Length and Area: Arc Length and Sector Area in Radians | 长度与面积：弧度制下的弧长与扇形面积

📚 Length and Area: Arc Length and Sector Area in Radians | 长度与面积：弧度制下的弧长与扇形面积

In A-Level Edexcel Mathematics, the topic of length and area often appears in the context of circles, sectors, and segments. The key idea is to measure angles in radians rather than degrees, because this produces much simpler formulas for arc length and sector area.

在 A-Level Edexcel 数学中，长度与面积这个主题经常出现在圆、扇形和弓形的问题里。核心思想是使用弧度制而非角度制来度量角，因为弧度制下的弧长公式和扇形面积公式要简洁得多。

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1. Why Radians? | 为什么使用弧度制

One radian is defined as the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. This definition directly connects angle, radius, and arc length.

1 弧度的定义是：由长度等于圆半径的圆弧在圆心处所对的角。这个定义直接将角度、半径和弧长联系在一起。

If the arc length is s and the radius is r, then the angle in radians is θ = s / r.

如果弧长为 s，半径为 r，则弧度制下的角为 θ = s / r。

θ (rad) = s / r

In A-Level problems, this definition forms the foundation for all length and area calculations involving sectors.

在 A-Level 题目中，这一定义是所有涉及扇形的长度和面积计算的基础。

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2. Converting Between Degrees and Radians | 角度与弧度的转换

Before using the formulas for arc length and sector area, you must be able to switch between degrees and radians. The conversion uses the fact that 180° is equal to π radians.

在使用弧长和扇形面积公式之前，你必须能够在角度制和弧度制之间进行转换。转换依据是 180° 等于 π 弧度。

To convert degrees to radians, multiply by π / 180.

将角度转换为弧度，乘以 π / 180。

θ (rad) = θ (deg) × π / 180

To convert radians to degrees, multiply by 180 / π.

将弧度转换为角度，乘以 180 / π。

θ (deg) = θ (rad) × 180 / π

Degrees | 角度	Radians | 弧度
30°	π / 6
45°	π / 4
60°	π / 3
90°	π / 2
180°	π
360°	2π

These exact values are very common in Edexcel exam questions, so you should memorise them.

这些精确值在 Edexcel 考试题中非常常见，你应该记住它们。

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3. Arc Length Formula l = rθ | 弧长公式 l = rθ

For a circle of radius r, the length of an arc that subtends an angle θ at the centre is simply the radius multiplied by the angle measured in radians.

对于半径为 r 的圆，在圆心处所对角为 θ 的圆弧长度，等于半径乘以以弧度为单位的角度。

l = rθ

For example, if r = 5 cm and θ = 1.2 rad, then l = 5 × 1.2 = 6 cm.

例如，如果 r = 5 cm，θ = 1.2 rad，则 l = 5 × 1.2 = 6 cm。

This formula only works when θ is in radians. In degrees, the formula would be l = θ / 360 × 2πr, which is less convenient.

这个公式只在 θ 以弧度为单位时成立。如果用角度制，公式就会变成 l = θ / 360 × 2πr，使用起来不太方便。

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4. Sector Area Formula A = ½ r²θ | 扇形面积公式 A = ½ r²θ

The area of a sector with angle θ radians and radius r is given by half the square of the radius multiplied by the angle.

半径为 r、角为 θ 弧度的扇形面积，等于半径的平方乘以角度再除以 2。

A = ½ r²θ

This comes from the fact that a sector is a fraction θ / 2π of the full circle, so its area is (θ / 2π) × πr² = ½ r²θ.

这是因为扇形占整个圆的比例为 θ / 2π，因此其面积为 (θ / 2π) × πr² = ½ r²θ。

For example, if r = 4 cm and θ = π / 2, then A = ½ × 16 × π / 2 = 4π cm².

例如，如果 r = 4 cm，θ = π / 2，则 A = ½ × 16 × π / 2 = 4π cm²。

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5. Alternative Sector Area Formula A = ½ rl | 扇形面积的另一种形式 A = ½ rl

Since arc length l = rθ, we can substitute this into the sector area formula to obtain an alternative version.

由于弧长 l = rθ，我们可以将其代入扇形面积公式，得到一个替代形式。

A = ½ rl

This form is useful when you are given the arc length l and radius r but not the angle θ. For example, if a sector has r = 6 cm and l = 9 cm, then A = ½ × 6 × 9 = 27 cm².

当你已知弧长 l 和半径 r，但不知道角度 θ 时，这个形式就非常有用。例如，一个扇形的 r = 6 cm，l = 9 cm，则 A = ½ × 6 × 9 = 27 cm²。

Both forms are equivalent because l = rθ, so ½ rl = ½ r(rθ) = ½ r²θ.

这两种形式是等价的，因为 l = rθ，所以 ½ rl = ½ r(rθ) = ½ r²θ。

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6. Segment Area: Sector Minus Triangle | 弓形面积：扇形面积减去三角形面积

A segment is the region bounded by a chord and its corresponding arc. To find the area of a segment, start with the sector area and subtract the area of the triangle formed by the two radii and the chord.

弓形是由一条弦及其对应的弧所围成的区域。要求弓形面积，先求扇形面积，再减去由两条半径和弦构成的三角形面积。

The triangle has sides r and r with included angle θ, so its area is ½ r² sin θ.

这个三角形有两条边 r 和 r，夹角为 θ，因此它的面积是 ½ r² sin θ。

A_segment = ½ r²θ − ½ r² sin θ = ½ r²(θ − sin θ)

For example, if r = 10 cm and θ = π / 3, then A_segment = ½ × 100 × (π / 3 − sin π / 3) = 50(π / 3 − √3 / 2) cm².

例如，如果 r = 10 cm，θ = π / 3，则 A_segment = ½ × 100 × (π / 3 − sin π / 3) = 50(π / 3 − √3 / 2) cm²。

Remember that the sector part requires θ in radians, while the triangle part uses the sine of the angle, which can be evaluated using radian mode on your calculator.

请记住，扇形部分要求 θ 以弧度为单位，而三角形部分使用的是该角的正弦值，在计算器上应使用弧度模式进行计算。

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7. Using Radians with Trigonometry | 弧度制与三角函数的结合

When solving problems involving segments or more complex geometrical shapes, you often need to evaluate trigonometric functions of angles given in radians.

在求解涉及弓形或更复杂几何图形的问题时，你经常需要计算以弧度给出的角的三角函数值。

Common exact values you should know include sin π / 6 = 1 / 2, cos π / 6 = √3 / 2, sin π / 3 = √3 / 2, and sin π / 2 = 1.

你应该熟悉的常见精确值包括 sin π / 6 = 1 / 2、cos π / 6 = √3 / 2、sin π / 3 = √3 / 2 和 sin π / 2 = 1。

Always check that your calculator is set to radians when the angle is given in radians. Using degree mode will give completely wrong answers in segment area calculations.

当角度以弧度给出时，务必检查计算器是否设置为弧度模式。如果使用角度模式，弓形面积的计算结果会完全错误。

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8. Solving Problems Involving Arc Length and Area | 涉及弧长与面积的综合问题

Exam questions often ask you to find the perimeter and area of a sector, or to work backwards from a given arc length or sector area to find the radius or angle.

考试题经常要求你求扇形的周长和面积，或者反过来，由已知的弧长或扇形面积求半径或角度。

The perimeter of a sector consists of the arc length plus the two radii.

扇形的周长由弧长加上两条半径组成。

P = l + 2r = rθ + 2r = r(θ + 2)

For example, a sector with radius 8 cm and angle 0.9 rad has arc length l = 8 × 0.9 = 7.2 cm, so its perimeter is 7.2 + 16 = 23.2 cm.

例如，一个半径为 8 cm、角为 0.9 rad 的扇形，其弧长为 l = 8 × 0.9 = 7.2 cm，因此周长为 7.2 + 16 = 23.2 cm。

You may also be asked to find the angle θ when given the sector area and radius: rearrange A = ½ r²θ to get θ = 2A / r².

你也可能被要求在已知扇形面积和半径的情况下求角度 θ：将 A = ½ r²θ 变形为 θ = 2A / r²。

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9. Common Mistakes | 常见错误

One of the most common mistakes is using degrees instead of radians in the formulas l = rθ and A = ½ r²θ. These formulas are only valid when θ is in radians.

最常见的错误之一是在公式 l = rθ 和 A = ½ r²θ 中使用角度制而不是弧度制。这些公式仅在 θ 为弧度时成立。

Another frequent error is forgetting to add the two radii when calculating the perimeter of a sector. The perimeter is not just the arc length.

另一个常见错误是计算扇形周长时忘记加上两条半径。周长不仅仅只是弧长。

Also, when finding the area of a segment, candidates sometimes use the wrong sign by adding the triangle area instead of subtracting it.

此外，在求弓形面积时，考生有时会把三角形面积加上而不是减去，导致符号错误。

Finally, always check whether your calculator is in radian or degree mode before performing any trigonometric calculation.

最后，在进行任何三角函数计算之前，务必检查计算器处于弧度模式还是角度模式。

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10. Exam-Style Worked Example | 考试风格例题

Worked example: A sector of a circle has radius 12 cm and angle π / 3 radians. Find the arc length, the sector area, the perimeter, and the area of the corresponding segment.

例题：一个圆的扇形半径为 12 cm，角为 π / 3 弧度。求弧长、扇形面积、周长以及相应弓形的面积。

Arc length: l = rθ = 12 × π / 3 = 4π cm.

弧长：l = rθ = 12 × π / 3 = 4π cm。

Sector area: A_sector = ½ r²θ = ½ × 144 × π / 3 = 24π cm².

扇形面积：A_sector = ½ r²θ = ½ × 144 × π / 3 = 24π cm²。

Perimeter: P = l + 2r = 4π + 24 cm.

周长：P = l + 2r = 4π + 24 cm。

Triangle area inside the sector: A_triangle = ½ r² sin θ = ½ × 144 × sin π / 3 = 72 × √3 / 2 = 36√3 cm².

扇形内部三角形面积：A_triangle = ½ r² sin θ = ½ × 144 × sin π / 3 = 72 × √3 / 2 = 36√3 cm²。

Segment area: A_segment = A_sector − A_triangle = 24π − 36√3 cm².

弓形面积：A_segment = A_sector − A_triangle = 24π − 36√3 cm²。

This type of multi-step problem is typical in Edexcel A-Level exams, so practise writing each step clearly.

这种多步骤的题目在 Edexcel A-Level 考试中非常典型，因此要练习把每一步都写清楚。

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11. Practice Checklist | 复习清单

Before the exam, make sure you can do all of the following confidently:

考试前，确保你能自信地完成以下所有内容：

Convert between degrees and radians using π / 180 and 180 / π.

使用 π / 180 和 180 / π 在角度与弧度之间进行转换。

Apply l = rθ and A = ½ r²θ with θ in radians.

在 θ 为弧度的前提下应用 l = rθ 和 A = ½ r²θ。

Use the alternative formula A = ½ rl for sector area.

使用扇形面积的替代公式 A = ½ rl。

Find the area of a segment as sector minus triangle.

用扇形面积减去三角形面积求弓形面积。

Work backwards from given length or area to find r or θ.

由已知的长度或面积反推 r 或 θ。

Set your calculator to the correct angle mode.

将计算器设置为正确的角度模式。

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12. Conclusion | 结论

The topic of length and area in radians is highly examinable in Edexcel A-Level Mathematics. Once you understand the definition of a radian and memorise the key formulas, most questions become straightforward.

弧度制下的长度与面积这一主题在 Edexcel A-Level 数学中考查频率很高。一旦你理解了弧度的定义并记住了关键公式，大多数题目就会变得简单直接。

Always work step by step, draw a clear diagram, and state which formula you are using. This will help you avoid common mistakes and gain full method marks.

始终一步一步地解题，画出清晰的示意图，并说明你正在使用的公式。这将帮助你避免常见错误，并获得完整的方法分。

Published by TutorHao | Mathematics Revision Series | aleveler.com

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