小数，否则答案中应保留 π。
Read the question carefully: sometimes you are given the diameter instead of the radius. Convert diameter to radius before using any formula.

仔细审题：有时题目给出的是直径而不是半径。在使用任何公式前，将直径转换为半径。

12. Exam-Style Questions | 考试风格题目

Let’s attempt a typical exam question that combines several concepts.

让我们尝试一道综合多个概念的典型考试题。

A sector of a circle has area 24 cm² and radius 6 cm. Find the arc length and the central angle in radians.

一个扇形的面积为 24 cm²，半径为 6 cm。求其弧长和圆心角（以弧度表示）。

First, find the angle using the area formula:

首先，利用面积公式求角度：

A = ½ r²θ ⇒ 24 = ½ × 6² × θ = 18θ

So θ = 24 / 18 = 4/3 radians.

所以 θ = 24 / 18 = 4/3 弧度。

Then the arc length is:

然后求弧长：

s = rθ = 6 × (4/3) = 8 cm

Alternatively, use A = ½ r s directly: 24 = ½ × 6 × s ⇒ s = 8 cm. Both routes work perfectly.

或者直接使用 A = ½ r s：24 = ½ × 6 × s ⇒ s = 8 cm。两种方法都完全可行。

13. Summary | 总结

The key formula for arc length in radians is s = rθ, where θ is measured in radians. If the angle is in degrees, use s = (θ / 360°) × 2πr. Always ensure units are consistent, and remember that the perimeter of a sector is 2r + s.

弧度制下弧长的关键公式是 s = rθ，其中 θ 以弧度为单位。如果角度以度数表示，使用 s = (θ / 360°) × 2πr。始终保持单位一致，并记住扇形的周长是 2r + s。

Understanding where the formula comes from helps you remember it and apply it flexibly. Practise with both radians and degrees, and be careful with units in exam conditions.

理解公式的来源有助于记忆并灵活运用。练习时同时使用弧度和角度，并在考试环境下注意单位。

With these tools, you can confidently solve any arc length problem in A-Level Mathematics.

有了这些方法，你可以自信地解决 A-Level 数学中任何涉及弧长的问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Arc Length Calculation | 弧长的计算