The Essential GCSE Maths Formula Handbook | GCSE数学考场必备公式汇总

📚 The Essential GCSE Maths Formula Handbook | GCSE数学考场必备公式汇总

This handbook compiles every vital formula you may need in your GCSE Maths exam, arranged by topic. Master these, understand when to use them, and you will walk into the examination hall with confidence. Each formula is presented clearly, followed by practical notes on application.

本手册按主题汇总了GCSE数学考试中你可能需要的所有核心公式。熟练掌握这些公式，明白它们的适用场景，你就能自信地走进考场。每条公式都清晰呈现，并附有实用的应用说明。

1. Number, Percentages and Ratio | 数字、百分比与比例

Percentages and ratio problems appear frequently across all three GCSE papers. The key is translating word problems into a mathematical operation with the correct multiplier.

百分比和比例问题在GCSE数学的三张试卷中频繁出现。关键在于将文字题转化为带正确乘数的数学运算。

Percentage change is calculated as:

Percentage change = (change ÷ original value) × 100%

To increase or decrease a quantity by a percentage, use the decimal multiplier method:

New value = original × (1 ± r)ⁿ

Here, r is the percentage change expressed as a decimal, and n is the number of repeated changes. For example, an increase of 4% per annum for 3 years uses 1.04³.

其中 r 是百分比变化写成小数的形式，n 是重复变化的次数。例如，每年增长4%持续3年，则使用 1.04³。

Compound interest formula: A = P(1 + r/100)ⁿ
复利公式：A = P(1 + r/100)ⁿ
Simple interest formula: I = PRT ÷ 100
单利公式：I = PRT ÷ 100

Remember to convert ratios carefully. If quantities are split in the ratio a:b, each part equals total ÷ (a+b).

注意在比例中小心换算。若总量按 a:b 分配，则每份等于总量 ÷ (a+b)。

2. Algebra — Expressions, Equations and the Quadratic Formula | 代数——表达式、方程与二次公式

Algebra is the backbone of GCSE Maths. The quadratic formula is on the formula sheet provided by exam boards, but you must know exactly how and when to apply it.

代数是GCSE数学的基石。二次公式虽已印在考局提供的公式表上，但你必须清楚何时以及如何运用它。

The quadratic formula solves equations of the form ax² + bx + c = 0:

x = [−b ± √(b² − 4ac)] ÷ 2a

The discriminant b² − 4ac determines the number of roots:

判别式 b² − 4ac 决定根的个数：

If b² − 4ac > 0: two distinct real roots
若 b² − 4ac > 0：两个不同的实数根
If b² − 4ac = 0: one repeated real root
若 b² − 4ac = 0：一个重复的实数根
If b² − 4ac < 0: no real roots
若 b² − 4ac < 0：没有实数根

You should also be fluent in expanding double brackets, solving simultaneous equations, and rearranging formulas with fractions.

你也需要熟练展开双重括号、解联立方程，以及处理含分数的公式变形。

3. Geometry — Perimeter, Area and Pythagoras’ Theorem | 几何——周长、面积与毕达哥拉斯定理

For 2D shapes, the most tested formulas involve triangles, rectangles, circles, trapeziums, and compound shapes. Units must always be consistent.

对于二维图形，最常考的是三角形、矩形、圆、梯形和组合图形的公式。单位必须始终保持一致。

Rectangle area: A = length × width
矩形面积：A = 长 × 宽
Triangle area: A = ½ × base × height
三角形面积：A = ½ × 底 × 高
Trapezium area: A = ½(a + b) × h
梯形面积：A = ½(a + b) × h
Perimeter: total distance around the shape
周长：图形外缘的总长度

For right-angled triangles, Pythagoras’ theorem relates the three sides:

对于直角三角形，毕达哥拉斯定理给出了三边关系：

a² + b² = c²

Here, c is the hypotenuse, the side opposite the right angle. Check whether the question asks you to find a shorter side or the hypotenuse, as rearranging is required.

其中 c 是斜边，即直角所对的边。注意题目要求的是短边还是斜边，因为这决定了你需要如何变形公式。

4. Coordinate Geometry and Linear Graphs | 坐标几何与直线图像

Linear graphs are central to the algebra and geometry crossover questions. The gradient–intercept form is the most common starting point.

线性图像是代数与几何交叉题型的核心。斜截式是最常见的切入点。

y = mx + c

Here, m is the gradient of the line, and c is the y-intercept. The gradient between two points (x₁, y₁) and (x₂, y₂) is:

其中 m 是直线的斜率，c 是 y 轴截距。两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率为：

m = (y₂ − y₁) ÷ (x₂ − x₁)

Parallel lines have equal gradients. Perpendicular lines have gradients whose product is −1. For perpendicular gradients, m₁ × m₂ = −1.

平行线斜率相等。垂直线斜率的乘积为 −1，即 m₁ × m₂ = −1。

The midpoint of two coordinates is found by averaging the x-coordinates and the y-coordinates.

两点坐标的中点通过将 x 坐标和 y 坐标分别取平均得到。

5. Trigonometry — Right-Angled Triangles | 三角学——直角三角形

SOH CAH TOA is the essential mnemonic for right-angled triangle trigonometry. Labels of opposite, adjacent, and hypotenuse must be correctly identified relative to the angle in question.

SOH CAH TOA 是解直角三角形三角函数的必备口诀。你必须根据所求角准确标出对边（opposite）、邻边（adjacent）和斜边（hypotenuse）。

sin θ = opposite ÷ hypotenuse

cos θ = adjacent ÷ hypotenuse

tan θ = opposite ÷ adjacent

Use the inverse functions sin⁻¹, cos⁻¹ and tan⁻¹ to find angles when side lengths are known.

当已知边长求角度时，使用反函数 sin⁻¹、cos⁻¹ 和 tan⁻¹。

Exact values for common angles are essential for non-calculator papers. Memorise the table below:

常用角的精确值是非计算器试卷的必考内容，请牢记下表：

θ	0°	30°	45°	60°	90°
sin θ	0	½	√2⁄2	√3⁄2	1
cos θ	1	√3⁄2	√2⁄2	½	0
tan θ	0	√3⁄3	1	√3	undefined

6. Circles — Area, Circumference and Sectors | 圆——面积、周长与扇形

Circle formulas are guaranteed to appear. Know them to the point of automatic recall.

圆的公式是必考内容。你需要将它们化为条件反射般的记忆。

Circumference: C = 2πr = πd
周长：C = 2πr = πd
Area of a circle: A = πr²
圆的面积：A = πr²
Arc length: L = (θ⁄360) × 2πr
弧长：L = (θ⁄360) × 2πr
Sector area: A = (θ⁄360) × πr²
扇形面积：A = (θ⁄360) × πr²

In each sector formula, θ is the central angle in degrees. If the angle is given in radians, replace 360° with 2π.

在扇形公式中，θ 是以度为单位表示圆心角。如果题目给出弧度制角度，则将 360° 换成 2π。

7. Volume and Surface Area of 3D Solids | 立体图形的体积与表面积

Prisms, cylinders, cones, spheres and pyramids each have distinct volume formulas. Identifying the shape class immediately narrows down the method.

棱柱、圆柱、圆锥、球和棱锥各有不同的体积公式。迅速判断图形类别能立刻缩小计算方法范围。

Cuboid volume: V = l × w × h
长方体体积：V = l × w × h
Prism volume: V = cross-sectional area × length
棱柱体积：V = 截面面积 × 长度
Cylinder volume: V = πr²h
圆柱体积：V = πr²h
Sphere volume: V = ⁴⁄₃πr³
球体积：V = ⁴⁄₃πr³
Cone volume: V = ⅓πr²h
圆锥体积：V = ⅓πr²h
Pyramid volume: V = ⅓ × base area × height
棱锥体积：V = ⅓ × 底面积 × 高

Surface area is the sum of all face areas. For a cylinder, SA = 2πr² + 2πrh. For a sphere, SA = 4πr².

表面积是各个面的面积总和。圆柱的表面积为 SA = 2πr² + 2πrh。球的表面积为 SA = 4πr²。

8. Sine Rule, Cosine Rule and the Area of Any Triangle | 正弦定理、余弦定理与任意三角形面积

When a triangle is not right-angled, you must switch to the sine rule or cosine rule. Decide based on the known information in the question.

当三角形不是直角三角形时，你必须使用正弦定理或余弦定理。根据题目中已知的信息来选择使用哪一个。

The sine rule is for pairs of angles and opposite sides:

正弦定理用于处理成对的角与对边：

a⁄sin A = b⁄sin B = c⁄sin C

The cosine rule connects three sides and one angle:

余弦定理将三条边与一个角联系起来：

a² = b² + c² − 2bc·cos A

It can be rearranged to find an angle:

它可以变形用来求角：

cos A = (b² + c² − a²) ÷ 2bc

The area of any triangle can be found when two sides and the included angle are known:

当已知两边及其夹角时，可用以下公式求任意三角形的面积：

Area = ½ab·sin C

9. Statistics — Mean, Median, Mode and Standard Form of Data | 统计——平均数、中位数、众数与数据标准形式

For grouped data, you calculate the mean using midpoints of intervals. The median is found by locating the middle value through cumulative frequency.

对于分组数据，你需要用组中值来计算平均数。中位数通过累积频率找到中间值的位置来确定。

Mean (raw data): sum of all values ÷ number of values
平均数（原始数据）：所有数值之和 ÷ 数值个数
Mean (grouped data): Σ(midpoint × frequency) ÷ Σfrequency
平均数（分组数据）：Σ(组中值 × 频数) ÷ Σ频数
Median: the middle value when data is ordered
中位数：数据排序后的中间值
Mode: the most frequently occurring value
众数：出现次数最多的数值

Interquartile range (IQR) measures spread: IQR = Upper quartile − Lower quartile. It is robust against outliers, unlike the full range.

四分位距（IQR）衡量数据的离散程度：IQR = 上四分位数 − 下四分位数。与全距不同，它不易受异常值影响。

10. Probability — Key Rules for Independent and Mutually Exclusive Events | 概率——独立事件与互斥事件的关键规则

Probability questions require both symbolic manipulation and logical reasoning. The two fundamental rules below cover most GCSE problems.

概率题既需要符号运算，也需要逻辑推理。以下两条基本规则覆盖了大多数GCSE题目。

For mutually exclusive events, the sum of probabilities is 1:

对于互斥事件，概率之和为 1：

P(A or B) = P(A) + P(B)

For independent events, multiply probabilities:

对于独立事件，概率相乘：

P(A and B) = P(A) × P(B)

The complement rule states: P(not A) = 1 − P(A). Conditional probability questions require multiplying along branches of a probability tree.

补事件规则为：P(非A) = 1 − P(A)。条件概率题需要沿概率树的分支进行乘法运算。

11. Vectors and Transformations | 向量与几何变换

Vectors describe both magnitude and direction. On a grid, a column vector is written as a 2×1 matrix:

向量同时描述大小和方向。在网格上，列向量可写作 2×1 的矩阵形式：

(x y) or (x, y) in column form ×

To add vectors, add corresponding components. To multiply a vector by a scalar, multiply each component by that scalar. When using column vectors, note that moving right and up gives positive components; moving left and down gives negative ones.

向量相加时，将对应分量相加。向量乘以标量时，将每个分量乘以该标量。使用列向量时，向右和向上移动为正分量，向左和向下则为负分量。

Transformations include translation, reflection, rotation and enlargement. For enlargement, the scale factor k relates linear dimensions; area scale factor is k², and volume scale factor is k³.

几何变换包括平移、反射、旋转和放大缩小。对于放大，比例因子 k 对应线性尺寸；面积比例因子为 k²，体积比例因子为 k³。

12. Final Exam Strategy — Ranking Formulas by Priority | 最后考场策略——按优先级排列公式

Not every formula carries equal weight. On exam day, allocate your revision time according to how often each topic appears and how many marks it tends to carry.

并非每个公式权重相同。在考场上，你应该根据各主题出现频率和分值占比来分配复习时间。

Priority	Topic	Typical Marks
High	Quadratic formula, percentages, circles, Pythagoras	×5–8
Medium	Trigonometry (right-angled), volume, sine/cosine rule	×3–6
Lower	Vectors, probability tree, midpoint of coordinates	×1–4

Always write down the formula first before substituting numbers. This awards method marks even if your final arithmetic is incorrect. Check whether the exam board gives a formula sheet — for Edexcel, AQA and OCR, the quadratic formula and some geometry formulas are provided, but the rest must come from memory.

务必先写出公式再代入数值。这样即使最终计算有误，也能获得方法分。确认你的考局是否提供公式表——爱德思（Edexcel）、AQA 和 OCR 会提供二次公式和部分几何公式，但其余内容必须凭记忆写出。

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The Essential GCSE Maths Formula Handbook | GCSE数学考场必备公式汇总