📚 IGCSE OCR Maths: Formula Summary Handbook | IGCSE OCR 数学：公式汇总手册
Having quick access to essential formulas can make a huge difference in your IGCSE OCR Maths revision. This handbook brings together all the key formulas you need to memorise, grouped by topic for easy reference. Use it alongside past papers to build speed and confidence before your exam.
快速回顾核心公式能显著提升你的 IGCSE OCR 数学复习效率。本手册汇集了你需要熟记的所有关键公式，并按主题分类，方便查阅。把它和历年真题结合使用，能在考前帮你提升答题速度和信心。
1. Number & Arithmetic Essentials | 数与算术基础公式
Percentage change gives the increase or decrease relative to the original value: Percentage change = (New value – Original value) / Original value × 100%.
百分比变化表示相对于原始值的增减：百分比变化 = (新值 – 原值) / 原值 × 100%。
For simple interest, the interest earned is proportional to the principal, rate, and time: I = Prt / 100, where P is principal, r is annual interest rate, and t is time in years.
单利计算中，获得的利息与本金、利率和时间成正比：I = Prt / 100，其中 P 为本金，r 为年利率，t 为年数。
Compound interest formula (often given in the exam but essential to understand): A = P(1 + r/100)ᵗ, where A is the final amount, P is principal, r is rate per period, and t is the number of periods.
复利公式（考试中有时会提供，但理解它依然很重要）：A = P(1 + r/100)ᵗ，A 为最终金额，P 为本金，r 为每期利率，t 为期数。
Reverse percentages: To find the original amount after a percentage increase or decrease, divide by the multiplier.
逆向百分比：已知增减后的值求原值，用最终值除以对应的乘数。
2. Algebraic Expressions & Factorisation | 代数表达式与因式分解
Expanding a product of two binomials: (a + b)(c + d) = ac + ad + bc + bd. Use the FOIL method to organise your work.
二项式乘积展开：(a + b)(c + d) = ac + ad + bc + bd。可使用 FOIL 法则有序地进行展开。
Difference of two squares: a² – b² = (a + b)(a – b). This is one of the most commonly tested factorisations.
平方差公式：a² – b² = (a + b)(a – b)。这是最常见的因式分解形式之一。
Perfect square expansions: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b².
完全平方式：(a + b)² = a² + 2ab + b²，(a – b)² = a² – 2ab + b²。
Factorising a quadratic x² + bx + c: find two numbers that multiply to c and add to b. For ax² + bx + c, use splitting the middle term or the formula method.
对二次式 x² + bx + c 因式分解：找出乘积为 c、和为 b 的两个数。对 ax² + bx + c 形式，可用“拆中项”法或求根公式。
3. Linear Equations & Graphs | 线性方程与图像
The equation of a straight line is most commonly expressed as y = mx + c, where m is the gradient and c is the y-intercept (the point where the line crosses the y-axis).
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距（直线与 y 轴的交点）。
Gradient between two points (x₁, y₁) and (x₂, y₂): m = (y₂ – y₁) / (x₂ – x₁). Be careful with negative coordinates.
两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率：m = (y₂ – y₁) / (x₂ – x₁)。注意坐标有负数时符号别错。
Midpoint of a line segment: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). The midpoint is the average of the x-coordinates and the average of the y-coordinates.
线段的中点：M = ((x₁ + x₂)/2, (y₁ + y₂)/2)。中点坐标是两端点横坐标的平均值和纵坐标的平均值。
Length of a line segment (distance between two points): d = √[(x₂ – x₁)² + (y₂ – y₁)²]. This comes directly from Pythagoras’ theorem.
线段长度（两点间距离）：d = √[(x₂ – x₁)² + (y₂ – y₁)²]。这直接来自勾股定理的应用。
4. Quadratic Equations | 二次方程
The solutions of the quadratic equation ax² + bx + c = 0 (a ≠ 0) can be found using the quadratic formula. The formula is given on the OCR formula sheet, but you must know how to apply it correctly.
二次方程 ax² + bx + c = 0 (a ≠ 0) 的解可以用求根公式得到。该公式在 OCR 公式表中提供，但你必须能正确代入数值。
x = [ –b ± √(b² – 4ac) ] / (2a)
Ensure you write the entire numerator over 2a and use brackets if substituting negative numbers.
务必把整个分子写在分母 2a 之上，代入负数时要用括号。
The expression under the square root, b² – 4ac, is called the discriminant. It determines the nature of the roots: if greater than 0, two real distinct roots; if equal to 0, one repeated real root; if less than 0, no real roots.
根号内的式子 b² – 4ac 称为判别式。它决定根的性质：大于 0 时有两个不等实根；等于 0 时有两个相等实根；小于 0 时无实根。
Completing the square: rewrite x² + bx as (x + b/2)² – (b/2)². This is essential for finding the turning point of a quadratic graph.
配方法：把 x² + bx 改写为 (x + b/2)² – (b/2)²。这用于求二次函数图像的顶点坐标。
5. Inequalities | 不等式
Solving linear inequalities follows the same rules as equations, but multiplying or dividing by a negative number reverses the inequality sign.
解线性不等式的步骤与解方程类似，但需注意：乘以或除以一个负数时，不等号方向要改变。
For example, –2x > 6 becomes x < –3 after dividing by –2. Always check a test point.
例如 –2x > 6 除以 –2 后得到 x < –3。永远可用一个测试点检验答案。
Graphing inequalities on a number line: use open circles for < and >, closed circles for ≤ and ≥. Shaded regions in 2D represent multiple inequalities.
在数轴上表示不等式：对 < 和 > 用空心圆点，对 ≤ 和 ≥ 用实心圆点。二维平面中的阴影区域表示同时满足多个不等式。
6. Sequences | 数列
An arithmetic (linear) sequence has a common difference d. The nth term is: uₙ = a + (n – 1)d, where a is the first term.
等差（线性）数列有公差 d。第 n 项公式为：uₙ = a + (n – 1)d，其中 a 是首项。
To find whether a number is a term in the sequence, set uₙ equal to that number and solve for n. If n is a positive integer, the number belongs to the sequence.
判断某数是否为数列的一项：令 uₙ 等于该数，解出 n。若 n 为正整数，则该数在数列中。
A quadratic sequence has a constant second difference. The nth term takes the form an² + bn + c. You can find a, b, and c by comparing given terms.
二次数列有恒定的二阶公差。其第 n 项形式为 an² + bn + c。可通过比较已知项列方程组求得 a, b, c 的值。
7. Geometry: Angles & Polygons | 几何：角与多边形
Sum of interior angles of an n-sided polygon: (n – 2) × 180°. Each interior angle in a regular polygon = [(n – 2) × 180°] / n.
n 边形内角和：(n – 2) × 180°。正多边形每个内角为 [(n – 2) × 180°] / n。
The sum of exterior angles of any convex polygon is always 360°, regardless of the number of sides.
任何凸多边形的外角和恒为 360°，与边数无关。
Angles on a straight line add up to 180°; angles around a point sum to 360°. Vertically opposite angles are equal.
直线上的邻角之和为 180°；绕一点的所有角之和为 360°。对顶角相等。
Corresponding and alternate angles between parallel lines are equal; co-interior (allied) angles add to 180°.
平行线间的同位角和内错角相等；同旁内角互补（和为 180°）。
8. Mensuration: Perimeter & Area | 测量：周长与面积公式
| Shape (形状) | Perimeter / Circumference (周长) | Area (面积) |
|---|---|---|
| Rectangle (矩形) | 2(l + w) | l × w |
| Triangle (三角形) | Sum of three sides | ½ × base × height |
| Parallelogram (平行四边形) | 2(a + b) | base × perpendicular height |
| Trapezium (梯形) | Sum of four sides | ½(a + b)h (a, b parallel) |
| Circle (圆) | Circumference = 2πr | πr² |
The area of a triangle formula Area = ½ ab sin C is useful when two sides and the included angle are known. This is sometimes provided in the exam but worth memorising.
当已知两边及其夹角时，使用三角形面积公式 面积 = ½ ab sin C。考试可能会提供该公式，但熟记能加快解题。
For a sector of a circle with radius r and angle θ°: arc length = (θ/360) × 2πr, sector area = (θ/360) × πr².
半径为 r、圆心角为 θ° 的扇形：弧长 = (θ/360) × 2πr，扇形面积 = (θ/360) × πr²。
9. Mensuration: Volume & Surface Area | 测量：体积与表面积
Prism volume = area of cross-section × length. This applies to cylinders, cuboids, and any uniform prism.
棱柱体积 = 底面积 × 长。这适用于圆柱、长方体和任何等截面棱柱。
Pyramid volume: V = ⅓ × base area × vertical height. The cone is a special pyramid with a circular base, so V = ⅓ πr²h.
棱锥体积：V = ⅓ × 底面积 × 垂直高度。圆锥是底面为圆形的特殊棱锥，故 V = ⅓ πr²h。
Sphere volume and surface area are given on the OCR formula sheet: V = ⁴⁄₃ πr³, Surface area = 4πr². Learn how to rearrange them.
球的体积和表面积在 OCR 公式表中给出：V = ⁴⁄₃ πr³，表面积 = 4πr²。要学会对公式进行变形求值。
Surface area of a cylinder: curved surface area = 2πrh, total = 2πr² + 2πrh. For a cone, curved surface area = πrl, where l is slant height, total = πr² + πrl.
圆柱表面积：侧面积 = 2πrh，总面积 = 2πr² + 2πrh。圆锥表面积：侧面积 = πrl（l 为斜高），总面积 = πr² + πrl。
10. Pythagoras & Trigonometry | 勾股定理与三角函数
In a right-angled triangle, a² + b² = c², where c is the hypotenuse. Use it to find missing sides or check for right angles.
在直角三角形中，a² + b² = c²，其中 c 为斜边。它可用于求未知边长或检验直角。
The three trigonometric ratios for right-angled triangles: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.
直角三角形中三个三角比：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
For non-right-angled triangles, the sine rule and cosine rule are usually provided, but are essential to use correctly.
对于非直角三角形，正弦定理和余弦定理通常会在公式表中给出，但你必须能正确选用。
Sine rule: a / sin A = b / sin B = c / sin C
Cosine rule: a² = b² + c² – 2bc cos A
The area formula ½ ab sin C from Section 8 is also useful here for triangle area without height.
第 8 节中的面积公式 ½ ab sin C 同样可用于无高三角形面积计算。
11. Probability | 概率
The probability of an event A is P(A) = number of favourable outcomes / total number of outcomes, when all outcomes are equally likely.
当所有结果等可能时，事件 A 的概率 P(A) = 有利结果数 / 总结果数。
The sum of probabilities of all mutually exclusive events is 1. For complementary events: P(A’) = 1 – P(A).
所有互斥事件概率之和为 1。对立事件满足 P(A’) = 1 – P(A)。
For independent events A and B: P(A and B) = P(A) × P(B). For mutually exclusive events: P(A or B) = P(A) + P(B).
若 A 与 B 相互独立，则 P(A 和 B) = P(A) × P(B)。若互斥，则 P(A 或 B) = P(A) + P(B)。
For non-mutually-exclusive events, the combined probability is P(A or B) = P(A) + P(B) – P(A and B).
若事件不互斥，其并事件概率为 P(A 或 B) = P(A) + P(B) – P(A 和 B)。
Tree diagrams: multiply along branches for combined events and add probabilities of appropriate outcomes.
树状图：沿分支相乘得联合概率，再将符合要求的结果概率相加。
12. Statistics | 统计
The mean of a dataset: Mean = (sum of all values) / (number of values). For grouped data, use midpoints of intervals.
数据集的平均数：均值 = (所有值的总和) / (值的个数)。对于分组数据，用各组的组中值进行计算。
The median is the middle value when data are ordered. If n is even, median is the average of the two middle values. The mode is the most frequent value.
中位数是排序后位于中间的数据值。若数据个数为偶数，则中位数为中间两数的平均值。众数是出现次数最多的值。
The range = highest value – lowest value. Interquartile range (IQR) = upper quartile (Q3) – lower quartile (Q1).
极差 = 最大值 – 最小值。四分位距 (IQR) = 上四分位数 (Q3) – 下四分位数 (Q1)。
To estimate the mean from a frequency table, use Σ(fx) / Σf, where x is the data value or midpoint, and f is frequency.
从频数表中估算平均数：用 Σ(fx) / Σf，x 为数据值或组中值，f 为频数。
For histograms with unequal class widths, frequency density = frequency / class width. The area of each bar represents frequency.
对于组距不等的直方图，频率密度 = 频数 / 组距。每个长方形的面积代表对应的频数。
The probability of an outcome estimated from a frequency table is called relative frequency, which stabilises as trials increase.
从频数表估计出的结果概率称为相对频率，当试验次数增加时它会趋于稳定。
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