📚 IGCSE Edexcel Maths: Complete Formula Handbook | IGCSE Edexcel 数学：公式汇总手册
This handbook compiles all the essential formulas required for the IGCSE Edexcel Mathematics syllabus. It is organised by topic to help you revise efficiently and ensure you are fully prepared for your exams.
本手册汇总了 IGCSE Edexcel 数学课程所需的所有重要公式，并按主题分类，帮助你高效复习，为考试做好充分准备。
1. Number | 数
Percentage change compares the difference to the original amount.
Percentage change = (difference / original) × 100
百分比变化是用差值除以原始值，再乘以100。
Compound interest calculates interest on both the principal and the accumulated interest.
A = P (1 + r/100)ⁿ
复利计算考虑了本金和累计利息的利息。
For repeated percentage change, the multiplier method is used.
New value = original × (multiplier)ⁿ
对于重复的百分比变化，使用乘数方法。
Speed, density and pressure are compound measures.
Speed = distance / time, Density = mass / volume, Pressure = force / area
速率、密度和压强都是复合量度。
Rounding to decimal places and significant figures uses standard rules; upper and lower bounds give the range of possible true values.
四舍五入到小数位或有效数字遵循标准规则；上界和下界给出了可能真实值的范围。
Numbers in standard form are written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer.
a × 10ⁿ with 1 ≤ a < 10
标准形式表示为 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。
2. Algebra – Basic Operations | 代数基础运算
When expanding brackets, multiply each term inside by the term outside.
a(b + c) = ab + ac
展开括号时，将括号外的项与括号内的每一项相乘。
To factorise, find the highest common factor (HCF) or use the difference of two squares.
a² – b² = (a – b)(a + b)
因式分解时，找出最大公因数或使用平方差公式。
The laws of indices help simplify expressions with powers.
aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1 / aⁿ
指数运算法则用于化简含有幂的表达式。
Surds can be simplified using √a × √b = √(ab) and rationalised by multiplying the numerator and denominator.
√a × √b = √(ab) , 1 / √a × (√a / √a) = √a / a
根式可化简，并可通过乘以适当因子进行有理化。
3. Solving Equations and Inequalities | 解方程与不等式
Linear equations are solved by isolating the variable.
ax + b = c → x = (c – b) / a
线性方程通过移项求解。
Quadratic equations ax² + bx + c = 0 can be solved by factorising, completing the square or using the quadratic formula.
x = (-b ± √(b² – 4ac)) / (2a)
二次方程可通过因式分解、配方法或二次公式求解。
To complete the square, write the expression in the form (x + p)² + q.
x² + bx = (x + b/2)² – (b/2)²
配方法是将表达式写成 (x + p)² + q 的形式。
Simultaneous equations can be solved by elimination or substitution.
联立方程可通过消元法或代入法求解。
When solving linear inequalities, remember to reverse the sign if multiplying or dividing by a negative number.
If -x > 3 then x < -3
解线性不等式时，如果乘以或除以负数，需要反转不等号。
4. Sequences | 数列
An arithmetic sequence has a constant difference between terms. The nth term is given by:
uₙ = a + (n-1)d
等差数列的相邻两项之差为常数，第 n 项公式如上。
For a quadratic sequence, the second difference is constant. The nth term is of the form an² + bn + c.
uₙ = an² + bn + c
二次数列的二级差为常数，第 n 项形式为 an² + bn + c。
A geometric sequence multiplies by a constant ratio r. The nth term is:
uₙ = arⁿ⁻¹
等比数列每一项乘以常数公比 r，第 n 项公式如上。
5. Graphs and Functions | 图形与函数
The gradient of a line between two points (x₁, y₁) and (x₂, y₂) is:
m = (y₂ – y₁) / (x₂ – x₁)
两点间直线的斜率计算公式如上。
The midpoint and distance between two points are useful in coordinate geometry.
Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2) , Distance = √((x₂-x₁)² + (y₂-y₁)²)
中点坐标和两点间距离公式在坐标几何中很常用。
The equation of a straight line can be written as y = mx + c, where m is the gradient and c is the y‑intercept.
y = mx + c
直线方程可写为 y = mx + c，其中 m 为斜率，c 为 y 轴截距。
Parallel lines have equal gradients; perpendicular lines have gradients whose product is -1.
m₁ = m₂ (parallel) , m₁ × m₂ = -1 (perpendicular)
平行线斜率相等；垂直线斜率乘积为 -1。
For a quadratic graph y = ax² + bx + c, the turning point and roots can be found using the methods above.
二次函数图像 y = ax² + bx + c 的转折点和根可用前述方法求得。
In distance–time graphs, gradient = speed; in speed–time graphs, gradient = acceleration and area under graph = distance travelled.
距离-时间图中，斜率表示速度；速度-时间图中，斜率表示加速度，曲线下面积表示行驶距离。
6. Geometry – Angles and Polygons | 几何 – 角与多边形
Angles on a straight line sum to 180°, angles around a point sum to 360°.
Angles on a line: a + b = 180° ; angles at a point: sum = 360°
直线上的角之和为 180°，一点周围的角之和为 360°。
Vertically opposite angles are equal.
Vertically opposite angles: a = b
对顶角相等。
Angles in a triangle sum to 180°; the exterior angle equals the sum of the two opposite interior angles.
Triangle: a + b + c = 180°, exterior = a + b
三角形内角和为 180°，外角等于两内对角之和。
For parallel lines, alternate angles, corresponding angles and co‑interior (allied) angles have special relationships.
Alternate: a = b ; Corresponding: c = d ; Co‑interior: e + f = 180°
平行线中的内错角、同位角和同旁内角有特定关系。
The sum of interior angles of an n‑sided polygon is (n-2) × 180°; each exterior angle of a regular polygon is 360°/n.
Sum interior = (n-2) × 180°, exterior = 360° / n
n 边形内角和为 (n-2)×180°，正多边形每个外角为 360°/n。
7. Mensuration – Perimeter, Area, Volume | 测量 – 周长、面积、体积
Perimeter is the distance around a shape.
周长是图形边界的总长度。
Key area formulas include:
重要面积公式有：
-
Rectangle: A = lw
矩形：A = 长×宽
-
Triangle: A = ½ bh
三角形：A = ½ ×底×高
-
Parallelogram: A = bh
平行四边形：A = 底×高
-
Trapezium: A = ½ (a + b)h
梯形：A = ½ (上底+下底)×高
-
Circle: A = πr², Circumference = 2πr or πd
圆：面积 = πr², 周长 = 2πr 或 πd
-
Sector: Area = (θ/360) × πr², Arc length = (θ/360) × 2πr
扇形：面积 = (θ/360) × πr², 弧长 = (θ/360) × 2πr
Volume formulas for common solids:
常见立体的体积公式：
-
Cube/Cuboid: V = lwh
立方体/长方体：V = 长×宽×高
-
Prism: V = area of cross‑section × length
棱柱：V = 横截面积 × 长度
-
Cylinder: V = πr²h
圆柱：V = πr²h
-
Pyramid: V = ⅓ × base area × height
棱锥：V = ⅓ ×底面积×高
-
Cone: V = ⅓ πr²h, Curved surface area = πrl (l = slant height)
圆锥：V = ⅓ πr²h, 侧面积 = πrl (l 为斜高)
-
Sphere: V = ⁴⁄₃ πr³, Surface area = 4πr²
球体：V = ⁴⁄₃ πr³, 表面积 = 4πr²
8. Pythagoras and Trigonometry | 毕达哥拉斯与三角学
In a right‑angled triangle, Pythagoras’ theorem states:
a² + b² = c²
在直角三角形中，毕达哥拉斯定理为：
The three trigonometric ratios are defined as:
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent
三个三角函数定义为：
Exact values for 0°, 30°, 45°, 60°, 90° should be memorised.
0°、30°、45°、60°、90° 的精确三角函数值需要熟记。
For any triangle, the sine rule and cosine rule apply:
a / sin A = b / sin B = c / sin C
a² = b² + c² – 2bc cos A
对于任意三角形，可用正弦定理和余弦定理。
Area of a triangle when two sides and the included angle are known:
Area = ½ ab sin C
已知两边及其夹角时三角形面积公式。
3D Pythagoras finds the diagonal of a cuboid:
d² = l² + w² + h²
立体几何中，长方体对角线满足 d² = 长² + 宽² + 高²。
9. Vectors and Transformations | 向量与变换
A vector can be represented as a column vector or with bold letter a. Addition and scalar multiplication are performed component‑wise.
a + b = … , ka = …
向量可表示为列向量或粗体字母，加减和数乘按分量进行。
The magnitude of vector a = (x, y) is |a| = √(x² + y²).
|(x, y)| = √(x² + y²)
向量的大小（模）为 √(x² + y²)。
Translation moves a shape by a vector (x y).
Move P to (Px+x, Py+y)
平移按向量 (x, y) 移动图形。
Reflection, rotation and enlargement are common transformations; the centre, angle, direction and scale factor must be stated.
反射、旋转和放大是常见变换，需要指明中心、角度、方向和比例因子。
Enlargement with scale factor k and centre at origin: (x, y) → (kx, ky).
(x, y) → (kx, ky)
以原点为中心、比例因子为 k 的放大： (x, y) → (kx, ky)。
Some transformations can be described using matrices, e.g. rotation by 90° anticlockwise about the origin: matrix [0 -1; 1 0].
| 0 | -1 |
| 1 | 0 |
某些变换可用矩阵描述，例如绕原点逆时针旋转 90° 的矩阵如上。
10. Statistics | 统计
Mean, median, mode and range are measures of central tendency and spread.
Mean = sum of values / number of values
平均数、中位数、众数和极差是集中趋势和离散程度的度量。
For grouped data, estimate the mean using midpoints.
Estimated mean = Σ(f × midpoint) / Σf
分组数据用组中值估算平均数。
The interquartile range (IQR) = upper quartile (Q₃) – lower quartile (Q₁).
IQR = Q₃ – Q₁
四分位距 (IQR) 为上四分位数减下四分位数。
Cumulative frequency graphs help find quartiles, median and percentiles.
累积频率图可用于寻找四分位数、中位数和百分位数。
In a histogram, frequency density = frequency / class width.
Frequency density = frequency / class width
直方图中，频率密度 = 频率 / 组距。
Scatter diagrams show correlation; a line of best fit can be drawn to make predictions.
散点图显示相关性，可画出最佳拟合线进行预测。
11. Probability | 概率
Probability of an event = number of favourable outcomes / total number of outcomes.
P(A) = n(A) / n(S)
事件概率 = 有利结果数 / 所有可能结果数。
For mutually exclusive events, P(A or B) = P(A) +
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