📚 Formula and Theorem Quick Reference Guide | 公式定理速查手册
This bilingual quick reference guide covers the most essential formulas and theorems from Year 10 mathematics, presented in both English and Chinese. It is designed to help OCR Chinese learners familiarise themselves with key mathematical terminology in Chinese, while also serving as a handy revision tool for core concepts. Each entry includes the standard formula, a brief explanation, and its Chinese equivalent so you can talk about maths confidently in two languages.
这本双语速查手册涵盖了 Year 10 数学中最关键的公式和定理，以中英双语呈现。它旨在帮助 OCR 中文学习者熟悉数学核心术语的中文表达，同时也是一份实用的知识点复习工具。每个条目都包含标准公式、简要说明及其中文对应表达，让你能够自信地用两种语言讨论数学。
1. Pythagoras’ Theorem | 勾股定理
In any right‑angled triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides. Written as c² = a² + b², where c represents the hypotenuse and a, b are the two shorter sides.
在任何直角三角形中，斜边（最长边）的平方等于另外两条直角边的平方和。公式写作 c² = a² + b²，其中 c 代表斜边，a 和 b 是两条直角边。
Chinese pronunciation: c píngfāng děngyú a píngfāng jiā b píngfāng.
中文读法：c 平方等于 a 平方加 b 平方。
2. Area Formulas | 面积公式
Square: A = s², where s is the side length. Rectangle: A = l × w, where l is length and w is width. Triangle: A = ½ bh, where b is base and h is height. Circle: A = πr², where r is the radius. Parallelogram: A = bh. Trapezium: A = ½ (a + b) h, where a and b are the lengths of the parallel sides.
正方形：A = s²，s 为边长。长方形：A = l × w，l 为长，w 为宽。三角形：A = ½ bh，b 为底，h 为高。圆：A = πr²，r 为半径。平行四边形：A = bh。梯形：A = ½ (a + b) h，a 和 b 为两条平行边的长度。
Chinese vocabulary: miànjī (area), dǐ (base), gāo (height), cháng (length), kuān (width), bànjìng (radius), píngxíng sìbiānxíng (parallelogram), tīxíng (trapezium).
中文词汇：面积、底、高、长、宽、半径、平行四边形、梯形。
3. Volume Formulas | 体积公式
Cube: V = s³, where s is the edge length. Cuboid: V = l × w × h. Cylinder: V = πr²h, where r is the radius of the base and h is the height. Cone: V = ⅓ πr²h. Sphere: V = ⁴⁄₃ πr³. Prism: V = area of cross‑section × length. Pyramid: V = ⅓ × base area × vertical height.
立方体：V = s³，s 为棱长。长方体：V = l × w × h。圆柱：V = πr²h，r 为底面半径，h 为高。圆锥：V = ⅓ πr²h。球：V = ⁴⁄₃ πr³。棱柱：V = 横截面积 × 长度。棱锥：V = ⅓ × 底面积 × 垂直高度。
Key terms: tǐjī (volume), léng cháng (edge length), yuán zhù (cylinder), yuán zhuī (cone), qiú (sphere), léng zhù (prism), léng zhuī (pyramid).
关键术语：体积、棱长、圆柱、圆锥、球、棱柱、棱锥。
4. Quadratic Formula | 二次公式
For a quadratic equation of the form ax² + bx + c = 0, where a ≠ 0, the solutions are given by: x = [⁻b ± √(b² − 4ac)] / (2a). The expression under the square root, Δ = b² − 4ac, is called the discriminant.
对于形如 ax² + bx + c = 0（a ≠ 0）的一元二次方程，解为：x = [⁻b ± √(b² − 4ac)] / (2a)。根号下的式子 Δ = b² − 4ac 被称为判别式。
Chinese statement: èr cì fāngchéng de qiúgēn gōngshì: x děngyú èr a fēnzhī fù b jiā jiǎn gēn hào b píngfāng jiǎn sì a c.
中文表述：二次方程的求根公式：x 等于 2a 分之 ⁻b 加减根号 b² − 4ac。
5. Trigonometric Ratios | 三角函数比
For a right‑angled triangle, the three primary trigonometric ratios are defined as: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. These are often remembered using the acronym SOH CAH TOA.
对于直角三角形，三个基本三角比定义为：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。常借助口诀 “SOH CAH TOA” 记忆。
Chinese terms: zhèngxián (sine), yúxián (cosine), zhèngqiē (tangent), duìbiān (opposite side), línbiān (adjacent side), xiébiān (hypotenuse).
中文术语：正弦、余弦、正切、对边、邻边、斜边。
6. Sine Rule and Cosine Rule | 正弦定理与余弦定理
For any triangle with sides a, b, c and opposite angles A, B, C: Sine Rule: a / sin A = b / sin B = c / sin C. Cosine Rule: a² = b² + c² − 2bc cos A. These rules are used to solve non‑right‑angled triangles.
对于任意三角形，边长为 a, b, c，对角为 A, B, C：正弦定理：a / sin A = b / sin B = c / sin C。余弦定理：a² = b² + c² − 2bc cos A。这两个定理用于解斜三角形。
Chinese names: zhèngxián dìnglǐ (Sine Rule), yúxián dìnglǐ (Cosine Rule). To read aloud: a bǐ sin A děngyú b bǐ sin B děngyú c bǐ sin C.
中文名称：正弦定理、余弦定理。读法：a 比 sin A 等于 b 比 sin B 等于 c 比 sin C。
7. Arithmetic Series | 等差数列
An arithmetic sequence has a common difference d. The nth term is given by uₙ = a + (n − 1)d, where a is the first term. The sum of the first n terms is Sₙ = n/2 [2a + (n − 1)d] or Sₙ = n/2 (a + l), where l is the last term.
等差数列有公差 d。第 n 项为 uₙ = a + (n − 1)d，其中 a 是首项。前 n 项和为 Sₙ = n/2 [2a + (n − 1)d] 或 Sₙ = n/2 (a + l)，l 为末项。
Vocabulary: děngchà shùliè (arithmetic sequence), gōngchā (common difference), qián n xiàng hé (sum of the first n terms), shǒuxiàng (first term), mòxiàng (last term).
词汇：等差数列、公差、前 n 项和、首项、末项。
8. Geometric Series | 等比数列
A geometric sequence has a common ratio r (r ≠ 0). The nth term is uₙ = a rⁿ⁻¹. The sum of the first n terms is Sₙ = a(1 − rⁿ) / (1 − r) for r ≠ 1. When |r| < 1, the infinite sum converges to S∞ = a / (1 − r).
等比数列有公比 r（r ≠ 0）。第 n 项为 uₙ = a rⁿ⁻¹。前 n 项和为 Sₙ = a(1 − rⁿ) / (1 − r)（r ≠ 1）。当 |r| < 1 时，无穷级数收敛于 S∞ = a / (1 − r)。
Key words: děngbǐ shùliè (geometric sequence), gōngbǐ (common ratio), wúqióng jíshù (infinite series).
关键词：等比数列、公比、无穷级数。
9. Circle Theorems | 圆定理
Some fundamental circle theorems: (1) The angle at the centre is twice the angle at the circumference subtended by the same arc. (2) The angle in a semicircle is a right angle. (3) Angles in the same segment are equal. (4) The opposite angles of a cyclic quadrilateral sum to 180°.
一些基本圆定理：（1）同弧所对的圆心角是圆周角的两倍。（2）半圆所对的圆周角是直角。（3）同弧上的圆周角相等。（4）圆内接四边形的对角和为 180°。
Chinese expressions: yuánxīn jiǎo (angle at the centre), yuánzhōu jiǎo (angle at the circumference), bàn yuán (semicircle), tóng hú (same arc), yuán nèi jiē sìbiānxíng (cyclic quadrilateral).
中文表达：圆心角、圆周角、半圆、同弧、圆内接四边形。
10. Probability Formulas | 概率公式
The probability of an event A is P(A) = number of favourable outcomes / total number of outcomes. For mutually exclusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B). The complement rule: P(not A) = 1 − P(A).
事件 A 的概率为 P(A) = 有利结果数 / 总结果数。对于互斥事件，P(A 或 B) = P(A) + P(B)。对于独立事件，P(A 与 B) = P(A) × P(B)。补集法则：P(非 A) = 1 − P(A)。
Chinese terms: gàilǜ (probability), shìjiàn (event), hùchì (mutually exclusive), dúlì (independent), bǔjí (complement).
中文术语：概率、事件、互斥、独立、补集。
11. Kinematics Formulas | 运动学公式
For motion in a straight line with constant acceleration: v = u + at; s = ut + ½ at²; v² = u² + 2as; s = ½ (u + v) t, where u = initial velocity, v = final velocity, a = acceleration, s = displacement, t = time.
对于匀加速直线运动：v = u + at；s = ut + ½ at²；v² = u² + 2as；s = ½ (u + v) t，其中 u 为初速度，v 为末速度，a 为加速度，s 为位移，t 为时间。
Key vocabulary: chū sùdù (initial velocity), mò sùdù (final velocity), jiāsùdù (acceleration), wèiyí (displacement), shíjiān (time).
关键用词：初速度、末速度、加速度、位移、时间。
12. Standard Index Laws | 指数运算法则
The laws of indices are essential for algebraic manipulation: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1 / aⁿ; a¹/ⁿ = ⁿ√a.
指数法则是代数运算的基础：aᵐ × aⁿ = aᵐ⁺ⁿ；aᵐ ÷ aⁿ = aᵐ⁻ⁿ；(aᵐ)ⁿ = aᵐⁿ；a⁰ = 1（a ≠ 0）；a⁻ⁿ = 1 / aⁿ；a¹/ⁿ = ⁿ√a。
Chinese reading: zhǐshù fǎzé. For example, a de m cìfāng chéng a de n cìfāng děngyú a de m jiā n cìfāng.
中文读法：指数法则。如：a 的 m 次方乘 a 的 n 次方等于 a 的 m + n 次方。
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