📚 Formula and Theorem Quick Reference Handbook | 公式定理速查手册
This Year 9 Edexcel Maths quick reference guide brings together all the essential formulas, theorems and rules you will encounter in Key Stage 3. Use it for daily revision, homework help and last-minute test preparation. Every entry is stated clearly in English and Chinese so you can learn the language of mathematics in both directions.
这本九年级 Edexcel 数学速查手册汇集了关键阶段 3 所有重要的公式、定理和法则，可用于日常复习、作业辅导和考前突击。每个条目都以中英双语清晰表述，帮助你双向掌握数学语言。
1. Angle Laws and Polygon Theorems | 角定律与多边形定理
Angles on a straight line add up to 180°. Angles around a point sum to 360°. Vertically opposite angles are equal.
平角为 180°，周角为 360°，对顶角相等。
For parallel lines: alternate angles are equal, corresponding angles are equal, and co-interior (allied) angles sum to 180°.
平行线中，内错角相等，同位角相等，同旁内角之和为 180°。
olygon is (n - 2) × 180°. The sum of exterior angles of any convex polygon is always 360°.
n 边形的内角和为 (n - 2) × 180°；任意凸多边形的外角和总是 360°。
Interior angle sum = (n − 2) × 180°
2. Pythagoras' Theorem | 毕达哥拉斯定理
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
在直角三角形中，斜边的平方等于两直角边的平方和。
a² + b² = c²
Here c represents the length of the hypotenuse, while a and b are the legs. Always check that the triangle has a right angle before using this theorem.
其中 c 代表斜边长，a 和 b 为直角边。使用前务必确认三角形含直角。
The converse is also true: if a² + b² = c² holds for a triangle, then it is right-angled.
逆定理同样成立：若三角形的三边满足 a² + b² = c²，则该三角形为直角三角形。
3. Area, Perimeter and Volume | 面积、周长与体积
parallelogram A = b × h, trapezium A = ½ (a + b) h, where a and b are the parallel sides.
常用面积公式：矩形 A = 长 × 宽，三角形 A = ½ × 底 × 高，平行四边形 A = 底 × 高，梯形 A = ½ (上底 + 下底) × 高。
| Rectangle / 矩形 | A = l × w |
| Triangle / 三角形 | A = ½ b h |
| Parallelogram / 平行四边形 | A = b h |
| Trapezium / 梯形 | A = ½ (a + b) h |
of a prism: V = area of cross-section × length. Surface area of a cuboid: 2(lw + lh + wh).
长方体体积 V = 长 × 宽 × 高；棱柱体积 V = 横截面积 × 长；长方体表面积 S = 2(长×宽 + 长×高 + 宽×高)。
4. Circle Facts | 圆的相关知识
Circumference of a circle: C = πd or C = 2πr, where d is the diameter and r is the radius.
圆的周长：C = πd 或 C = 2πr，其中 d 为直径，r 为半径。
button on your calculator for accuracy, or approximate π as 3.14 or 22/7 where instructed.
圆的面积：A = πr²。计算时尽量使用计算器上的 π 键以提高精度，若题目要求可近似为 3.14 或 22/7。
C = 2πr A = πr²
Key terms: radius – distance from centre to edge; diameter – distance across the circle through the centre (d = 2r); arc – part of the circumference; sector – 'pizza slice' region bounded by two radii and an arc.
关键术语：半径 – 圆心到圆周的距离；直径 – 穿过圆心的弦（d = 2r）；弧 – 圆周的一部分；扇形 – 由两条半径和一条弧围成的“披萨切片”。
5. Fractions, Decimals and Percentages | 分数、小数与百分比
To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a percentage, multiply by 100 and add the % sign.
分数化小数：分子除以分母。小数化百分数：乘以 100 并添加 % 符号。
Percentage of an amount: x% of A = (x/100) × A. Percentage change = (change ÷ original) × 100%.
求一个数的百分之几：x% of A = (x/100) × A。百分数变化 = (变化量 ÷ 原量) × 100%。
Increase A by p%: A × (1 + p/100). Decrease A by p%: A × (1 − p/100).
增加 p%：A × (1 + p/100)；减少 p%：A × (1 − p/100)。
Ordering fractions, decimals and percentages is easier when you convert them all to the same form, often decimals.
比较分数、小数和百分比大小时，先将它们转化为同一种形式（通常化为小数）会更方便。
6. Ratio and Proportion | 比与比例
To share an amount in a given ratio, first find the total number of parts. Divide the amount by that total to find the value of one part, then multiply by each share's parts.
按给定比例分配数量时，先计算总份数，用总量除以总份数得到一份的量，再乘以各自的份数。
roportional, y = kx; the ratio y/x is constant. For inverse proportion, y = k/x or xy = k.
两个量成正比例时：y = kx，比值 y/x 为定值。成反比例时：y = k/x 或 xy = k。
p. Maps and scale drawings use scale factor = drawing length / actual length (same units).
归一法：先求出一份的量，再推至所需份数。地图和比例尺图的缩放因子 = 图纸长度 ÷ 实际长度（单位统一）。
7. Powers and Roots | 幂与根
Squaring: a² = a × a. Square root: √a is the non‑negative number whose square is a. Cubing: a³ = a × a × a. Cube root: ³√a³ = a.
平方：a² = a × a；平方根：√a 表示平方等于 a 的非负数。立方：a³ = a × a × a；立方根：³√a³ = a。
aᵐ × aⁿ = aᵐ⁺ⁿ (aᵐ)ⁿ = aᵐⁿ
dex laws: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. Also a⁰ = 1 (a ≠ 0) and a⁻ⁿ = 1/aⁿ.
指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ。同时 a⁰ = 1（a ≠ 0），a⁻ⁿ = 1/aⁿ。
a × 10ⁿ where 1 ≤ a < 10 and n is an integer. Useful for very large or very small numbers.
标准形式：a × 10ⁿ，其中 1 ≤ a < 10，n 为整数，常用于表示很大或很小的数。
8. Expanding and Factorising | 代数展开与因式分解
stributive law: a(b + c) = ab + ac. For two binomials: (a + b)(c + d) = ac + ad + bc + bd.
去括号运用乘法分配律：a(b + c) = ab + ac。二项式相乘：(a + b)(c + d) = ac + ad + bc + bd。
al products: (a + b)² = a² + 2ab + b², (a − b)² = a² − 2ab + b², (a + b)(a − b) = a² − b².
特殊乘积：(a + b)² = a² + 2ab + b²，(a − b)² = a² − 2ab + b²，(a + b)(a − b) = a² − b²。
Factorising is the reverse of expanding. Always look for the highest common factor first. For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b.
因式分解是展开的逆过程。首先提取最大公因数。对于 x² + bx + c 型的二次式，寻找两个数，其积为 c、和为 b。
9. Solving Equations | 解方程
To solve a linear equation, perform the same operation on both sides to isolate the variable. Always check your answer by substitution.
解一元一次方程时，通过对方程两边施以相同运算来分离变量。务必代入检验。
Example: 3x + 7 = 19 → subtract 7: 3x = 12 → divide by 3: x = 4.
例如：3x + 7 = 19 → 两边减 7：3x = 12 → 两边除以 3：x = 4。
For equations with brackets, expand first or divide if possible. For equations containing fractions, multiply every term by the lowest common denominator.
含括号的方程可先展开或先尽可能相除；含分数的方程则各项乘以最简公分母。
ratics to solve: if x² + 5x + 6 = 0, factorise as (x + 2)(x + 3) = 0, so x = −2 or x = −3.
通过因式分解解二次方程：如 x² + 5x + 6 = 0，分解为 (x + 2)(x + 3) = 0，故 x = −2 或 x = −3。
10. Straight Line Graphs | 直线图形
is the gradient (steepness), and c is the y-intercept (where the line crosses the y-axis).
直线方程通常写作 y = mx + c。m 为斜率（坡度），c 为 y 轴截距（直线与 y 轴交点的纵坐标）。
m = (change in y) / (change in x) = (y₂ − y₁) / (x₂ − x₁)
To plot a straight line, create a table of x-values, calculate the corresponding y-values, plot the points and draw the line.
画直线图形时，先列出 x 值的表格，计算相应的 y 值，描点并连线。
the same gradient. Perpendicular lines have gradients whose product is −1 (m₁ × m₂ = −1).
平行线的斜率相等；垂线的斜率满足 m₁ × m₂ = −1。
11. Statistics and Probability | 统计与概率
en data is ordered. Mode is the most frequent value. Range = highest value − lowest value.
平均数 = 所有数值之和 ÷ 数据个数。中位数是排序后位于中间的数值。众数是出现次数最多的值。极差 = 最大值 − 最小值。
avourable outcomes) ÷ (total number of possible outcomes), assuming equally likely outcomes.
事件的概率 = （有利结果数）÷（所有可能结果总数），前提是每个结果出现的可能性相同。
Probability scale: 0 (impossible) to 1 (certain). The sum of probabilities of all mutually exclusive outcomes is 1.
概率尺度范围从 0（不可能）到 1（必然发生）。所有互斥事件概率之和为 1。
In sample spaces and tree diagrams, multiply along branches for combined events, and add branch probabilities for 'or' statements.
在样本空间和树状图中，沿分支相乘求复合事件的概率，相加各分支概率求“或”事件的概率。
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