📚 IGCSE WJEC Maths: Formula Summary Handbook | IGCSE WJEC 数学：公式汇总手册
This handbook brings together all the key formulas you need to master for the IGCSE WJEC Mathematics specification. Use it for daily revision, quick checks and last-minute practice. Every formula is presented in a clear, exam-ready format with paired English and Chinese explanations to reinforce understanding.
本手册汇集了 IGCSE WJEC 数学课程中你需掌握的所有核心公式。可用于日常复习、快速查阅和考前冲刺。每条公式均以清晰的考试格式呈现，并配有中英文双语解释，帮助加深理解。
1. Number and Arithmetic | 数与算术
Essential formulas for working with fractions, percentages, standard form and ratio form the foundation of many IGCSE questions. These tools help you manipulate numbers accurately and efficiently under time pressure.
分数、百分数、标准形式和比率的公式是许多 IGCSE 题目的基础。这些工具能帮助你在紧张的时间限制下准确、高效地处理数字。
A percentage change is always calculated as: change divided by original value, then multiplied by 100.
百分比变化始终这样计算：变化量除以原始值，再乘以 100。
Percentage change = (Change ÷ Original) × 100%
Standard form writes a large or small number as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer.
标准形式将很大或很小的数写成 A × 10ⁿ，其中 1 ≤ A < 10 且 n 为整数。
e.g. 45 000 = 4.5 × 10⁴, 0.0032 = 3.2 × 10⁻³
For compound percentages, such as repeated percentage increase or decrease, use a multiplier raised to the power of the number of periods.
对于重复百分比增长或减少的复合百分数问题，将乘数提升到周期数次幂。
Final amount = Original × (1 ± r/100)ⁿ, where r is the rate and n the number of periods
When sharing a quantity in a given ratio, first find the total number of parts, then divide the total quantity by that sum, and multiply by each share.
按给定比率分配数量时，先求出总份数，再用总数除以总份数，然后乘以每一份额。
For ratio a : b, share = (a/(a+b)) × Total and (b/(a+b)) × Total
2. Algebra and Equations | 代数与方程
Algebra is the backbone of higher-tier GCSE Mathematics. WJEC exam papers expect you to expand brackets, factorise quadratics, rearrange formulas and solve linear, simultaneous and quadratic equations with fluency.
代数是 GCSE 数学提高层级的核心。WJEC 试卷要求你能熟练展开括号、因式分解二次式、变换公式以及求解线性方程、联立方程和二次方程。
To solve a quadratic equation ax² + bx + c = 0 by formula, substitute the coefficients into the quadratic formula.
用公式法解二次方程 ax² + bx + c = 0 时，将系数代入求根公式。
x = (−b ± √(b² − 4ac)) / (2a)
The difference of two squares factorises instantly as (a + b)(a − b). Always check for a common factor first.
平方差可立即分解为 (a + b)(a − b)。务必先寻找公因子。
a² − b² = (a + b)(a − b)
For a linear equation in the form y = mx + c, m gives the gradient and c the y-intercept. Parallel lines share the same m; perpendicular lines satisfy m₁ × m₂ = −1.
对于线性方程 y = mx + c，m 表示斜率，c 表示 y 轴截距。平行线斜率相同；垂直线满足 m₁ × m₂ = −1。
m = (y₂ − y₁) / (x₂ − x₁)
When solving simultaneous equations, you can eliminate one variable by making coefficients equal and adding or subtracting the equations.
解联立方程时，可通过使某一变量的系数相等后相加或相减来消元。
3. Graphs and Functions | 图像与函数
Functions and their graphs appear in many WJEC topics, from plotting straight lines to interpreting quadratic, cubic and reciprocal curves. Recognising key features such as intercepts, turning points and asymptotes is vital.
函数及其图像出现在许多 WJEC 主题中，从绘制直线到解读二次、三次和倒数曲线。识别截距、转折点和渐近线等关键特征至关重要。
The gradient of a straight line between two points (x₁, y₁) and (x₂, y₂) is the change in y over the change in x.
两点 (x₁, y₁) 和 (x₂, y₂) 间直线的斜率等于 y 的变化量除以 x 的变化量。
m = (y₂ − y₁) / (x₂ − x₁)
The equation of a line can be written using point-slope form when you know a point and the gradient.
若已知一点和斜率，可以用点斜式写出直线方程。
y − y₁ = m(x − x₁)
For a quadratic function y = x² + bx + c, the line of symmetry is x = −b/2. Completing the square gives the turning point.
对于二次函数 y = x² + bx + c，对称轴为 x = −b/2。完成平方可求得转折点。
Vertex form: y = (x + p)² + q, turning point at (−p, q)
Exponential growth and decay functions take the form y = a × bˣ; if b > 1 it is growth, if 0 < b < 1 it is decay.
指数增长与衰减函数的形式为 y = a × bˣ；若 b > 1 则为增长，若 0 < b < 1 则为衰减。
4. Sequences | 数列
WJEC asks you to find the nth term of linear and quadratic sequences, and to use it to generate terms. Understanding the structure of a sequence saves time and avoids counting errors.
WJEC 要求你求出线性和二次数列的第 n 项，并用它生成项。理解数列的结构能节省时间并避免计数错误。
For an arithmetic (linear) sequence with first term a and common difference d, the nth term is given by a simple rule.
对于首项为 a、公差为 d 的等差（线性）数列，第 n 项由一个简单规律给出。
nth term = a + (n − 1)d
The sum of the first n terms of an arithmetic series can be found using the first and last term or the first term and common difference.
等差数列前 n 项和可利用首项与末项，或首项与公差求得。
Sₙ = n/2 (a + l) or Sₙ = n/2 [2a + (n − 1)d]
A quadratic sequence has a second difference that is constant. The nth term has the form An² + Bn + C. Find A = half the second difference, then solve for B and C.
二次数列具有恒定的二次差。第 n 项的形式为 An² + Bn + C。令 A = 二次差的一半，再解出 B 和 C。
If second difference = 2, then A = 1; nth term = n² + Bn + C
Always test your nth term with the first few terms to confirm it is correct.
务必用前几项检验你的第 n 项公式，确认无误。
5. Geometry: Angles and Polygons | 几何：角与多边形
Angle facts underpin many geometric reasoning questions. WJEC expects you to apply rules for angles on a straight line, around a point, in triangles and in parallel lines, as well as interior and exterior angles of polygons.
角的事实是许多几何推理题的基础。WJEC 要求你运用关于直线上的角、一点周围的角、三角形内角和平行线中角的规则，以及多边形的内角和外角。
Angles around a single point sum to 360°. Adjacent angles on a straight line sum to 180°.
一点周围的角之和为 360°。一条直线上的邻角之和为 180°。
Sum of angles on a straight line = 180°
The sum of interior angles of any polygon with n sides is (n − 2) × 180°. This comes from splitting the polygon into n − 2 triangles.
任何 n 边多边形的内角和为 (n − 2) × 180°。这是因为可将多边形分割成 n − 2 个三角形。
Sum of interior angles = (n − 2) × 180°
The exterior angle of a regular polygon is 360° divided by the number of sides n. The interior and exterior angle at each vertex add up to 180°.
正多边形的一个外角等于 360° 除以边数 n。每个顶点处的内角和外角之和为 180°。
Exterior angle = 360° / n
Angles in a triangle always add to 180°, and angles in a quadrilateral add to 360°.
三角形的内角和始终为 180°，四边形的内角和为 360°。
6. Circle Theorems | 圆定理
WJEC IGCSE Mathematics includes the classic circle theorems. You need to be able to quote and apply them, often in multi-step problems that mix angle facts with algebra.
WJEC IGCSE 数学包含经典的圆定理。你需要能够引用并应用它们，通常出现于混合角度知识和代数的多步骤问题中。
The angle at the centre of a circle is twice the angle at the circumference, subtended by the same arc.
圆心角是同弧所对圆周角的两倍。
Angle at centre = 2 × Angle at circumference
The angle in a semicircle is a right angle (90°), because it is subtended by a diameter.
半圆中的圆周角是直角（90°），因为它对着直径。
Angle in a semicircle = 90°
Angles in the same segment of a circle are equal. They stand on the same chord and the same arc.
同弧上的圆周角相等。它们位于同一条弦和同一段弧上。
Angles in the same segment are equal
Opposite angles in a cyclic quadrilateral add up to 180°. This property is often used to find missing angles.
圆内接四边形的对角和为 180°。这一性质常用于求未知角。
A + C = 180°, B + D = 180°
The radius to a tangent meets the tangent at 90°. The angle between a tangent and a chord equals the angle in the alternate segment.
半径与切线相交成 90°。弦切角等于内对角。
7. Mensuration: Perimeter, Area, Volume | 测量：周长、面积、体积
These formulas are the most frequently needed in calculator and non-calculator papers. Learn both the 2D and 3D formulas thoroughly, including those for arcs, sectors, cylinders, cones, pyramids, spheres and frustums.
这些公式在可用和不可用计算器的试卷中都需要最频繁使用。彻底掌握二维和三维公式，包括弧长、扇形、圆柱、圆锥、棱锥、球和圆台的公式。
Here is a concise table of essential perimeter, area and volume formulas you must know.
下表简明列出了你必须掌握的关键周长、面积和体积公式。
| Shape | Formula |
| Rectangle area | A = l × w |
| Triangle area | A = ½ × base × height |
| Trapezium area | A = ½ (a + b) h |
| Circle circumference | C = 2πr = πd |
| Circle area | A = πr² |
| Arc length (angle A°) | Arc = (A/360) × 2πr |
| Sector area (angle A°) | Area = (A/360) × πr² |
| Cylinder volume | V = πr²h |
| Cylinder curved surface area | CSA = 2πrh |
| Cone volume | V = ⅓πr²h |
| Cone curved surface area | CSA = πrl (l = slant height) |
| Sphere volume | V = ⁴⁄₃πr³ |
| Sphere surface area | SA = 4πr² |
| Pyramid volume | V = ⅓ × base area × height |
For a prism, volume is simply the area of the uniform cross-section multiplied by the length. This applies to triangular prisms, cuboids and any prism.
棱柱的体积等于均匀横截面面积乘以长度。这适用于三棱柱、长方体以及任何棱柱。
Vprism = Area of cross-section × length
8. Trigonometry | 三角学
Trigonometry connects angles with side ratios in right-angled and non-right-angled triangles. WJEC requires confidence with sine, cosine and tangent, as well as the sine rule, cosine rule and area formula.
三角学将角与直角三角形和任意三角形中的边长比联系起来。WJEC 要求你熟练掌握正弦、余弦、正切以及正弦定理、余弦定理和面积公式。
In a right-angled triangle, label sides relative to the given angle θ: opposite (opp), adjacent (adj) and hypotenuse (hyp). Then SOH CAH TOA gives the ratios.
在直角三角形中，相对于给定角 θ 标记对边 (opp)、邻边 (adj) 和斜边 (hyp)。然后运用 SOH CAH TOA 记忆比值。
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
The sine rule is used to find missing sides or angles in any triangle when you have a matching pair of side and angle.
正弦定理可在已知一对边与角对应关系时，用于求任意三角形中未知的边或角。
a/sin A = b/sin B = c/sin C
The cosine rule links all three sides of a triangle with one angle. Use it when you know two sides and the included angle (SAS) or all three sides (SSS).
余弦定理将三角形的三边与一个角联系起来。当已知两边及其夹角 (SAS) 或三边 (SSS) 时可使用它。
a² = b² + c² − 2bc cos A
The area of any triangle can be found using two sides and the sine of the included angle.
任意三角形的面积可用两边及其夹角的正弦值求得。
Area = ½ ab sin C
To solve an equation like sin x = k, use the inverse function and consider the symmetry of the sine curve to find all solutions within 0° ≤ x ≤ 360°.
解如 sin x = k 的方程时，使用反函数并考虑正弦曲线的对称性，以求出 0° ≤ x ≤ 360° 内的所有解。
9. Vectors and Transformations | 向量与变换
Vectors describe movement with both magnitude and direction. WJEC questions often combine vector arithmetic, geometric proofs and column vectors. Transformations include translations, reflections, rotations and enlargements.
向量描述具有大小和方向的移动。WJEC 题目通常结合向量运算、几何证明和列向量。变换包括平移、反射、旋转和放大。
A vector is written as a column vector (x over y) or as a combination of i and j unit vectors. The magnitude is found using Pythagoras.
向量写作列向量 (x, y 竖排) 或 i 与 j 单位向量的组合。其模长用勾股定理求得。
If v = (a, b), then |v| = √(a² + b²)
Vector addition and scalar multiplication follow simple arithmetic rules. The resultant vector is found by adding the corresponding components.
向量的加法和标量乘法遵循简单运算法则。合成向量通过对应分量相加求得。
(a, b) + (c, d) = (a + c, b + d); k × (a, b) = (ka, kb)
The position vector of a point P relative to the origin O is OP. The vector from A to B is OB − OA.
点 P 相对于原点 O 的位置向量为 OP。从 A 到 B 的向量等于 OB − OA。
AB = OB − OA
For an enlargement with centre (0,0) and scale factor k, every point (x, y) maps to (kx, ky). For any other centre, use vector rays.
对于以 (0,0) 为中心、k 为比例因子的放大，每点 (x, y) 映射到 (kx, ky)。对于其他中心，使用向量射线法。
10. Probability | 概率
Probability is used to measure the chance of an event happening. WJEC expects you to calculate probabilities from equally likely outcomes, use tree diagrams for successive events, and apply the AND/OR rules.
概率用于衡量事件发生的可能性。WJEC 要求你从等可能结果中计算概率，使用树图处理连续事件，并运用“与”/“或”法则。
The basic formula for probability uses the number of favourable outcomes divided by the total number of possible outcomes.
概率的基本公式是用有利结果数除以所有可能结果总数。
P(Event) = Number of favourable outcomes / Total number of outcomes
For independent events A and B, the probability that both occur is the product of their individual probabilities (AND rule).
对于独立事件 A 和 B，两者同时发生的概率等于各自概率的乘积（与法则）。
P(A and B) = P(A) × P(B)
For mutually exclusive events, the probability that either A or B occurs is the sum of their probabilities (OR rule). For overlapping events, subtract the intersection once.
对于互斥事件，A 或 B 发生的概率等于各自概率之和（或法则）。对于有重叠的事件，需减去一次重叠部分。
P(A or B) = P(A) + P(B) − P(A and B)
Tree diagrams help visualise combined events. Multiply along branches for successive events, and add the probabilities of different paths for combined outcomes.
树图有助于可视化复合事件。沿着分支相乘得到连续事件的概率，并将不同路径的概率相加得到复合结果的概率。
Total probability of all branches = 1
Conditional probability considers the likelihood of an event given that another event has already occurred.
条件概率考虑在另一事件已发生的条件下某事件发生的可能性。
P(A|B) = P(A and B) / P(B)
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导