GCSE OCR Maths: Formula Summary Handbook | GCSE OCR 数学：公式汇总手册

📚 GCSE OCR Maths: Formula Summary Handbook | GCSE OCR 数学：公式汇总手册

This handbook presents the essential formulas needed for GCSE OCR Mathematics. Whether you are sitting the Foundation or Higher tier, a solid grasp of these mathematical tools will boost your confidence and accuracy. Use this guide alongside your exam practice to ensure you can apply every formula correctly under timed conditions.

本手册汇总了GCSE OCR数学所需的核心公式。无论你参加基础层还是高等层考试，扎实掌握这些数学工具将提升你的信心与做题准确性。请将本指南配合真题训练使用，确保你能够限时条件下正确运用每一个公式。

1. Number | 数与算术

Multiplying fractions: a/b × c/d = ac / bd

分数乘法：a/b × c/d = ac / bd

Dividing fractions: a/b ÷ c/d = a/b × d/c = ad / bc

分数除法：a/b ÷ c/d = a/b × d/c = ad / bc

Percentage change: Percentage change = (change ÷ original value) × 100%

百分比变化：百分比变化 = (变化量 ÷ 原始值) × 100%

Percentage increase/decrease multiplier: For an increase of r%, multiply by (1 + r/100); for a decrease, multiply by (1 – r/100).

百分比增减乘数：增加 r% 时乘以 (1 + r/100)；减少 r% 时乘以 (1 – r/100)。

Simple interest: I = P × r × t / 100 (where P = principal, r = annual rate %, t = years)

单利：I = P × r × t / 100（其中 P 为本金，r 为年利率%，t 为年数）

Compound interest (Higher tier): Total amount = P × (1 + r/100)^n, where n is the number of compounding periods.

复利（高等层）：总金额 = P × (1 + r/100)^n，其中 n 为复利计算期数。

2. Algebra: The Basics | 代数基础

Quadratic formula (Higher tier): For ax² + bx + c = 0, the solutions are given by

x = [–b ± √(b² – 4ac)] / (2a)

二次方程求根公式（高等层）：对于 ax² + bx + c = 0，解为

x = [–b ± √(b² – 4ac)] / (2a)

Difference of two squares: a² – b² = (a + b)(a – b)

平方差公式：a² – b² = (a + b)(a – b)

Arithmetic sequence nth term: u_n = a + (n – 1)d, where a is the first term and d is the common difference.

等差数列第 n 项：u_n = a + (n – 1)d，其中 a 为首项，d 为公差。

Laws of indices: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m–n); (a^m)^n = a^(mn); a^0 = 1; a^(–n) = 1/(a^n).

指数定律：a^m × a^n = a^(m+n)；a^m ÷ a^n = a^(m–n)；(a^m)^n = a^(mn)；a^0 = 1；a^(–n) = 1/(a^n)。

3. Graphs and Functions | 图形与函数

Equation of a straight line: y = mx + c, where m is the gradient and c is the y-intercept.

直线方程：y = mx + c，其中 m 为斜率，c 为 y 轴截距。

Gradient between two points: m = (y₂ – y₁) / (x₂ – x₁)

两点间斜率：m = (y₂ – y₁) / (x₂ – x₁)

Distance between two points (Higher tier): d = √[(x₂ – x₁)² + (y₂ – y₁)²]

两点间距离（高等层）：d = √[(x₂ – x₁)² + (y₂ – y₁)²]

Equation of a circle (Higher tier): (x – a)² + (y – b)² = r², centre (a, b), radius r.

圆的方程（高等层）：(x – a)² + (y – b)² = r²，圆心 (a, b)，半径 r。

4. Ratio, Proportion and Rates of Change | 比率、比例与变化率

Ratio equality: If a : b = c : d, then ad = bc.

比例等式：若 a : b = c : d，则 ad = bc。

Direct proportion: y = kx, where k is the constant of proportionality.

正比例：y = kx，其中 k 为比例常数。

Inverse proportion: y = k / x.

反比例：y = k / x。

Repeated percentage change: Final value = original × (1 ± r/100)^n, same structure as compound interest.

重复百分比变化：最终值 = 原始值 × (1 ± r/100)^n，结构与复利相同。

5. Geometry: Angles and Polygons | 几何：角与多边形

Sum of interior angles of an n-sided polygon: (n – 2) × 180°

n 边形内角和：(n – 2) × 180°

Sum of exterior angles (any polygon): 360°

任意多边形外角和：360°

Interior angle of a regular n-sided polygon: (n – 2) × 180° / n

正 n 边形每个内角：(n – 2) × 180° / n

Exterior angle of a regular polygon: 360° / n

正多边形每个外角：360° / n

6. Perimeter, Area and Volume | 周长、面积与体积

Rectangle: Area A = l × w, Perimeter P = 2(l + w)

矩形：面积 A = 长 × 宽，周长 P = 2(长 + 宽)

Triangle: Area A = ½ × base × height

三角形：面积 A = ½ × 底 × 高

Parallelogram: Area A = base × perpendicular height

平行四边形：面积 A = 底 × 垂直高

Trapezium: Area A = ½ × (a + b) × h, where a and b are the parallel sides.

梯形：面积 A = ½ × (a + b) × h，其中 a、b 为平行边。

Circle: Area A = πr², Circumference C = 2πr = πd

圆：面积 A = πr²，周长 C = 2πr = πd

Prism: Volume = area of cross-section × length

棱柱：体积 = 横截面积 × 长

Cylinder: Volume V = πr²h

圆柱：体积 V = πr²h

Pyramid and cone (Higher tier): Volume = ⅓ × base area × height

棱锥与圆锥（高等层）：体积 = ⅓ × 底面积 × 高

Sphere (Higher tier): Volume V = ⁴⁄₃ πr³, Surface area A = 4πr²

球（高等层）：体积 V = ⁴⁄₃ πr³，表面积 A = 4πr²

7. Pythagoras and Trigonometry | 毕达哥拉斯定理与三角学

Pythagoras’ theorem: a² + b² = c² (where c is the hypotenuse of a right‑angled triangle).

毕达哥拉斯定理：a² + b² = c²（其中 c 为直角三角形的斜边）。

Basic trigonometric ratios (right‑angled triangle): sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

基本三角比（直角三角形）：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

Exact trig values to remember:

Angle θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	½	√3/2	1/√3
45°	√2/2	√2/2	1
60°	√3/2	½	√3
90°	1	0	undefined

需要记忆的精确三角值：见上表，角度θ分别对应0°、30°、45°、60°、90°的正弦、余弦和正切值。

Sine rule (Higher tier): a/sin A = b/sin B = c/sin C (use for finding sides or angles in any triangle).

正弦定理（高等层）：a/sin A = b/sin B = c/sin C（用于任意三角形的边或角求解）。

Cosine rule (Higher tier): a² = b² + c² – 2bc cos A (to find a side), or cos A = (b² + c² – a²) / (2bc) (to find an angle).

余弦定理（高等层）：a² = b² + c² – 2bc cos A（求边），或 cos A = (b² + c² – a²) / (2bc)（求角）。

Area of a triangle using trigonometry (Higher tier): Area = ½ ab sin C

利用三角学求三角形面积（高等层）：面积 = ½ ab sin C

8. Circles: Arc Length and Sector Area | 圆：弧长与扇形面积

Arc length: l = (θ/360) × 2πr, where θ is the angle at the centre in degrees.

弧长：l = (θ/360) × 2πr，其中 θ 是以度为单位的圆心角。

Sector area: A = (θ/360) × πr²

扇形面积：A = (θ/360) × πr²

Segment area (Higher tier): Segment area = sector area – area of triangle. You would normally be given guidance in the exam.

弓形面积（高等层）：弓形面积 = 扇形面积 – 三角形面积。考试中通常会给出提示。

9. Probability | 概率

Basic probability: P(event) = number of favourable outcomes / total number of possible outcomes, for equally likely outcomes.

基本概率：若等可能结果，P(事件) = 有利结果数 / 所有可能结果数。

Complementary events: P(A’) = 1 – P(A)

互补事件：P(A’) = 1 – P(A)

Mutually exclusive events: P(A or B) = P(A) + P(B)

互斥事件：P(A 或 B) = P(A) + P(B)

Independent events: P(A and B) = P(A) × P(B)

独立事件：P(A 和 B) = P(A) × P(B)

Expected frequency: Expected frequency = probability × number of trials

期望频率：期望频率 = 概率 × 试验次数

10. Statistics | 统计

Mean: Mean = Σx / n (sum of all data values divided by the number of values).

平均数：平均数 = Σx / n（所有数据值之和除以数据个数）。

Range: Range = highest value – lowest value

极差：极差 = 最大值 – 最小值

Interquartile range (IQR): IQR = upper quartile – lower quartile

四分位距：IQR = 上四分位数 – 下四分位数

Frequency density (histograms): Frequency density = frequency ÷ class width

频数密度（直方图）：频数密度 = 频数 ÷ 组距

11. Compound Measures and Kinematics | 复合度量与运动学

Speed, distance, time: speed = distance / time, distance = speed × time, time = distance / speed.

速度、距离、时间：速度 = 距离 / 时间，距离 = 速度 × 时间，时间 = 距离 / 速度。

Density, mass, volume: density = mass / volume

密度、质量、体积：密度 = 质量 / 体积

Pressure, force, area: pressure = force / area

压强、力、面积：压强 = 力 / 面积

SUVAT equations for constant acceleration (Higher tier): v = u + at, s = ut + ½ at², v² = u² + 2as, s = ½ (u + v) t (where u = initial velocity, v = final velocity, a = acceleration, t = time, s = displacement).

匀加速运动公式（高等层）：v = u + at，s = ut + ½ at²，v² = u² + 2as，s = ½ (u + v) t（u 为初速度，v 为末速度，a 为加速度，t 为时间，s 为位移）。

12. Vectors | 向量

Vector addition: If a = (x₁, y₁) and b = (x₂, y₂), then a + b = (x₁ + x₂, y₁ + y₂).

向量加法：如果 a = (x₁, y₁) 且 b = (x₂, y₂)，则 a + b = (x₁ + x₂, y₁ + y₂)。

Scalar multiplication: k × a = (k x₁, k y₁)

标量乘法：k × a = (k x₁, k y₁)

Magnitude of a vector (Higher tier): |a| = √(x² + y²) for a = (x, y).

向量的大小（高等层）：对于 a = (x, y)，|a| = √(x² + y²)。

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GCSE OCR Maths: Formula Summary Handbook | GCSE OCR 数学：公式汇总手册