Areas of Sectors and Segments | 扇形与弓形的面积

📚 Areas of Sectors and Segments | 扇形与弓形的面积

In Edexcel A Level Mathematics, the topic “Areas of sectors and segments” builds directly on radian measure, arc length and basic trigonometry. Students often meet the sector area formula A = ½ r² θ, but the real challenge is using it correctly to find segment areas, especially when questions mix radians, degrees and exact values.

在 Edexcel A Level 数学中，“扇形与弓形的面积”这一主题直接建立在弧度制、弧长和基础三角学之上。同学们通常会遇到扇形面积公式 A = ½ r² θ，但真正的难点在于正确使用它来求弓形面积，尤其是当题目混合弧度、角度和精确值时。

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1. Radian Measure and Sector Definition | 弧度制与扇形定义

A sector is the region bounded by two radii and an arc of a circle. The angle at the centre, θ, can be measured in degrees or radians. In A Level work, radian measure is standard because it simplifies the arc length and area formulas.

扇形是由两条半径和一段圆弧围成的区域。圆心角 θ 可以用角度或弧度来度量。在 A Level 中，弧度制是标准，因为它能简化弧长和面积公式。

One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. Therefore 2π radians = 360° and π radians = 180°. You should be fluent in converting between the two systems.

一弧度是指圆心角所对的弧长等于半径时的角。因此 2π 弧度 = 360°，π 弧度 = 180°。你必须熟练掌握两种制度之间的转换。

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

For a circle of radius r, an angle θ at the centre in radians cuts off an arc of length l given by:

对于半径为 r 的圆，圆心角 θ（以弧度计）所对的弧长 l 为：

l = rθ

This formula only works when θ is in radians. If you are given degrees, first convert to radians using θ_rad = θ_deg × π/180. Then substitute.

该公式只在 θ 为弧度时成立。如果题目给出的角度是度数，需要先用 θ_rad = θ_deg × π/180 转换为弧度，再代入。

Exam questions often ask for arc length as a separate part or combine it with perimeter of a sector. The perimeter of a sector is the arc length plus twice the radius, that is P = rθ + 2r = r(θ + 2).

考试题经常单独考查弧长，或将其与扇形周长结合。扇形周长为弧长加两条半径，即 P = rθ + 2r = r(θ + 2)。

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

The area of a sector with radius r and central angle θ in radians is:

半径为 r、圆心角为 θ（弧度）的扇形面积为：

A = ½ r² θ

This formula comes from the proportion of the full circle: sector area = (θ / 2π) × πr² = ½ r² θ. Notice that the units of θ must be radians because the fraction θ/(2π) is dimensionless.

该公式来自整个圆的比例关系：扇形面积 = (θ / 2π) × πr² = ½ r² θ。注意 θ 的单位必须是弧度，因为比值 θ/(2π) 是无量纲的。

If θ is given in degrees, the equivalent formula is A = (θ/360) × πr². However, in A Level exam papers, radian-based questions are much more common, and using ½ r² θ directly is expected.

如果 θ 以度数给出，等效公式为 A = (θ/360) × πr²。但在 A Level 考试中，基于弧度的问题更为常见，使用 ½ r² θ 直接计算是符合预期的。

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4. Converting Between Degrees and Radians | 角度与弧度转换

To convert from degrees to radians, multiply by π/180. To convert from radians to degrees, multiply by 180/π. Common conversions you should memorise include 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π.

从度数转换为弧度，乘以 π/180；从弧度转换为度数，乘以 180/π。你应该熟记常见转换：30° = π/6，45° = π/4，60° = π/3，90° = π/2，180° = π，360° = 2π。

When a question asks for an exact answer, leave your working in terms of π. For example, a sector with angle 60° and radius 5 cm has area A = ½ × 5² × π/3 = 25π/6 cm², not a decimal.

当题目要求精确答案时，保留 π 的形式。例如，圆心角 60°、半径 5 cm 的扇形面积为 A = ½ × 5² × π/3 = 25π/6 cm²，而不是写成小数。

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5. Triangle Area Inside the Sector | 扇形内的三角形面积

A segment is the region between a chord and its corresponding arc. To find the area of a minor segment, you subtract the area of the isosceles triangle formed by the two radii and the chord from the area of the sector.

弓形是弦与对应弧之间的区域。求劣弓形面积时，你要从扇形面积中减去由两条半径和弦组成的等腰三角形的面积。

The triangle has two sides equal to r and included angle θ. Its area is:

这个等腰三角形的两条边等于 r，夹角为 θ。其面积为：

A_triangle = ½ r² sin θ

Again, θ is in radians for consistency, and your calculator must be in radian mode when evaluating sin θ. This formula works for any θ, but for a minor segment we usually have 0 < θ < π.

同样，θ 采用弧度制以保持一致，计算 sin θ 时计算器必须处于弧度模式。该公式对任意 θ 都成立，但劣弓形通常满足 0 < θ < π。

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6. Area of a Minor Segment | 劣弓形面积

The area of a minor segment is the sector area minus the triangle area:

劣弓形面积为扇形面积减去三角形面积：

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

Here θ must be in radians and must be the angle of the minor sector, usually less than π. For example, if r = 6 cm and θ = 1.2 rad, then sector area = ½ × 36 × 1.2 = 21.6 cm², triangle area = 18 sin 1.2 ≈ 16.8 cm², and segment area ≈

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