Formula Summary Handbook | 公式汇总手册

📚 Formula Summary Handbook | 公式汇总手册

This handbook brings together the essential formulae you need for GCSE WJEC Mathematics, covering both Foundation and Higher tier topics. Use it as a quick reference to reinforce your understanding and build confidence before the exam.

这本手册汇总了GCSE WJEC数学中必备的关键公式，覆盖基础层和高等层的核心内容。你可以把它当作考前快速复习的参考，帮助加深理解、增强信心。

1. Number | 数

Percentage change compares the difference to the original amount:

Percentage Change = (Difference ÷ Original) × 100%

百分比变化用差值除以原值再乘100%表示。

For compound interest, the amount after n years is given by:

A = P(1 + r/100)ⁿ

复利计算中，n年后的本息和为 A = P(1 + r/100)ⁿ，P为本金，r为年利率。

A multiplier for a percentage increase of p% is 1 + p/100; for a decrease, it is 1 − p/100.

增加p%时的乘数为1 + p/100，减少p%时的乘数为1 − p/100。

To add or subtract fractions, use a common denominator; for example, a/b ± c/d = (ad ± bc)/bd.

分数加减法要先通分，例如 a/b ± c/d = (ad ± bc)/bd。

When multiplying numbers in standard form, multiply the coefficients and add the exponents: (a × 10ᵐ) × (b × 10ⁿ) = ab × 10ᵐ⁺ⁿ.

标准形式相乘时，系数相乘，指数相加：(a × 10ᵐ) × (b × 10ⁿ) = ab × 10ᵐ⁺ⁿ。

2. Algebra: Expressions and Equations | 代数：表达式与方程

The solution to a quadratic equation ax² + bx + c = 0 is found using the quadratic formula:

x = (−b ± √(b² − 4ac)) / (2a)

二次方程 ax² + bx + c = 0 的解由求根公式给出：x = (−b ± √(b² − 4ac)) / (2a)。

The discriminant Δ = b² − 4ac tells us the nature of the roots: if Δ > 0, two distinct real roots; if Δ = 0, one repeated root; if Δ < 0, no real roots.

判别式 Δ = b² − 4ac 可判断根的情况：Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 无实根。

To factorise a quadratic of the form x² + bx + c, find two numbers that multiply to c and add to b.

对形如 x² + bx + c 的二次式因式分解，需找到乘积为 c、和为 b 的两个数。

Completing the square: write x² + bx + c in the form (x + p)² + q, where p = b/2 and q = c − p².

配方法：将 x² + bx + c 写成 (x + p)² + q，其中 p = b/2，q = c − p²。

For a linear equation in the form y = mx + c, m is the gradient and c is the y-intercept.

直线方程 y = mx + c 中，m 为斜率，c 为 y 轴截距。

3. Sequences | 序列

For an arithmetic sequence with first term a and common difference d, the nth term is:

aₙ = a + (n − 1)d

对于首项为 a、公差为 d 的等差数列，第 n 项为 aₙ = a + (n − 1)d。

The sum of the first n terms of an arithmetic series is:

Sₙ = n/2 (2a + (n − 1)d) or Sₙ = n/2 (a + l)

等差数列前 n 项和为 Sₙ = n/2 (2a + (n − 1)d) 或 Sₙ = n/2 (a + l)，其中 l 为末项。

A geometric sequence has nth term aₙ = arⁿ⁻¹, where r is the common ratio and a is the first term.

等比数列的第 n 项为 aₙ = arⁿ⁻¹，a 为首项，r 为公比。

For a geometric series with |r| < 1, the sum to infinity is S∞ = a / (1 − r).

当 |r| < 1 时，无穷等比级数的和为 S∞ = a / (1 − r)。

A Fibonacci-type sequence has each term as the sum of the two preceding terms: uₙ = uₙ₋₁ + uₙ₋₂.

斐波那契型数列中每一项为前两项之和：uₙ = uₙ₋₁ + uₙ₋₂。

4. Graphs and Coordinate Geometry | 图形与坐标几何

The gradient of a straight line between (x₁, y₁) and (x₂, y₂) is m = (y₂ − y₁) / (x₂ − x₁).

过 (x₁, y₁) 和 (x₂, y₂) 两点的直线斜率 m = (y₂ − y₁) / (x₂ − x₁)。

Lines with gradients m₁ and m₂ are parallel if m₁ = m₂, and perpendicular if m₁m₂ = −1.

若斜率满足 m₁ = m₂，则两直线平行；若 m₁m₂ = −1，则两直线垂直。

The distance between two points is:

d = √[(x₂ − x₁)² + (y₂ − y₁)²]

两点间距离公式为 d = √[(x₂ − x₁)² + (y₂ − y₁)²]。

The midpoint of the line segment joining (x₁, y₁) and (x₂, y₂) is ((x₁ + x₂)/2, (y₁ + y₂)/2).

连接两点的线段中点坐标为 ((x₁ + x₂)/2, (y₁ + y₂)/2)。

The equation of a circle with centre (a, b) and radius r is (x − a)² + (y − b)² = r².

以 (a, b) 为圆心、r 为半径的圆方程为 (x − a)² + (y − b)² = r²。

For a quadratic graph y = ax² + bx + c, the turning point occurs at x = −b/(2a).

二次函数 y = ax² + bx + c 的顶点横坐标为 x = −b/(2a)。

5. Mensuration: Perimeter, Area, Surface Area and Volume | 测量：周长、面积、表面积和体积

Areas of common shapes:

常见图形面积公式：

Rectangle: A = l × w

A = l × w

矩形面积：长乘宽，A = l × w。

Triangle: A = ½ × base × height

A = ½ b h

三角形面积：底乘高的一半，A = ½ b h。

Parallelogram: A = b h

A = b h

平行四边形面积：底乘高，A = b h。

Trapezium: A = ½ (a + b) h, where a and b are the parallel sides.

A = ½ (a + b) h

梯形面积：上底加下底乘以高除以2，A = ½ (a + b) h。

Circle: circumference C = 2πr or πd, area A = πr².

C = 2πr = πd, A = πr²

圆：周长 C = 2πr 或 πd，面积 A = πr²。

Sector: arc length = θ/360 × 2πr, area = θ/360 × πr² (θ in degrees).

Arc length = (θ/360) × 2πr, Sector area = (θ/360) × πr²

扇形：弧长 = θ/360 × 2πr，面积 = θ/360 × πr²（θ 用度数）。

Volumes of 3D shapes:

立体体积公式：

Prism: volume = area of cross-section × length.

棱柱：体积 = 底面积 × 长度。

Cylinder: V = πr²h.

V = πr²h

圆柱：V = πr²h。

Cone: V = ⅓ πr²h.

V = ⅓ πr²h

圆锥：V = ⅓ πr²h。

Sphere: volume V = ⁴⁄₃ πr³, surface area A = 4πr².

V = ⁴⁄₃ πr³, A = 4πr²

球体：体积 V = ⁴⁄₃ πr³，表面积 A = 4πr²。

Pyramid: volume = ⅓ × base area × vertical height.

V = ⅓ × base area × h

棱锥：体积 = ⅓ × 底面积 × 高。

6. Angles and Polygons | 角与多边形

Sum of interior angles of an n-sided polygon:

Sum = (n − 2) × 180°

n边形内角和为 (n − 2) × 180°。

For a regular polygon, each interior angle = (n − 2) × 180° / n, and each exterior angle = 360° / n.

正多边形每个内角为 (n − 2) × 180° / n，每个外角为 360° / n。

The sum of exterior angles of any convex polygon is always 360°.

任何凸多边形的外角和总是360°。

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

平行线被截线所截，同位角相等，内错角相等，同旁内角互补（和为180°）。

In a triangle, the exterior angle equals the sum of the two opposite interior angles.

三角形的一个外角等于与它不相邻的两个内角之和。

7. Pythagoras’ Theorem and Trigonometry | 毕达哥拉斯定理与三角学

In a right-angled triangle with legs a, b and hypotenuse c:

a² + b² = c²

直角三角形中，两直角边为 a、b，斜边为 c，则 a² + b² = c²。

The three trigonometric ratios for an acute angle θ are:

sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent

锐角 θ 的三角函数定义为：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。

In any triangle ABC, the sine rule relates sides and sines of opposite angles:

a / sin A = b / sin B = c / sin C

任意三角形中，正弦定理：a / sin A = b / sin B = c / sin C。

The cosine rule links three sides and an included angle:

a² = b² + c² − 2bc cos A

余弦定理：a² = b² + c² − 2bc cos A，其中 A 是边 b 和 c 的夹角。

The area of a triangle can be found using two sides and the included angle:

Area = ½ ab sin C

三角形面积可用公式 ½ ab sin C 计算，其中 C 为边 a 和 b 的夹角。

Exact trigonometric values for key angles should be memorised, such as sin 30° = ½, cos 60° = ½, tan 45° = 1.

应熟记特殊角精确三角函数值，如 sin 30° = ½，cos 60° = ½，tan 45° = 1。

8. Vectors | 向量

A vector is often written as a column vector:

(x, y)ᵀ

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Formula Summary Handbook | 公式汇总手册