📚 Year 10 Edexcel Mathematics: Quick-Reference Formula & Theorem Handbook | Year 10 Edexcel 数学：公式定理速查手册
This quick-reference handbook brings together all the essential formulas and theorems you will need for Year 10 Edexcel Mathematics. Use it to revise, check your understanding, and build confidence before tests. Every section pairs English explanations with Chinese translations to support EAL learners and deepening comprehension.
这本速查手册汇集了 Year 10 Edexcel 数学所需的所有核心公式和定理。可用于复习、自查并在考试前建立信心。每个小节都配有中英文对照解释，帮助 EAL 学生深入理解。
1. Algebraic Expressions & Manipulation | 代数表达式与运算
When expanding brackets, multiply each term inside by the term outside. The same principle applies when multiplying two binomials using the FOIL method.
去括号时，将括号外的项乘以括号内的每一项。利用 FOIL 法则展开两个二项式时原理相同。
- (a + b)(c + d) = ac + ad + bc + bd
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
- (a + b)(a - b) = a² - b²
Factorising is the reverse process of expanding. Always look for the highest common factor first, then recognise special patterns such as difference of two squares.
因式分解是展开的逆过程。首先提取最大公因式，然后识别特殊模型，如平方差公式。
x² - y² = (x + y)(x - y)
2. Linear Equations & Straight-Line Graphs | 直线方程与图像
t line is usually written as y = mx + c, where m is the gradient and c is the y-intercept.
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距。
To find the gradient between two points (x₁, y₁) and (x₂, y₂), use the formula:
求两点 (x₁, y₁) 与 (x₂, y₂) 之间的斜率，使用公式：
m = (y₂ - y₁) / (x₂ - x₁)
pendicular lines have gradients that are negative reciprocals of each other: m₁ × m₂ = -1.
平行线斜率相等。互相垂直的直线斜率互为负倒数：m₁ × m₂ = -1。
3. Quadratic Equations & Functions | 二次方程与函数
utions can be found by factorising, completing the square, or using the quadratic formula.
二次方程的标准形式为 ax² + bx + c = 0。可以通过因式分解、配方法或求根公式求解。
x = [-b ± √b² - 4ac] / (2a)
two distinct real roots, Δ = 0 gives one repeated root, and Δ < 0 gives no real roots.
判别式 Δ = b² - 4ac 决定了根的情况：Δ > 0 时有两个不等实根，Δ = 0 时有一个重根，Δ < 0 时无实根。
For a quadratic in completed square form y = a(x - h)² + k, the vertex is at (h, k).
对于顶点式 y = a(x - h)² + k，顶点坐标为 (h, k)。
4. Pythagoras' Theorem & Trigonometry | 毕达哥拉斯定理与三角学
Pythagoras' theorem only applies to right-angled triangles: the square of the hypotenuse is equal to the sum of the squares of the other two sides.
毕达哥拉斯定理仅适用于直角三角形：斜边的平方等于两直角边的平方和。
a² + b² = c²
The three trigonometric ratios for a right-angled triangle are defined as:
直角三角形的三个三角比定义如下：
sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent
To remember these, use the mnemonic SOH CAH TOA. When finding an angle, you use the inverse functions sin⁻¹, cos⁻¹ and tan⁻¹.
可用记忆口诀 SOH CAH TOA。求角度时需使用反函数 sin⁻¹、cos⁻¹ 和 tan⁻¹。
The sine rule and cosine rule are used for non-right-angled triangles, but these are introduced in later stages. In Year 10 you may begin using basic trigonometric methods.
正弦定理与余弦定理适用于非直角三角形，会在后续阶段学习。Year 10 可初步接触基础三角方法。
5. Area, Perimeter & Volume | 面积、周长与体积
Here are the essential formulas for 2D shapes and 3D solids. All measurements should be in consistent units.
以下是二维图形和三维立体的必备公式。所有测量值应使用统一单位。
| Shape/形状 | Area/Volume Formula/公式 |
| Rectangle/矩形 | Area = length × width |
| Triangle/三角形 | Area = ½ × base × height |
| Parallelogram/平行四边形 | Area = base × perpendicular height |
| Trapezium/梯形 | Area = ½ (a + b) × h |
| Circle/圆 | Area = πr² Circumference = 2πr 或 πd |
| Sector/扇形 | Area = (θ/360) × πr² Arc length = (θ/360) × 2πr |
| Prism/棱柱 | Volume = area of cross-section × length |
| Cylinder/圆柱 | Volume = πr²h Curved surface area = 2πrh |
he volume is 1/3 × base area × height. These are often tested in problem-solving contexts.
请记住，圆锥或棱锥的体积 = 1/3 × 底面积 × 高。这类题目常在应用题中出现。
6. Circle Theorems | 圆的定理
Circle theorems describe angle relationships within a circle. You must be able to state them and apply them to find missing angles.
圆的定理描述了圆内角与弧的关系。你需要能够陈述这些定理并用于求未知角。
- The angle at the centre is twice the angle at the circumference subtended by the same arc.
- 同弧所对的圆心角是圆周角的两倍。
- Angles in the same segment are equal.
- 同弦上的圆周角相等。
- The angle in a semicircle is always 90°.
- 半圆内的圆周角恒为 90°。
- Opposite angles in a cyclic quadrilateral sum to 180°.
- 圆内接四边形的对角之和为 180°。
- The tangent at any point on a circle is perpendicular to the radius at that point.
- 圆的切线垂直于过切点的半径。
- From an external point, two tangents drawn to a circle are equal in length.
- 从圆外一点引圆的两条切线长度相等。
Using these theorems together with basic angle facts (angles on a straight line sum to 180°, angles around a point sum to 360°) is a key skill.
将这些定理与基本角度事实（直线上的角之和为 180°，一点周围的角之和为 360°）结合使用是一项关键技能。
7. Ratios, Percentages & Proportionality | 比、百分比与比例
To share a quantity in a given ratio, first add the parts to find the total number of shares, then divide.
按给定比例分配一个量时，先将各项相加得到总份数，再分配。
Percentage change is calculated as: (change ÷ original) × 100%. For repeated percentage changes, use a multiplier.
百分比变化计算为：（变化量 ÷ 原值）× 100%。涉及多次百分比变化时，使用乘数。
Percentage increase: New = Original × (1 + r)
Percentage decrease: New = Original × (1 - r)
inverse proportion means y = k/x. The constant k can be found from given pairs of values.
正比例表示 y = kx；反比例表示 y = k/x。比例常数 k 可通过已知数值对求出。
When comparing ratios or proportions, always simplify to the form 1 : n or n : 1 where helpful.
比较比例时，常化简为 1 : n 或 n : 1 的形式。
8. Probability | 概率
The probability of an event A is found by dividing the number of favourable outcomes by the total number of possible outcomes, provided all outcomes are equally likely.
事件 A 的概率等于有利结果数除以所有可能结果总数，前提是所有结果等可能。
P(A) = number of favourable outcomes / total number of outcomes
Probability always lies between 0 and 1 inclusive. The sum of probabilities of all mutually exclusive outcomes is 1.
概率值始终在 0 到 1 之间（含）。所有互斥结果的概率之和为 1。
events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B).
对于独立事件，P(A 与 B) = P(A) × P(B)。对于互斥事件，P(A 或 B) = P(A) + P(B)。
Tree diagrams help organise outcomes in multistage experiments. Multiply along branches for 'and', and add the probabilities of the relevant branches for 'or'.
树形图有助于整理多步试验的结果。沿分支相乘求“与”的概率，将所需分支概率相加求“或”的概率。
9. Statistics – Averages & Spread | 统计 – 平均数与离差
The three common averages are the mean, median and mode. The mean is the sum of all values divided by the number of values.
三种常见的平均数是平均数、中位数和众数。平均数 = 所有数值之和 ÷ 数据个数。
Mean = Σx / n
The median is the middle value when data is ordered. If there are two middle values, the median is their average. The mode is the most frequently occurring value.
中位数是排序后处于中间位置的数值。若有两个中间值，则中位数为两者的平均数。众数是出现次数最多的数值。
ghest value - lowest value. Other measures like interquartile range are studied in depth but the range is the simplest measure of variability.
极差衡量数据分散程度：极差 = 最大值 - 最小值。其他如四分位距会更深入学习，但极差是变异性最简单度量。
From a frequency table, estimated mean = Σ(f × midpoint) / Σf.
对于频数表，估算平均数 = Σ（频数 × 组中值）÷ 总频数。
10. Compound Measures & Conversions | 复合单位与换算
Speed, density and pressure are common compound measures that combine two units.
速度、密度和压强是常见的复合单位，结合了两个基本单位。
Speed = Distance ÷ Time
Density = Mass ÷ Volume
Pressure = Force ÷ Area
e In each case, rearranging the formula is essential. For example, Distance = Speed × Time, and Time = Distance ÷ Speed.
每种情况下，公式变形都很关键。例如：路程 = 速度 × 时间，时间 = 路程 ÷ 速度。
0 m, 1 m = 100 cm, 1 cm = 10 mm, 1 tonne = 1000 kg, 1 litre = 1000 ml, 1 litre = 1000 cm³.
常用公制换算：1 千米 = 1000 米，1 米 = 100 厘米，1 厘米 = 10 毫米，1 吨 = 1000 千克，1 升 = 1000 毫升，1 升 = 1000 立方厘米。
ilo- (×1000), centi- (×0.01), milli- (×0.001). This will help in converting units quickly.
记住前缀：千（×1000），厘（×0.01），毫（×0.001）。这会有助于快速换算单位。
11. Angles & Polygon Properties | 角度与多边形性质
polygon is (n - 2) × 180°. Each interior angle of a regular polygon is (n - 2) × 180° / n.
n 边形的内角和为 (n - 2) × 180°。正多边形每个内角大小为 (n - 2) × 180° / n。
s of any polygon always sum to 360°. In a regular polygon, each exterior angle = 360° / n.
任意多边形的外角和恒为 360°。正多边形的每个外角 = 360° / n。
Angles on a straight line sum to 180°, angles at a point sum to 360°, and vertically opposite angles are equal.
直线上的邻角之和为 180°，一点周围的周角之和为 360°，对顶角相等。
12. Index Laws & Standard Form | 指数法则与标准形式
When multiplying powers with the same base, add the indices. When dividing, subtract the indices. Raising a power to a power means multiplying indices.
同底数幂相乘，指数相加。相除时指数相减。幂的乘方等于指数相乘。
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ
ny non-zero number to the power 0 is 1. A negative index means the reciprocal: a⁻ⁿ = 1/aⁿ.
任何非零数的 0 次幂为 1。负指数表示倒数：a⁻ⁿ = 1/aⁿ。
10ⁿ where 1 ≤ A < 10 and n is an integer. It is widely used in science and engineering.
标准形式将很大或很小的数写为 A × 10ⁿ，其中 1 ≤ A < 10，n 为整数。广泛用于科学和工程领域。
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