📚 GCSE WJEC Maths: Circular Motion & Circles Revision | GCSE WJEC 数学：圆周运动 考点精讲
In GCSE WJEC Mathematics, the topic often referred to as 'circular motion' revolves around the key properties of circles, including their geometry, measurement, and equations. Understanding how points and angles behave on a circle is essential for solving a wide range of problems. This guide breaks down all the vital concepts you need to master, from circumference and area to arc length, sector area, circle theorems, and the equation of a circle. Whether you are studying Foundation or Higher tier, these revision notes will help you move confidently around the circle.
在 WJEC GCSE 数学中，通常被称为“圆周运动”的专题实际上涵盖了圆的核心性质，包括圆的几何、度量和方程。理解点和角在圆上的行为对于求解各类问题至关重要。本指南将分解从周长和面积到弧长、扇形面积、圆周定理以及圆的方程等所有你需要掌握的关键概念。无论你学习基础层还是高阶，这些复习笔记都将帮助你自信地“绕圆而行”。
1. The Circle & Its Terminology | 圆的概念与术语
s lie on the circle. The diameter is a special chord that passes through the centre and is twice the radius: d = 2r. An arc is a part of the circumference. A sector is the region enclosed by two radii and an arc, while a segment is the region between a chord and an arc.
圆定义为平面内到定点（圆心）距离等于定长（半径 r）的所有点的集合。围绕圆的距离称为周长。从圆心到圆上任意一点的线段是半径。弦是两个端点都在圆上的线段。直径是一条通过圆心的特殊弦，长度为半径的两倍：d = 2r。弧是圆周的一部分。扇形是由两条半径和一段弧所围成的区域，而弓形是弦和弧之间的区域。
At GCSE level, you need to be comfortable with these terms because they form the foundation for all circle calculations and theorems.
在 GCSE 阶段，你需要熟悉这些术语，因为它们构成了所有圆的计算和定理的基础。
2. Circumference and Area | 周长与面积
hese formulas and be able to apply them in both exact form (leaving answers in terms of π) and approximate decimal form. WJEC often asks for answers to be given in terms of π unless a specific degree of accuracy is requested.
圆的周长 C 公式为：C = 2πr 或 C = πd。π（圆周率）是一个常数，约等于 3.14 或 22/7。圆所围成的面积 A 是：A = πr²。你必须熟记这些公式，并能够以精确形式（用 π 表示）和近似小数形式应用它们。WJEC 常要求用 π 表示答案，除非明确指出精确到一定小数位数。
example, to find the area of a circle with radius 5 cm, you compute A = π × 5² = 25π cm².
例如，求半径为 5 厘米的圆的面积，计算 A = π × 5² = 25π 平方厘米。
Be careful: if you are given the diameter, halve it first to find the radius before plugging it into the area formula.
注意：如果给出的是直径，先除以 2 求出半径，再代入面积公式。
3. Arc Length | 弧长
rcle corresponds to 360°. You can think of it as (fraction of the circle) × circumference.
弧是圆周的一部分。弧长与它所对的圆心角成正比。公式为：弧长 = (θ / 360) × 2πr，其中 θ 是圆心角的度数。这是因为整个圆对应 360°。你可以把它理解为（圆的分数部分）× 周长。
cm has a central angle of 72°, the arc length = (72/360) × 2π × 10 = (1/5) × 20π = 4π cm.
如果一个半径为 10 厘米的圆，圆心角为 72°，那么弧长 = (72/360) × 2π × 10 = (1/5) × 20π = 4π 厘米。
In WJEC exam questions, you might be given the arc length and asked to find the radius or the angle. Always set up the equation and solve algebraically.
在 WJEC 考试题中，可能会给出弧长让你求半径或角度。务必列出方程并代数求解。
4. Area of a Sector | 扇形面积
The sector is like a 'pizza slice' ucial for solving problems involving shaded regions. Remember that the perimeter of a sector includes the arc length plus the two radii, so be prepared for perimeter-of-sector questions.
扇形就像圆的“披萨片”。其面积也与圆心角成正比：扇形面积 = (θ / 360) × πr²。这个公式对于求解阴影区域问题至关重要。记住，扇形的周长包含弧长加上两条半径，所以要准备回答扇形的周长问题。
sector with radius 8 cm and angle 45° has area = (45/360) × π × 8² = (1/8) × 64π = 8π cm².
一个半径为 8 厘米、角度为 45° 的扇形，面积 = (45/360) × π × 8² = (1/8) × 64π = 8π 平方厘米。
In Foundation tier, the angle is often a simple fraction like 90°, 180°, or 60°; Higher tier may involve more complex angles and require exact answers in terms of π.
在基础层，角度通常是简单分数如 90°、180° 或 60°；高阶可能涉及更复杂的角度，并要求用 π 给出精确答案。
5. Chords, Tangents and Segment | 弦、切线与弓形
e radius and the angle. The formula is: Area of Minor Segment = (θ/360)πr² − (1/2)r² sin θ, but this is rarely examined at GCSE—often you can use geometric reasoning with special angles.
一条弦将圆分成两个弓形：大弓形和小弓形。弓形的面积可以通过扇形面积减去由两条半径和弦构成的三角形面积来求得。对于 WJEC 高阶，你可能会遇到已知半径和角度求弓形面积的问题。公式为：小弓形面积 = (θ/360)πr² − (1/2)r² sin θ，但这在 GCSE 很少考——通常可以使用特殊角的几何推理。
A tangent is a straight line that touches the circle at exactly one point. The key fact is that the radius to the point of tangency is perpendicular to the tangent. This property is fundamental in circle theorems.
切线是与圆恰好接触于一点的直线。关键事实是：连接切点与圆心的半径垂直于该切线。这一性质是圆周定理的基础。
6. Circle Theorems – Part 1: Angle Relationships | 圆周定理（一）角关系
WJEC GCSE requires you to know and apply several circle theorems. First, the angle at the centre is twice any angle at the circumference subtended by the same arc. Second, angles in the same segment of a circle are equal. Third, the angle in a semicircle (angle subtended by a diameter) is always a right angle (90°). These theorems allow you to find missing angles in complex diagrams.
WJEC GCSE 要求你掌握并应用若干圆周定理。首先，同弧所对的圆心角是圆周角的两倍。其次，同一弦上同弧所对的圆周角相等。第三，半圆上的圆周角（直径所对的圆周角）始终是直角（90°）。这些定理能帮你求解复杂图形中的未知角。
If ∠AOB at the centre is 100°, then any angle ACB on the circumference on the same arc AB equals 50°.
如果圆心角 ∠AOB 为 100°，则同弧 AB 上的所有圆周角 ACB 都等于 50°。
The fact that the angle in a semicircle is 90° is particularly useful for proving that a triangle is right-angled when a diameter is involved.
半圆上的角为 90° 这一事实在涉及直径时极为有用，常用于证明三角形为直角三角形。
7. Circle Theorems – Part 2: Cyclic Quadrilaterals & Tangents | 圆周定理（二）圆内接四边形与切线
A cyclic quadrilateral is a four-sided figure with all vertices on the circumference. The opposite angles in a cyclic quadrilateral add up to 180°. For tangents, the angle between a tangent and a chord through the point of contact equals the angle in the alternate segment. Additionally, tangents drawn from the same external point are equal in length. These theorems need to be recognised and quoted in proof questions.
圆内接四边形是所有顶点都在圆周上的四边形。圆内接四边形的对角互补，和为 180°。对于切线，切线与过切点的弦所夹的角等于弦切角（alternate segment）。另外，从同一点引出的两条切线长度相等。这些定理在证明题中需要识别并引用。
If ∠ABC is an angle between a tangent at B and chord BC, then it equals the angle in the alternate segment, such as ∠BDC where D is any point on the circle on the opposite side of BC.
例如，如果 ∠ABC 是切点 B 处的切线与弦 BC 的夹角，那么它等于 alternate segment 中的角，比如与 BC 异侧的圆周角 ∠BDC。
8. Equation of a Circle | 圆的方程（高阶）
² + y² = r². You also need to recognise and use the equation for any centre (a, b): (x − a
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