📚 IGCSE Mathematics: Core Formulas and Exam Applications | IGCSE数学：核心公式汇总与考点应用
This article compiles the most essential formulas for IGCSE Mathematics, organised by topic, with clear explanations of how they are applied in typical exam questions. Mastering these formulas is the key to solving problems quickly and accurately.
本文按专题整理了 IGCSE 数学中最核心的公式，并结合典型考点说明其应用方法。熟练掌握这些公式，是快速、准确解题的关键。
1. Algebra: Expanding and Factorising | 代数：展开与因式分解
The ability to expand and factorise algebraic expressions is fundamental. You must know the difference of two squares and perfect square expansions by heart.
展开与因式分解是代数的基础。你必须牢记平方差公式和完全平方公式。
Difference of two squares: a² − b² = (a + b)(a − b)
平方差公式：a² − b² = (a + b)(a − b)
Perfect square expansion: (a ± b)² = a² ± 2ab + b²
完全平方公式：(a ± b)² = a² ± 2ab + b²
Common factor first: 6x² + 9x = 3x(2x + 3)
先提公因式：6x² + 9x = 3x(2x + 3)
a² − b² = (a + b)(a − b) (±) (a ± b)² = a² ± 2ab + b²
When factorising a quadratic ax² + bx + c, look for two numbers that multiply to ac and add to b. In exams, always check whether a common factor can be taken out first.
因式分解二次式 ax² + bx + c 时，要找两个数，使其乘积等于 ac、和等于 b。考试中，务必先检查是否有公因式可提。
2. Algebra: Linear Equations and Inequalities | 代数：线性方程与不等式
Linear equations and inequalities appear in every IGCSE paper. Solving them requires balancing both sides and reversing the sign when multiplying or dividing by a negative number.
线性方程和不等式出现在每份 IGCSE 试卷中。解方程时需要保持两边平衡，而在乘以或除以负数时要改变不等号方向。
If ax + b = c, then x = (c − b) ÷ a
若 ax + b = c，则 x = (c − b) ÷ a
Inequality rule: if a < b and c < 0, then ac > bc
不等式性质：若 a < b 且 c < 0，则 ac > bc
2x + 5 = 13 → x = 4 ; 3(x − 2) ≤ 9 → x ≤ 5
For word problems, define the unknown, form an equation, and solve it step by step. Always check your solution in the original equation.
对于应用题，先设未知数，再列方程并逐步求解。最后务必将解代回原方程检验。
3. Simultaneous Equations and Quadratics | 联立方程与二次方程
You should be able to solve simultaneous linear equations by elimination or substitution, and solve quadratic equations by factorising, completing the square, or using the quadratic formula.
你需要会用消元法或代入法解联立线性方程，并会用因式分解、配方法或求根公式解二次方程。
Quadratic formula: x = [−b ± √b² − 4ac] / 2a
求根公式：x = [−b ± √b² − 4ac] / 2a
Discriminant: Δ = b² − 4ac decides the number of real roots
判别式：Δ = b² − 4ac 决定实根的个数
If Δ > 0, two distinct real roots; Δ = 0, one repeated root; Δ < 0, no real roots
若 Δ > 0，有两个不等实根；Δ = 0，有一个重根；Δ < 0，没有实根
x = (−b ± √b² − 4ac) / 2a
When one equation is linear and one is quadratic, substitute the linear expression into the quadratic. Then solve the resulting quadratic. Do not forget to give both x and y values in pairs.
当方程组中一个是一次方程、一个是二次方程时，将一次式代入二次式，再解所得二次方程。注意成对写出 x 和 y 的值。
4. Coordinate Geometry | 坐标几何
Coordinate geometry tests your ability to work with straight lines. Key formulas include gradient, midpoint, length, and equation of a line.
坐标几何考查对直线的运用。关键公式包括斜率、中点、长度和直线方程。
Gradient: m = (y₂ − y₁) / (x₂ − x₁)
斜率：m = (y₂ − y₁) / (x₂ − x₁)
Midpoint: ((x₁ + x₂)/2, (y₁ + y₂)/2)
中点：((x₁ + x₂)/2, (y₁ + y₂)/2)
Length: √[(x₂ − x₁)² + (y₂ − y₁)²]
长度：√[(x₂ − x₁)² + (y₂ − y₁)²]
Equation of a line: y − y₁ = m(x − x₁) or y = mx + c
直线方程：y − y₁ = m(x − x₁) 或 y = mx + c
m = (y₂ − y₁)/(x₂ − x₁)
Parallel lines have equal gradients; perpendicular lines have gradients whose product is −1. Use these facts to solve many geometry problems.
平行线的斜率相等；垂直线的斜率乘积为 −1。利用这些性质可以解决许多几何问题。
5. Mensuration: Perimeter, Area and Volume | 度量：周长、面积与体积
Mensuration formulas are essential for IGCSE. You need to know the area and volume of common shapes and solids, including circles, triangles, cylinders, prisms, cones and spheres.
度量公式是 IGCSE 的必考内容。你需要掌握常见平面图形和立体图形的面积与体积，包括圆、三角形、圆柱、棱柱、圆锥和球。
| Shape | 图形 | Formula | 公式 |
| Triangle | 三角形 | A = ½ × base × height |
| Circle | 圆 | A = πr², C = 2πr |
| Cylinder | 圆柱 | V = πr²h, curved area = 2πrh |
| Cone | 圆锥 | V = ⅓πr²h, curved area = πrl |
| Sphere | 球 | V = 4⁄3πr³, surface area = 4πr² |
units. When a question involves a prism, remember Volume = cross-sectional area × length.
始终注意单位。当题目涉及棱柱时，记住体积 = 横截面积 × 长度。
6. Trigonometry: Right-Angled Triangles | 三角学：直角三角形
Right-angled triangle trigonometry uses the ratios sine, cosine and tangent. You must memorise the relationships between sides and angles.
直角三角形三角学使用正弦、余弦和正切比。你必须牢记边角之间的关系。
H-TOA: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
Sin θ = 对边/斜边，Cos θ = 邻边/斜边，Tan θ = 对边/邻边
Pythagoras: a² + b² = c² (c is the hypotenuse)
勾股定理：a² + b² = c²（c 为斜边）
sin θ = O/H, cos θ = A/H, tan θ = O/A
Use trigonometry to find an unknown side or angle. For inverse operations, use sin⁻¹, cos⁻¹ or tan⁻¹ on your calculator.
用三角比求未知边或角。求角时使用计算器上的 sin⁻¹、cos⁻¹ 或 tan⁻¹ 功能。
7. Further Trigonometry: Sine and Cosine Rules | 进阶三角学：正弦定理与余弦定理
For non-right-angled triangles, the sine rule and cosine rule are indispensable. Remember the conditions for using each rule.
对于非直角三角形，正弦定理和余弦定理不可或缺。记住每种定理的适用条件。
Sine rule: a/sin A = b/sin B = c/sin C
正弦定理：a/sin A = b/sin B = c/sin C
Cosine rule: c² = a² + b² − 2ab cos C
余弦定理：c² = a² + b² − 2ab cos C
Area of triangle: Area = ½ab sin C
三角形面积：面积 = ½ab sin C
c² = a² + b² − 2ab cos C
The sine rule is useful when you know two angles and one side, or two sides and a non-included angle. The cosine rule is used when you know three sides or two sides and the included angle.
已知两角一边或两边及其中一边对角时用正弦定理；已知三边或两边及其夹角时用余弦定理。
8. Vectors and Transformations | 向量与几何变换
Vectors represent movement and direction. You need to add vectors, multiply by a scalar, and find the magnitude of a vector. Transformations include reflection, rotation, translation and enlargement.
向量表示移动和方向。你需要会向量的加法、数乘以及求模长。几何变换包括反射、旋转、平移和放大。
Vector addition: (a,b) + (c,d) = (a+c, b+d)
向量加法：(a,b) + (c,d) = (a+c, b+d)
Magnitude: |(x,y)| = √x² + y²
模长：|(x,y)| = √x² + y²
Enlargement: scale factor k, area scale factor k²
放大：线性比例因子 k，面积比例因子 k²
|v| = √x² + y²
For transformations, always state the type, the centre/line of reflection/rotation, the angle and direction, and the scale factor.
描述变换时，必须说明变换类型、中心/旋转中心/对称轴、角度与方向，以及比例因子。
9. Mensuration: Arc Length and Sector Area | 度量：弧长与扇形面积
Arc length and sector area are frequent exam topics. They are fractions of the full circumference and area, based on the angle θ in degrees.
弧长和扇形面积是常见考点。它们是圆周长和圆面积按角度 θ（度）取比例的部分。
Arc length = θ/360 × 2πr
弧长 = θ/360 × 2πr
Sector area = θ/360 × πr²
扇形面积 = θ/360 × πr²
Arc length = (θ/360) × 2πr ; Sector area = (θ/360) × πr²
If the angle is given in radians, replace θ/360 by θ/(2π). However, IGCSE usually uses degrees unless stated otherwise.
如果角度以弧度给出，则将 θ/360 替换为 θ/(2π)。但 IGCSE 通常使用角度制，除非题目另有说明。
10. Probability | 概率
Probability measures the chance of an event. The total probability of all outcomes is 1. You should understand mutually exclusive and independent events.
概率度量事件发生的可能性。所有结果的总概率为 1。你需要理解互斥事件和独立事件。
P(event) = number of favourable outcomes / total outcomes
P(事件) = 有利结果数 / 总结果数
For mutually exclusive events: P(A or B) = P(A) + P(B)
互斥事件：P(A 或 B) = P(A) + P(B)
For independent events: P(A and B) = P(A) × P(B)
独立事件：P(A 且 B) = P(A) × P(B)
Complement rule: P(not A) = 1 − P(A)
对立事件：P(非 A) = 1 − P(A)
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
Use tree diagrams for multi-stage experiments. Remember to multiply along branches and add for alternative paths.
多阶段试验用树状图。记住：沿分支相乘、不同路径相加。
11. Statistics: Mean, Median, Mode and Range | 统计：平均数、中位数、众数与极差
Statistics questions ask you to calculate measures of central tendency and spread. You must also interpret data from tables and charts.
统计题要求计算集中趋势和离散程度的度量。你还必须会解读表格和图表中的数据。
Mean = sum of all values / number of values
平均数 = 所有值之和 / 值的个数
Median is the middle value when data are ordered
中位数是将数据排序后的中间值
Mode is the most frequent value
众数是出现次数最多的值
Range = largest value − smallest value
极差 = 最大值 − 最小值
Mean = Σx / n
For grouped data, use mid-interval values to estimate the mean. Also be able to draw and interpret histograms, bar charts and pie charts.
对于分组数据，用组中值估计平均数。还需会绘制和解读直方图、柱状图和饼图。
12. Graphs of Functions | 函数图像
Understanding graphs is vital. You should recognize linear, quadratic, cubic, reciprocal and exponential graphs, and find gradients and areas under curves.
理解图像至关重要。你需要辨认线性、二次、三次、反比例和指数函数图像，并会求斜率及曲线下的面积。
Linear: y = mx + c, straight line, gradient m
线性：y = mx + c，直线，斜率 m
Quadratic: y = ax² + bx + c, parabola, vertex at x = −b/(2a)
二次：y = ax² + bx + c，抛物线，对称轴 x = −b/(2a)
Reciprocal: y = k/x, two branches, asymptotes at axes
反比例：y = k/x，两支曲线，以坐标轴为渐近线
Exponential: y = aˣ, growth or decay depending on a
指数：y = aˣ，a 决定增长还是衰减
y = ax² + bx + c → vertex x = −b/(2a)
To find the gradient of a curve at a point, draw a tangent and calculate its slope. The area under a curve can be estimated by counting squares or using the trapezium rule.
求曲线在某点的斜率，可作切线并计算其斜率。曲线下的面积可通过数方格或梯形法则估算。
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