KS3 Cambridge Advanced Mathematics: Formula & Theorem Quick Reference | KS3 Cambridge 进阶数学：公式定理速查手册

📚 KS3 Cambridge Advanced Mathematics: Formula & Theorem Quick Reference | KS3 Cambridge 进阶数学：公式定理速查手册

This quick reference handbook compiles the essential formulas and theorems for the Cambridge KS3 Advanced Mathematics syllabus. Use it to review key concepts in algebra, geometry, handling data, and problem-solving – all presented in a concise bilingual format to support your independent study.

这本速查手册汇集了剑桥KS3进阶数学课程的核心公式与定理。内容涵盖代数、几何、数据处理和问题解决，采用简洁的中英双语排版，便于你独立复习和随时查阅。

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1. Number and Operations | 数与运算

Index Laws

When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.

同底数幂相乘时，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。

When dividing powers with the same base, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

同底数幂相除时，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。

When raising a power to another power, multiply the exponents: (aᵐ)ⁿ = aᵐⁿ.

幂的幂时，指数相乘：(aᵐ)ⁿ = aᵐⁿ。

Any non-zero number raised to the power of zero equals 1: a⁰ = 1 (a ≠ 0).

任何非零数的零次幂等于1：a⁰ = 1（a ≠ 0）。

Negative exponents indicate reciprocals: a⁻ⁿ = 1 / aⁿ.

负指数表示倒数：a⁻ⁿ = 1 / aⁿ。

Roots and Fractional Exponents

The square root of a number can be written as a fractional exponent: √a = a¹/². Similarly, the cube root is ∛a = a¹/³.

平方根可以写成分数指数形式：√a = a¹/²；立方根可写为∛a = a¹/³。

Standard Form

A number in standard form is written as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. For example, 4500 = 4.5 × 10³ and 0.0076 = 7.6 × 10⁻³.

一个数的标准形式写作 a × 10ⁿ，其中 1 ≤ a < 10，n 为整数。例如 4500 = 4.5 × 10³，0.0076 = 7.6 × 10⁻³。

Fractions

Addition/Subtraction: a/b + c/d = (ad + bc) / bd. Multiplication: a/b × c/d = (a × c) / (b × d). Division: a/b ÷ c/d = a/b × d/c.

加法/减法：a/b + c/d = (ad + bc) / bd；乘法：a/b × c/d = (a × c) / (b × d)；除法：a/b ÷ c/d = a/b × d/c。

Percentage Change

Increase: New value = Original × (1 + r/100). Decrease: New value = Original × (1 − r/100).

增加：新值 = 原值 × (1 + r/100)；减少：新值 = 原值 × (1 − r/100)。

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2. Algebraic Expressions | 代数表达式

Expanding Brackets

Distribute each term inside the bracket: a(b + c) = ab + ac.

将括号外的项与括号内每一项相乘：a(b + c) = ab + ac。

For two binomials: (a + b)(c + d) = ac + ad + bc + bd. Remember to combine like terms afterwards.

两个二项式相乘：(a + b)(c + d) = ac + ad + bc + bd。记得合并同类项。

Special Expansions

Square of a sum: (a + b)² = a² + 2ab + b².

和的平方：(a + b)² = a² + 2ab + b²。

Square of a difference: (a − b)² = a² − 2ab + b².

差的平方：(a − b)² = a² − 2ab + b²。

Difference of two squares: a² − b² = (a + b)(a − b).

平方差公式：a² − b² = (a + b)(a − b)。

Factorising

Look for a common factor: ab + ac = a(b + c). For quadratics like x² + bx + c, find two numbers that multiply to c and add to b.

寻找公因式：ab + ac = a(b + c)。对于 x² + bx + c 型的二次式，找出两个乘积为c且和为b的数进行因式分解。

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3. Linear Equations | 线性方程

To solve ax + b = c, first subtract b from both sides: ax = c − b, then divide by a: x = (c − b) / a (a ≠ 0).

求解 ax + b = c，先将等式两边同时减去 b：ax = c − b，然后两边除以 a：x = (c − b) / a（a ≠ 0）。

If the equation contains brackets, expand them first. Collect like terms on each side, then apply inverse operations to isolate the variable.

如果方程含有括号，应先去括号。将同类项移到同一边，然后通过逆运算分离出变量。

Example: Solve 3(x − 2) = 15. Expand to 3x − 6 = 15. Add 6: 3x = 21. Divide by 3: x = 7.

例题：解 3(x − 2) = 15。展开得 3x − 6 = 15，两边加6：3x = 21，再除以3：x = 7。

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4. Ratio, Proportion and Rates | 比、比例和率

A ratio compares two quantities. The ratio a:b can be written as a fraction a/b. To divide a quantity in a given ratio, find the total number of parts and multiply.

比用于比较两个量。a:b 可以写成分数 a/b。按给定比分配一个量时，先计算总份数，再对应相乘。

Direct Proportion: y is directly proportional to x if y = kx, where k is the constant of proportionality. The graph is a straight line through the origin.

正比例：若 y 与 x 成正比，则 y = kx，其中 k 是比例常数，图像为过原点的直线。

Inverse Proportion: y is inversely proportional to x if y = k/x. As x increases, y decreases.

反比例：若 y 与 x 成反比，则 y = k/x。x 增大时 y 减小。

Rates like speed are expressed as distance per unit time: speed = distance / time.

率如速率表示为距离除以时间：速度 = 距离 / 时间。

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5. Geometry: Angles and Shapes | 几何：角与形状

Angle Facts

Vertically opposite angles are equal. Angles on a straight line sum to 180°. Angles around a point sum to 360°.

对顶角相等。平角等于180°。周角等于360°。

When a transversal crosses parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior (allied) angles sum to 180°.

平行线被一直线所截：同位角相等，内错角相等，同旁内角互补（和为180°）。

Polygons

Sum of interior angles of an n-sided polygon = (n − 2) × 180°.

n 边形的内角和 = (n − 2) × 180°。

Sum of exterior angles of any convex polygon = 360°.

任何凸多边形的外角和均为360°。

Triangle angle sum = 180°. Isosceles triangles have two equal sides and two equal base angles. Equilateral triangles have all sides and angles equal (60° each).

三角形内角和为180°。等腰三角形有两边相等，两底角相等。等边三角形三边相等，三个角均为60°。

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6. Perimeter, Area and Volume | 周长、面积和体积

Use the followingtable to recall key plane figure formulas. All lengths are in the same unit.

使用下表快速回忆常见平面图形的相关公式。所有长度单位需统一。

Shape	Perimeter / Circumference	Area
Rectangle	2(l + w)	l × w
Triangle	a + b + c	½ × b × h
Parallelogram	2(a + b)	b × h
Trapezium	Sum of 4 sides	½ (a + b)h
Circle	C = 2πr or πd	A = πr²

3D Shapes – Volume and Surface Area

Cuboid volume: V = l × w × h. Cylinder volume: V = πr²h. Curved surface area of cylinder: A = 2πrh. Sphere volume: V = (4/3)πr³ (extension).

长方体体积：V = l × w × h；圆柱体积：V = πr²h；圆柱侧面积：A = 2πrh；球体积：V = (4/3)πr³（拓展内容）。

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7. Pythagoras’ Theorem | 勾股定理

In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides.

a² + b² = c²

where c is the hypotenuse and a, b are the legs.

在直角三角形中，斜边（直角所对的边）的平方等于两条直角边的平方和。即 a² + b² = c²，其中 c 是斜边，a、b 是直角边。

To find the length of the hypotenuse: c = √(a² + b²). To find a leg: a = √(c² − b²).

求斜边长：c = √(a² + b²)；求直角边长：a = √(c² − b²)。

Use the converse: if a² + b² = c², then the triangle is right-angled.

逆定理同样成立：若三角形三边满足 a² + b² = c²，则该三角形是直角三角形。

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8. Statistics: Averages and Charts | 统计：平均数和图表

Mean, Median, Mode and Range

Mean = (sum of all data values) / (number of data values).

平均数 = 所有数据值之和 ÷ 数据个数。

Median is the middle value when data are ordered. For an even number of values, median = average of the two middle numbers.

中位数是将数据排序后位于中间位置的数。若数据个数为偶数，中位数是中间两个数的平均值。

Mode is the value that appears most often. Range = maximum value − minimum value.

众数是出现次数最多的数据值。极差 = 最大值 − 最小值。

Representing Data

Use bar charts for discrete data, pie charts for proportions, and line graphs for trends. Stem-and-leaf diagrams help display the shape while retaining individual data values.

离散数据用条形图，比例用饼图，趋势用折线图。茎叶图既能展示分布形态又能保留原始数据。

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9. Probability | 概率

Probability of an event A: P(A) = (number of favourable outcomes) / (total number of possible outcomes), where all outcomes are equally likely.

事件 A 的概率：P(A) = 有利结果的个数 / 所有等可能结果的总数。

Probability scale: 0 means impossible, 1 means certain. A probability is always between 0 and 1 inclusive.

概率值在0到1之间。0表示不可能事件，1表示必然事件。

For mutually exclusive events A and B: P(A or B) = P(A) + P(B).

互斥事件 A 或 B 的概率：P(A 或 B) = P(A) + P(B)。

For independent events A and B: P(A and B) = P(A) × P(B).

独立事件 A 和 B 同时发生的概率：P(A 和 B) = P(A) × P(B)。

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10. Sequences | 数列

A sequence is an ordered list of numbers that follows a rule. Each number is called a term.

数列是按一定规则排列的一列数；每个数称为项。

Arithmetic Progressions (Linear Sequences)

An arithmetic sequence has a constant difference between consecutive terms. The nth term formula is given by:

aₙ = a₁ + (n − 1)d

where a₁ is the first term, d is the common difference, and n is the term position.

等差数列相邻两项的差相等。第 n 项公式：aₙ = a₁ + (n − 1)d，其中 a₁ 为首项，d 为公差，n 为项数。

To find whether a number is a term in the sequence, set aₙ equal to that number and solve for n; n must be a positive integer.

判断某数是否属于数列，可将该数代入 aₙ 并解出 n；若 n 为正整数，则该数是数列的项。

Using Differences

If the first differences are not constant, check the second

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