📚 Year 11 AQA Mathematics: Quick-reference Formula and Theorem Handbook | Year 11 AQA 数学：公式定理速查手册
This essential handbook gathers every key formula, theorem, and technique you must recall for the AQA GCSE Mathematics examinations. Use it to reinforce your memory, check understanding, and build confidence before your next assessment. Each section pairs core facts with worked-style notes so you can see how the Maths works in practice.
这本速查手册汇总了 AQA GCSE 数学考试必须牢记的所有关键公式、定理和技巧。你可以用它来强化记忆、检验理解并在下一次评估前建立信心。每一节都将核心事实与示例式笔记配对，帮助你理解数学在实际中如何运作。
1. Quadratic Formula | 一元二次方程求根公式
For any quadratic equation written in the form ax² + bx + c = 0, the solutions for x are given by the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. This formula works even when the quadratic does not factorise easily. Remember that the expression under the square root, b² − 4ac, is called the discriminant. If the discriminant is positive, there are two distinct real roots; if zero, one repeated root; if negative, no real roots.
对于任何写成 ax² + bx + c = 0 形式的一元二次方程，x 的解由求根公式给出：x = (−b ± √(b² − 4ac)) / 2a。即使二次式不能轻易因式分解，该公式也同样有效。记住根号下的表达式 b² − 4ac 叫做判别式。若判别式为正，则有两个不同的实数根；若为零，则有一个重根；若为负，则没有实数根。
2. Straight Line: Gradient and y-intercept | 直线：斜率与截距
The equation of any straight line can be written as y = mx + c, where m represents the gradient (steepness) and c is the y-intercept (the point where the line crosses the y-axis). To find the gradient between two points (x₁, y₁) and (x₂, y₂), use m = (y₂ − y₁) / (x₂ − x₁). Parallel lines have the same gradient. For perpendicular lines, the product of their gradients is −1 ( m₁ × m₂ = −1 ).
任何直线的方程都能写成 y = mx + c，其中 m 表示斜率（倾斜程度），c 是 y 轴截距（直线与 y 轴的交点）。要计算两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率，使用 m = (y₂ − y₁) / (x₂ − x₁)。平行线具有相同的斜率。对于互相垂直的直线，它们斜率的乘积为 −1 ( m₁ × m₂ = −1 )。
3. Pythagoras’ Theorem | 勾股定理
In any 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. This is written as a² + b² = c², where c is the hypotenuse. You can also rearrange to find a shorter side: a² = c² − b². Always label the sides correctly before substituting values. Pythagoras’ theorem is used to check if a triangle is right-angled and to solve problems involving distance on a coordinate grid.
在任何直角三角形中，斜边（直角所对的边）的平方等于另外两条边的平方和。这记作 a² + b² = c²，其中 c 为斜边。你也可以变形来求较短边：a² = c² − b²。代入数值前务必正确标出各边。勾股定理用于检验三角形是否为直角三角形，并解决坐标网格上的距离问题。
4. Trigonometric Ratios | 三角比
The three trigonometric ratios for angles in right-angled triangles are defined as: sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent. A common mnemonic is SOH CAH TOA. These ratios can find missing sides and angles. When the unknown is an angle, use the inverse functions sin⁻¹, cos⁻¹, tan⁻¹. Values of sin, cos and tan for 0°, 30°, 45°, 60° and 90° must be memorised.
直角三角形中角的三个三角比定义为：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。记忆口诀是 SOH CAH TOA。这些比值可用来求未知的边长或角度。当未知量为角度时，使用反函数 sin⁻¹、cos⁻¹、tan⁻¹。必须记住 0°、30°、45°、60°和 90° 的正弦、余弦和正切值。
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
For non-right-angled triangles, the sine rule a / sin A = b / sin B = c / sin C and cosine rule a² = b² + c² − 2bc cos A are essential. Use the sine rule for two angles and a side, or two sides and a non-included angle. Use the cosine rule when you have three sides or two sides and the included angle.
对于非直角三角形，正弦定理 a / sin A = b / sin B = c / sin C 和余弦定理 a² = b² + c² − 2bc cos A 至关重要。当已知两角一边，或两边及其中一边的对角时使用正弦定理；当已知三边，或两边及其夹角时使用余弦定理。
5. Area and Volume Formulas | 面积与体积公式
Quickly recall these common area formulas: rectangle = length × width, triangle = ½ × base × vertical height, parallelogram = base × vertical height, trapezium = ½ (a + b) × h where a and b are the parallel sides, circle = πr², and circumference = 2πr or πd. For volumes: prism = area of cross-section × length, cylinder = πr²h, pyramid = ⅓ × area of base × vertical height, cone = ⅓πr²h, sphere = ⁴⁄₃πr³. Surface area of a sphere = 4πr², curved surface area of a cone = πrl where l is the slant height.
快速回顾以下常见面积公式：矩形 = 长 × 宽，三角形 = ½ × 底 × 垂直高，平行四边形 = 底 × 垂直高，梯形 = ½ (a + b) × h，其中 a 和 b 为平行边，圆 = πr²，周长 = 2πr 或 πd。体积方面：棱柱 = 横截面积 × 长，圆柱 = πr²h，棱锥 = ⅓ × 底面积 × 垂直高，圆锥 = ⅓πr²h，球体 = ⁴⁄₃πr³。球体表面积 = 4πr²，圆锥侧面积 = πrl，其中 l 为斜高。
6. Circle Theorems | 圆定理
Memorising circle theorems saves time in the exam. Key facts: the angle in a semicircle is always 90°. The angle at the centre is twice the angle at the circumference standing on the same arc. Angles in the same segment are equal. Opposite angles in a cyclic quadrilateral sum to 180°. The angle between a tangent and a chord equals the angle in the alternate segment. A radius meets a tangent at 90°. Two tangents from an external point are equal in length.
记住圆定理能节省考试时间。关键事实：半圆中的角始终是 90°。圆心角是同一弧所对圆周角的两倍。同弧上的圆周角相等。圆内接四边形对角和为 180°。切线与弦的夹角等于弦另一侧的圆周角（交替弓形定理）。半径与切线成 90°。从圆外一点引出的两条切线长度相等。
7. Probability Formulas | 概率公式
For single events, probability = number of favourable outcomes / total number of possible outcomes. Probabilities range from 0 to 1. For mutually exclusive events, P(A or B) = P(A) + P(B). For independent events, P(A and B) = P(A) × P(B). A useful tool is the probability tree diagram, where you multiply along branches and add between outcomes. Remember that the sum of probabilities of all branches from a single node is 1.
对于单一事件，概率 = 有利结果的数量 / 所有可能结果的总数。概率取值范围为 0 到 1。对于互斥事件，P(A 或 B) = P(A) + P(B)。对于独立事件，P(A 且 B) = P(A) × P(B)。一个实用的工具是概率树图，你沿着分支相乘，并在不同结果之间相加。记住，从一个节点出发的所有分支概率之和为 1。
8. Averages and Spread | 平均数与分散程度
The three measures of central tendency are mean, median and mode. Mean = sum of values ÷ number of values. Median = middle value when ordered. Mode = most frequent value. For grouped frequency tables, estimate the mean using class midpoints. Range = highest value − lowest value. Quartiles and percentiles divide data into four and one hundred parts; the interquartile range (IQR = Q₃ − Q₁) measures spread. A box plot displays these five values visually.
三种集中趋势的度量是平均值、中位数和众数。平均值 = 数值总和 ÷ 数据个数。中位数 = 排序后居中的值。众数 = 出现最频繁的值。对于分组频数表，使用组中点来估算平均值。极差 = 最大值 − 最小值。四分位数和百分位数将数据分成四等份和一百等份；四分位距 (IQR = Q₃ − Q₁) 衡量分散程度。箱线图将上述五个统计量可视化。
9. Transformations | 变换
Transformations of shapes include translation (sliding), reflection (mirroring), rotation (turning) and enlargement (resizing). In a translation, every point moves by the same vector ( a b ) (as a column). Reflection requires a mirror line, given by an equation such as x = 2 or y = x. Rotation needs a centre, angle and direction (clockwise or anticlockwise). Enlargement is defined by a scale factor and a centre of enlargement. When the scale factor is negative, the image appears inverted.
图形的变换包括平移（滑动）、反射（镜像）、旋转（转动）和放大（改变大小）。平移中每一点按照同一列向量 ( a b ) 移动。反射需要一个镜像线，用方程给出，如 x = 2 或 y = x。旋转需要中心、角度和方向（顺时针或逆时针）。放大由比例因子和放大中心定义。当比例因子为负时，图像会倒转。
10. Proportion and Ratios | 比例与比率
Direct proportion means as one quantity increases, the other increases at the same rate: y = kx. Inverse proportion means y = k / x, so as x increases, y decreases. For ratios, divide a quantity in the ratio a : b by finding the total number of parts (a + b) and calculating the value of each part. When working with compound measures like speed (distance / time), density (mass / volume) or pressure (force / area), use consistent units and set up proportional equations if needed.
正比例意味着一个量增加时，另一个量以相同速率增加：y = kx。反比例意味着 y = k / x，因此当 x 增加时 y 减少。对于比率，按 a : b 分配某个量，先求出总份数 (a + b)，再计算每份的值。处理复合量度如速度（距离/时间）、密度（质量/体积）或压强（力/面积）时，要使用一致的单位，必要时建立比例方程。
11. Laws of Indices | 指数定律
The laws of indices simplify expressions involving powers. When multiplying like bases, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ. When dividing, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ. Raising a power to a power: (aᵐ)ⁿ = aᵐ×ⁿ. Negative exponents mean reciprocals: a⁻ⁿ = 1 / aⁿ. Fractional exponents represent roots: a¹/₂ = √a, a¹/₃ = ³√a, and in general aᵐ/ⁿ = (ⁿ√a)ᵐ. Any non-zero number to the power zero is 1: a⁰ = 1.
指数定律可以简化含有乘方的表达式。同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。相除时指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。乘方的乘方：(aᵐ)ⁿ = aᵐ×ⁿ。负指数表示倒数：a⁻ⁿ = 1 / aⁿ。分数指数表示方根：a¹/₂ = √a，a¹/₃ = ³√a，一般地 aᵐ/ⁿ = (ⁿ√a)ᵐ。任何非零数的零次幂等于 1：a⁰ = 1。
12. Standard Form | 标准形
Standard form writes very large or very small numbers as a × 10ⁿ, where 1 ≤ a < 10 and n is an integer. For example, 45 000 000 = 4.5 × 10⁷, and 0.000 000 32 = 3.2 × 10⁻⁷. To add or subtract, convert to ordinary numbers or the same power of 10 first. To multiply, multiply the 'a' values and add the powers of 10. To divide, divide the 'a' values and subtract the powers of 10. Standard form appears frequently in science and measures contexts.
标准形用 a × 10ⁿ 表示很大或很小的数字，其中 1 ≤ a < 10 且 n 为整数。例如，45 000 000 = 4.5 × 10⁷，0.000 000 32 = 3.2 × 10⁻⁷。进行加减运算时，先转换为普通数字或将 10 的指数化为相同。相乘时，将 'a' 值相乘并将 10 的指数相加。相除时，将 'a' 值相除并将 10 的指数相减。标准形在科学和计量场景中经常出现。
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