📚 Year 9 Cambridge Maths: Formula & Theorem Quick Reference | Year 9 Cambridge 数学：公式定理速查手册
This quick reference handbook covers all the essential formulae and theorems you will encounter in the Cambridge Lower Secondary Mathematics Stage 9 curriculum. It serves as a handy revision tool to help you recall key concepts, from laws of indices to probability. Each section pairs clear English statements with Chinese translations so that you can study bilingually and reinforce your understanding.
本速查手册涵盖 Cambridge 初中数学 Stage 9 课程中的所有核心公式与定理。它可作为你复习备考的便携工具，帮助你回顾从指数法则到概率的关键知识点。每个部分均配有清晰的英文说明与中文翻译，让你在中英双语学习中加深理解。
1. Laws of Indices | 指数法则
When multiplying powers with the same base, add their indices: aᵐ × aⁿ = aᵐ⁺ⁿ.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。
en dividing powers with the same base, subtract the indices: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (where a ≠ 0).
同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（a ≠ 0）。
To raise a power to another power, multiply the indices: (aᵐ)ⁿ = aᵐⁿ.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。
When raising a product to a power, apply the power to each factor: (ab)ⁿ = aⁿ bⁿ.
积的乘方，每个因数分别乘方：(ab)ⁿ = aⁿ bⁿ。
Any non‑zero number raised to the power zero equals 1: a⁰ = 1 (a ≠ 0).
任何非零数的零次幂都等于1：a⁰ = 1（a ≠ 0）。
e exponent means the reciprocal of the positive power: a⁻ⁿ = 1 / aⁿ, and (a/b)⁻ⁿ = (b/a)ⁿ.
负指数表示相应正指数幂的倒数：a⁻ⁿ = 1 / aⁿ，(a/b)⁻ⁿ = (b/a)ⁿ。
2. Standard Form | 标准形式
er written in standard form is expressed as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
用标准形式表示一个数时，应写成 a × 10ⁿ 的形式，其中 1 ≤ a < 10，n 为整数。
For large numbers, n is positive; for small numbers, n is negative.
对于很大的数，n 为正；对于很小的数，n 为负。
To multiply numbers in standard form, multiply the a‑values and add the powers of 10: (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ.
标准形式下乘法：数字部分相乘，10的指数相加：(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ。
To divide numbers in standard form, divide the a‑values and subtract the powers of 10: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a / b) × 10ᵐ⁻ⁿ.
标准形式下除法：数字部分相除，10的指数相减：(a × 10ᵐ) ÷ (b × 10ⁿ) = (a / b) × 10ᵐ⁻ⁿ。
3. Percentages | 百分数
a percentage of a number, multiply the number by the percentage expressed as a decimal: 30% of 200 = 0.30 × 200 = 60.
求一个数的百分之几，将该数乘以百分数的小数形式：200的30% = 0.30 × 200 = 60。
Percentage increase = (increase / original amount) × 100%.
增加的百分率 =（增加量 / 原始数量）× 100%。
Percentage decrease = (decrease / original amount) × 100%.
减少的百分率 =（减少量 / 原始数量）× 100%。
To calculate a value after a percentage change, use a multiplier: for an increase, multiply by 1 + r; for a decrease, multiply by 1 − r, where r is the percentage as a decimal.
计算百分数变化后的值，使用乘数：增加则乘以 1 + r，减少则乘以 1 − r，其中 r 为百分数的小数形式。
Reverse percentage: if a value after a p% increase is given, the original = given value / (1 + p/100).
逆向百分数：已知增加 p% 后的值，则原始值 = 给定值 /（1 + p/100）。
4. Expanding and Factorising Algebraic Expressions | 代数展开与因式分解
Distributive law: a(b + c) = ab + ac and a(b − c) = ab − ac.
分配律：a(b + c) = ab + ac，a(b − c) = ab − ac。
Product of two binomials (FOIL): (x + a)(x + b) = x² + ax + bx + ab = x² + (a + b)x + ab.
两个二项式相乘（FOIL）：(x + a)(x + b) = x² + ax + bx + ab = x² + (a + b)x + ab。
Difference of two squares: (a + b)(a − b) = a² − b².
平方差公式：(a + b)(a − b) = a² − b²。
Perfect squares: (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b².
完全平方：(a + b)² = a² + 2ab + b²；(a − b)² = a² − 2ab + b²。
Factorising is the reverse of expanding. Always look for a common factor first, then consider quadratic forms such as x² + bx + c.
因式分解是展开的逆过程。先考虑提取公因式，再考虑形如 x² + bx + c 的二次三项式。
+ bx + c, find two numbers that multiply to c and add to b: x² + 5x + 6 = (x + 2)(x + 3).
因式分解 x² + bx + c 时，找到乘积为 c、和为 b 的两个数：x² + 5x + 6 = (x + 2)(x + 3)。
5. Solving Linear Equations | 解一元线性方程
operation on both sides to isolate the unknown. For example: 2x + 3 = 11 → 2x = 8 → x = 4.
解方程时，在等式两边同时进行相同的运算以解出未知数。例如：2x + 3 = 11 → 2x = 8 → x = 4。
f the unknown appears on both sides, collect like terms: 5x + 2 = 3x + 6 → 2x = 4 → x = 2.
若未知数出现在等式两边，先移项合并同类项：5x + 2 = 3x + 6 → 2x = 4 → x = 2。
Always check your solution by substituting it back into the original equation.
务必把解代入原方程检验。
6. Linear Inequalities | 一元一次不等式
Solving inequalities follows the same rules as equations, with one crucial difference: when multiplying or dividing by a negative number, reverse the inequality sign.
解不等式与解方程步骤相同，但有一个关键区别：当乘或除以一个负数时，必须反转不等号方向。
Example: −2x ≤ 6 → divide by −2 → x ≥ −3.
示例：−2x ≤ 6 → 除以 −2 → x ≥ −3。
an be represented on a number line: an open circle for < or >, a closed circle for ≤ or ≥.
不等式可以在数轴上表示：用空心圆表示 < 或 >，用实心圆表示 ≤ 或 ≥。
7. Simultaneous Linear Equations | 联立二元一次方程组
Two methods are used: elimination and substitution.
常用两种方法：消元法和代入法。
one variable. Example: 2x + y = 7 and x − y = 2 → adding gives 3x = 9 → x = 3, then y = 1.
消元法：将两个方程相加或相减以消去一个未知数。例：2x + y = 7 和 x − y = 2，相加得 3x = 9 → x = 3，再得 y = 1。
xample: y = 2x from first equation, substitute into x + y = 6 → x + 2x = 6 → x = 2, y = 4.
代入法：从一个方程中将一个未知数用另一个表示，再代入另一方程。例：由 y = 2x 代入 x + y = 6 → x + 2x = 6 → x = 2, y = 4。
8. Sequences: Finding the nth Term | 数列：求第 n 项
For a linear (arithmetic) sequence, the term‑to‑ th term can be written as: Tₙ = dn + b, where d is the common difference and b is the zero‑th term (T₀).
对于线性（等差）数列，相邻两项的差为常数。第 n 项可写成：Tₙ = dn + b，其中 d 为公差，b 为第零项（即 T₀）。
Example: sequence 5, 8, 11, 14, … common difference d = 3. T₀ = 2 (since T₁ = 5 = 3×1 + 2), so the nth term is Tₙ = 3n + 2.
例：数列 5, 8, 11, 14, … 的公差 d = 3，T₀ = 2（因为 T₁ = 5 = 3×1 + 2），故第 n 项为 Tₙ = 3n + 2。
For a quadratic sequence, the second differences are constant, but Year 9 only requires recognising simple patterns or deriving nth term for linear sequences.
对于二次数列，二阶差为常数，但 Year 9 阶段仅需识别简单规律或求线性数列的第 n 项。
9. Geometry: Angle Properties and Polygons | 几何：角的性质与多边形
Angles on a straight line add up to 180°.
平角等于180°。
Angles around a point add up to 360°.
周角等于360°。
Vertically opposite angles are equal.
对顶角相等。
Angles in a triangle sum to 180°.
三角形内角和为180°。
The sum of the interior angles of an n‑sided polygon is (n − 2) × 180°.
n 边形的内角和为 (n − 2) × 180°。
The sum of the exterior angles of any convex polygon is always 360°, irrespective of the number of sides.
任何凸多边形的外角和恒为360°，与边数无关。
For a regular n‑d polygon, each interior angle = [(n − 2) × 180°] / n, and each exterior angle = 360° / n.
对于正 n 边形，每个内角 = [(n − 2) × 180°] / n，每个外角 = 360° / n。
10. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角函数
In a right‑ ls the sum of the squares of the other two sides: c² = a² + b², where c is the hypotenuse.
在直角三角形中，斜边的平方等于两条直角边的平方和：c² = a² + b²（c 为斜边）。
The three trigonometric ratios for an acute angle θ in a right‑angled triangle are:
直角三角形中锐角 θ 的三个三角函数比：
sin θ = opposite / hypotenuse
正弦 sin θ = 对边 / 斜边
cos θ = adjacent / hypotenuse
余弦 cos θ = 邻边 / 斜边
tan θ = opposite / adjacent
正切 tan θ = 对边 / 邻边
To find a missing side or angle, you can use the appropriate ratio and, if necessary, the inverse trigonometric functions.
要求未知边长或角度，可选择合适的三角函数比，必要时使用反三角函数。
11. Perimeter, Area, and Volume | 周长、面积与体积
Below is a quick reference table for the perimeter and area of common 2D shapes, and the surface area and volume of 3D solids.
以下是常见二维图形的周长与面积，以及常见三维体的表面积与体积速查表。
| Shape 形状 | Perimeter / Circumference 周长 | Area 面积 |
|---|---|---|
Rectangle 矩形 | P = 2(l + w) P = 2(长 + 宽) | A = l × w A = 长 × 宽 |
Triangle 三角形 | P = a + b + c P = 三边之和 | A = 1/2 × base × height A = 1/2 × 底 × 高 |
Parallelogram 平行四边形 | P = 2(a + b) and a, b are adjacent sides P = 2(a + b)，a,b为邻边 | A = base × height A = 底 × 高 |
Trapezium 梯形 | P = sum of all four sides P Published by TutorHao | Year 9 Mathematics Revision Series | aleveler.com 广告 / Sponsored 【现货】剑桥国际高中教材 Cambridge IGCSE & O Level Additional Mathemati… 数学 备考教辅 · 正版现货 · 与本文内容相关 查看详情 / View on Taobao参考价 ¥198.0更多咨询请联系16621398022（同微信） 备考资讯每日更新，关注不错过 考纲变化 · 科目分析 · 学习方法  在微信里长按二维码识别关注 或截图后用「扫一扫 → 相册」选择本图 公众号：tutorhao |