📚 IGCSE AQA Maths Formula Compendium | IGCSE AQA 数学公式汇总手册
This handbook compiles essential formulas for the AQA IGCSE Mathematics specification. It covers core and extended tiers, giving you a quick reference for arithmetic, algebra, geometry, statistics, and compound measures. Use it alongside your revision to memorise and apply formulas with confidence.
本手册汇集了 AQA IGCSE 数学教学大纲的必备公式。它涵盖核心和扩展部分，为您提供算术、代数、几何、统计和复合测量公式的快速参考。与复习相结合，熟记并自信地应用这些公式。
1. Arithmetic and Percentages | 算术与百分比
To increase a quantity by r%, multiply by the factor (1 + r/100). For a decrease, use (1 – r/100).
要增加 r%，乘以增长因子 (1 + r/100)；减少 r%，乘以 (1 – r/100)。
Simple interest is calculated as I = P × r × t / 100, where P is the principal, r is the annual rate and t is the time in years. The total amount is A = P + I.
简单利息：I = P × r × t / 100，其中 P 为本金，r 为年利率，t 为年数。总额 A = P + I。
Compound interest is given by A = P(1 + r/100)ⁿ, where n is the number of years.
复利公式为 A = P(1 + r/100)ⁿ，其中 n 为年数。
Percentage profit or loss equals (change / original) × 100%.
百分比利润或亏损等于 (变化值 / 原值) × 100%。
2. Direct and Inverse Proportion | 正反比例
When y is directly proportional to x, we write y = kx, where k is the constant of proportionality.
若 y 与 x 成正比，写为 y = kx，其中 k 为比例常数。
If y is inversely proportional to x, the relationship is y = k/x.
若 y 与 x 成反比，关系式为 y = k/x。
You can find k by substituting a given pair of values. The graph of direct proportion is a straight line through the origin; the graph of inverse proportion is a rectangular hyperbola.
代入已知的一组值可以求出 k。正比例图像是一条过原点的直线；反比例图像是等轴双曲线。
3. Algebraic Formulae and Quadratic Equations | 代数公式与二次方程
(a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b².
(a + b)² = a² + 2ab + b²，(a – b)² = a² – 2ab + b²。
The difference of two squares factors as a² – b² = (a – b)(a + b).
平方差因式分解：a² – b² = (a – b)(a + b)。
To solve a quadratic equation ax² + bx + c = 0, use the quadratic formula:
x = (-b ± √b² – 4ac) / (2a)
求解二次方程 ax² + bx + c = 0 时，使用二次公式。
The expression under the square root, b² – 4ac, is called the discriminant. It tells you how many real roots the quadratic has.
根号下的表达式 b² – 4ac 叫做判别式，它决定了二次函数实根的个数。
4. Sequences | 数列
For an arithmetic progression (AP), the nᵗʰ term is aₙ = a₁ + (n – 1)d, where a₁ is the first term and d is the common difference.
等差数列的第 n 项为 aₙ = a₁ + (n – 1)d，其中 a₁ 为首项，d 为公差。
The sum of the first n terms of an AP is Sₙ = n/2 [2a₁ + (n – 1)d] or Sₙ = n/2 (a₁ + aₙ).
等差数列前 n 项和：Sₙ = n/2 [2a₁ + (n – 1)d] 或 Sₙ = n/2 (a₁ + aₙ)。
For a geometric progression (GP), the nᵗʰ term is aₙ = a rⁿ⁻¹, where a is the first term and r is the common ratio.
等比数列第 n 项为 aₙ = a rⁿ⁻¹，其中 a 为首项，r 为公比。
The sum of the first n terms (finite GP) is Sₙ = a(1 – rⁿ) / (1 – r) for r ≠ 1.
有限等比数列前 n 项和（r ≠ 1）：Sₙ = a(1 – rⁿ) / (1 – r)。
If |r| < 1, the sum to infinity is S∞ = a / (1 – r).
当 |r| < 1 时，无穷等比数列的和为 S∞ = a / (1 – r)。
5. Equations and Inequalities | 方程与不等式
olate the variable using inverse operations. For example, from 2x + 3 = 11 you find x = 4.
解一次方程时，通过逆运算移项使变量孤立。例如由 2x + 3 = 11 可得 x = 4。
Quadratic equations can be solved by factorising, completing the square, or the quadratic formula. When factorising, set each bracket to zero.
二次方程可通过因式分解、配方法或二次公式求解。因式分解后，令每个括号为零即可。
Inequalities are solved like equations, but reverse the sign when multiplying or dividing by a negative number. Represent solutions on a number line.
不等式的解法与方程类似，但当乘以或除以负数时，不等号方向要改变。解集可用数轴表示。
6. Geometry and Measures | 几何与测量
Area of a rectangle = length × width; Area of a triangle = ½ × base × height.
矩形面积 = 长 × 宽；三角形面积 = ½ × 底 × 高。
Area of a parallelogram = base × vertical height; Area of a trapezium = ½ (a + b)h, where a and b are the parallel sides.
平行四边形面积 = 底 × 垂直高；梯形面积 = ½ (a + b)h，其中 a 和 b 为两平行边。
Circle area = πr²; Circumference = 2πr or πd.
圆的面积 = πr²；周长 = 2πr 或 πd。
Volume of a prism = cross-sectional area × length; Cylinder volume = πr²h.
棱柱体积 = 横截面积 × 长；圆柱体积 = πr²h。
Pyramid/cone volume = ⅓ × base area × vertical height; Sphere volume = ⁴⁄₃ πr³.
棱锥/圆锥体积 = ⅓ × 底面积 × 垂直高；球体体积 = ⁴⁄₃ πr³。
Surface area of a sphere = 4πr²; Curved surface area of a cylinder = 2πrh.
球的表面积 = 4πr²；圆柱侧面积 = 2πrh。
Arc length s = θ/360 ×
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