📚 Year 8 Edexcel Mathematics: Formula & Theorem Quick Revision Guide | Year 8 Edexcel 数学：公式定理速查手册
This quick revision guide compiles the essential formulas, rules, and theorems you need for the Year 8 Edexcel Mathematics curriculum. Use it to check your understanding of key topics before tests and exams.
本速查手册汇总了 Year 8 Edexcel 数学课程中必须掌握的公式、法则和定理，供你在测验和考试前快速回顾和查漏补缺。
1. Number Properties and Operations | 数的性质与运算
Addition and multiplication are commutative: a + b = b + a and a × b = b × a.
加法和乘法满足交换律：a + b = b + a 且 a × b = b × a。
Addition and multiplication are associative: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
加法和乘法满足结合律：(a + b) + c = a + (b + c) 且 (a × b) × c = a × (b × c)。
Multiplication distributes over addition: a × (b + c) = a × b + a × c. This is the distributive law.
乘法对加法满足分配律：a × (b + c) = a × b + a × c。
When multiplying or dividing two numbers with the same sign, the answer is positive. When the signs differ, the answer is negative.
两个同号数相乘或相除，结果为正；异号数相乘或相除，结果为负。
2. Fractions, Decimals, and Percentages | 分数、小数和百分比
To convert a fraction to a decimal, divide the numerator by the denominator. For example, 3/8 = 0.375.
分数化小数：用分子除以分母。例如 3/8 = 0.375。
To convert a decimal to a fraction, write the decimal as the numerator over a power of 10 (e.g. 0.6 = 6/10) and simplify.
小数化分数：以 10 的幂为分母写出分数（如 0.6 = 6/10），再约分。
To increase a value by r%, multiply by (1 + r/100). To decrease by r%, multiply by (1 – r/100).
增加 r%，乘以 (1 + r/100)；减少 r%，乘以 (1 – r/100)。
3. Laws of Indices | 指数定律
When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。
When dividing powers with the same base, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (a ≠ 0).
同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ（a ≠ 0）。
When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ.
幂的乘方，指数相乘：(aᵐ)ⁿ = aᵐⁿ。
Any non-zero number raised to the power of 0 is 1: a⁰ = 1.
任何非零数的 0 次幂均为 1：a⁰ = 1。
A negative exponent indicates the reciprocal: a⁻ⁿ = 1 / aⁿ.
负指数表示倒数：a⁻ⁿ = 1 / aⁿ。
4. Algebraic Expressions and Equations | 代数表达式与方程式
To expand a bracket, multiply the term outside by each term inside: a(b + c) = ab + ac.
去括号时，将括号外的项乘以括号内的每一项：a(b + c) = ab + ac。
Factorising is the reverse of expanding. Look for a common factor: ab + ac = a(b + c).
因式分解是展开的逆过程，提取公因式：ab + ac = a(b + c)。
To solve a linear equation like ax + b = c, perform inverse operations in reverse order. Keep the balance by doing the same to both sides.
解形如 ax + b = c 的一次方程，按照逆运算顺序进行，并在等号两边同时操作以保持平衡。
When solving inequalities, reverse the inequality sign if you multiply or divide by a negative number.
解不等式时，如果两边同时乘以或除以一个负数，不等号方向要反转。
5. Linear Graphs and Slope | 线性图形与斜率
The equation of a straight line is y = mx + c, where m is the slope (gradient) and c is the y-intercept.
直线方程为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
The slope m between two points (x₁, y₁) and (x₂, y₂) is calculated as: m = (y₂ − y₁) / (x₂ − x₁).
两点 (x₁, y₁) 和 (x₂, y₂) 间的斜率公式：m = (y₂ − y₁) / (x₂ − x₁)。
Parallel lines have equal slopes.
平行线斜率相等。
y = mx + c
6. Angles and Polygons | 角度与多边形
Vertically opposite angles are equal. Angles on a straight line add up to 180°.
对顶角相等；平角（直线上的角）之和为 180°。
When a transversal crosses parallel lines, corresponding angles are equal, alternate interior angles are equal, and co-interior angles sum to 180°.
平行线被截线所截时，同位角相等、内错角相等、同旁内角互补（和为 180°）。
The sum of interior angles of a triangle is 180°. For any polygon with n sides, the sum of interior angles is (n − 2) × 180°.
三角形内角和为 180°；n 边形内角和为 (n − 2) × 180°。
The sum of exterior angles of any convex polygon is always 360°.
任意凸多边形的外角和均为 360°。
7. Perimeter, Area, and Volume | 周长、面积和体积
Below are common area and perimeter formulas. For π, use 3.14 or the π button on your calculator.
以下是常用面积与周长公式。圆周率 π 取 3.14 或使用计算器上的 π 键。
| Shape 形状 | Perimeter / Circumference 周长 | Area 面积 |
|---|---|---|
| Rectangle 矩形 (l × w) | P = 2(l + w) | A = l × w |
| Triangle 三角形 (b, h) | Sum of sides | A = ½ × b × h |
| Parallelogram 平行四边形 (b, h) | Sum of sides | A = b × h |
| Trapezium 梯形 (a, b parallel, h) | Sum of sides | A = ½(a + b) × h |
| Circle 圆 (r) | C = 2πr or πd | A = πr² |
The volume of a prism is found by multiplying the area of its cross-section by its length (height).
棱柱体积 = 底面积 × 高（长度）。
Vprism = Area of cross‑section × length
8. Pythagoras’ Theorem | 勾股定理
In a right-angled triangle, the square of the hypotenuse (c, the longest side) equals the sum of the squares of the other two sides: a² + b² = c².
在直角三角形中，斜边（最长边）的平方等于两条直角边的平方和：a² + b² = c²。
You can rearrange to find a shorter side: a = √(c² − b²). Always check that the triangle is right‑angled before applying the theorem.
可通过变形求直角边：a = √(c² − b²)。使用定理前务必确认三角形为直角三角形。
a² + b² = c²
9. Ratio and Proportion | 比例与比率
To simplify a ratio, divide all parts by their highest common factor. For example, 8:12 simplifies to 2:3.
化简比：将比的各项除以它们的最大公因数。例如 8:12 化简为 2:3。
When sharing a quantity in a given ratio, add the ratio parts to find the total number of shares, then divide accordingly.
按比例分配时，先求出比例的总份数，再用总量除以总份数，最后乘以各项份数。
Two quantities are in direct proportion if their ratio stays constant: y = kx, where k is the constant of proportionality.
两个量成正比例时，它们的比值保持不变：y = kx，其中 k 为比例常数。
10. Statistics and Probability | 统计与概率
The mean (average) is calculated by adding all data values and dividing by the number of values. Median is the middle value when data are ordered. Mode is the most frequent value. Range = highest value − lowest value.
平均数（均值）是所有数据之和除以数据个数。中位数是排序后位于中间的数值。众数是出现次数最多的数值。极差 = 最大值 − 最小值。
The probability of an event is: Number of favourable outcomes / Total number of possible outcomes. Probabilities always lie between 0 (impossible) and 1 (certain).
事件发生的概率 = 有利结果数 / 所有可能结果数。概率始终在 0（不可能）到 1（必然）之间。
The sum of probabilities of all mutually exclusive outcomes of an experiment is 1.
互斥事件的概率之和为 1。
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