Circle Circumference, Area, and Sector Formulas | 圆的周长、面积与扇形公式

📚 Circle Circumference, Area, and Sector Formulas | 圆的周长、面积与扇形公式

Circles are one of the most fundamental shapes in geometry, and their properties appear throughout the IB Mathematics curriculum. In this article, we will explore the key formulas for the circumference and area of a circle, as well as the arc length and sector area formulas, with worked examples and common pitfalls to avoid.

圆是几何学中最基本的图形之一，其相关性质贯穿 IB 数学课程。本文将系统讲解圆的周长与面积公式，以及弧长与扇形面积公式，并配有例题和常见易错点分析，帮助同学们牢固掌握这些核心考点。

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1. The Number π and the Origin of the Formulas | 圆周率 π 与公式的由来

For any circle, the ratio of its circumference C to its diameter d is a constant, denoted by the Greek letter π. This means C / d = π, so C = πd. Since the diameter is twice the radius, d = 2r, we also obtain C = 2πr.

对于任意圆，周长 C 与直径 d 的比值是一个常数，记为希腊字母 π。因此 C / d = π，即 C = πd。由于直径等于半径的两倍，即 d = 2r，所以我们得到 C = 2πr。

C = 2πr 或 C = πd

The area A of a circle of radius r is given by A = πr². This formula can be understood by imagining the circle cut into many thin sectors and rearranged into an approximate rectangle of length πr and width r.

半径为 r 的圆的面积公式为 A = πr²。我们可以这样直观理解：将圆切成许多细小的扇形，再拼成一个近似长方形，其长为 πr，宽为 r，从而得到面积公式。

A = πr²

The value of π is approximately 3.14159, but in IB exams you should leave your answer in terms of π unless otherwise instructed or unless an approximate value is specifically requested.

π 的近似值为 3.14159，但在 IB 考试中，除非题目特别要求近似值，否则应保留 π 的形式作为精确答案。

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2. Circumference: Full Circle | 周长：完整圆

The circumference is the total distance around a circle. It is a linear measurement, so its units are the same as the radius or diameter (e.g., cm, m).

周长是绕圆一周的总距离，属于长度测量，因此其单位与半径或直径相同（如厘米、米）。

Example 1: A circle has a radius of 7 cm. Find its circumference.

例 1：已知圆的半径为 7 cm，求其周长。

C = 2πr = 2 × π × 7 = 14π cm ≈ 43.98 cm

If the diameter is given instead, use C = πd. For example, a circle with diameter 10 m has circumference 10π m ≈ 31.42 m.

如果题目给出的是直径，则可使用 C = πd。例如，直径为 10 m 的圆，其周长为 10π m ≈ 31.42 m。

• Always ensure the radius and diameter are in the same unit before calculating.

  计算前务必确保半径和直径的单位一致。

• When asked for exact value, keep π in the answer.

  当要求精确值时，答案中保留 π。

• When asked for an approximation, use a suitable value of π (usually 3.14 or the calculator’s π).

  当要求近似值时，使用合适的 π 值（通常为 3.14 或计算器上的 π）。

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3. Area of a Circle | 圆的面积

The area of a circle measures the total region enclosed by the circle. It is a two-dimensional measurement, so its units are squared (e.g., cm², m²).

圆的面积衡量的是圆所围成的平面区域的大小，属于二维度量，因此单位为平方单位（如 cm²、m²）。

Example 2: Find the area of a circle with radius 5 cm.

例 2：求半径为 5 cm 的圆的面积。

A = πr² = π × 5² = 25π cm² ≈ 78.54 cm²

If the diameter is given, first halve it to find the radius, then apply the area formula. A common mistake is to substitute the diameter directly into A = πr², which would give four times the correct area.

如果题目给出的是直径，需要先除以 2 得到半径，再代入面积公式。常见的错误是直接将直径代入 A = πr²，这样得到的结果会是正确面积的 4 倍。

Example 3: A circle has a diameter of 12 cm. Find its area.

例 3：已知圆的直径为 12 cm，求其面积。

r = 12 ÷ 2 = 6 cm, A = π × 6² = 36π cm² ≈ 113.10 cm²

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4. Arc Length of a Sector | 扇形的弧长

A sector is a portion of a circle bounded by two radii and an arc. The arc length is the distance along the curved edge of the sector. If the sector has central angle θ (in degrees), the arc length is a fraction θ/360 of the full circumference.

扇形是由两条半径和一段弧围成的圆的一部分。弧长是沿着扇形弯曲边缘的距离。若扇形的圆心角为 θ（以度为单位），则弧长等于圆周长的 θ/360。

Arc length = θ/360 × 2πr = θ/180 × πr

This formula works because the full circle has 360° and circumference 2πr. A sector with angle θ subtends exactly θ/360 of the whole circle.

该公式成立的原因在于整个圆对应 360°，周长为 2πr。圆心角为 θ 的扇形恰好对应整个圆的 θ/360。

Example 4: A sector has radius 8 cm and central angle 60°. Find the arc length.

例 4：扇形半径为 8 cm，圆心角为 60°，求弧长。

Arc length = 60/360 × 2π × 8 = 1/6 × 16π = 8π/3 cm ≈ 8.38 cm

Note: In some IB courses, especially at university level or in higher-level IB, angles may be measured in radians. The formula then becomes much simpler: arc length = rθ, where θ is measured in radians.

注意：在某些 IB 课程中，尤其是大学预科高级别课程中，角度可能以弧度制表示。此时弧长公式会变得更为简洁：弧长 = rθ，其中 θ 以弧度为单位。

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5. Area of a Sector | 扇形的面积

Similarly, the area of a sector is a fraction θ/360 of the full circle’s area. Since the full circle has area πr², the sector area is given by:

同理，扇形的面积等于整个圆面积的 θ/360。由于整个圆的面积为 πr²，扇形面积公式为：

Sector area = θ/360 × πr²

Using radians, the formula becomes sector area = ½ r²θ, which is often much more convenient.

若使用弧度制，公式变为扇形面积 = ½ r²θ，这在计算中往往更加方便。

Example 5: A sector has radius 6 cm and central angle 90°. Find its area.

例 5：扇形半径为 6 cm，圆心角为 90°，求其面积。

Sector area = 90/360 × π × 6² = 1/4 × 36π = 9π cm² ≈ 28.27 cm²

It is also common to ask for the perimeter of a sector, which is the sum of the arc length and the two radii: P = arc length + 2r.

题目还常要求扇形的周长，它等于弧长加上两条半径：P = 弧长 + 2r。

Example 6: For the sector in Example 4, find the perimeter.

例 6：对例 4 中的扇形，求其周长。

P = 8π/3 + 2 × 8 = 8π/3 + 16 ≈ 24.38 cm

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6. Segment of a Circle | 弓形（圆的一部分）

A segment of a circle is the region between an arc and the chord connecting its endpoints. Its area can be found by subtracting the area of the triangle formed by the two radii and the chord from the area of the sector.

圆的弓形是弧与其两端点所连弦之间的区域。弓形面积可通过扇形面积减去由两条半径与弦构成的三角形面积得到。

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