📚 Year 11 AQA Maths: Quick Reference Formulas & Theorems | AQA 数学 Year 11 公式定理速查手册
This handbook brings together every essential formula and theorem for the Year 11 AQA Mathematics GCSE examination. Use it to reinforce your memory, check your working, and approach problems with confidence across algebra, geometry, trigonometry, statistics and more.
本手册汇集了 Year 11 AQA 数学 GCSE 考试中所有必考的公式和定理。用于巩固记忆、检查解题过程，并让你在代数、几何、三角、统计等板块中胸有成竹。
1. Quadratic Equations and the Formula | 二次方程与求根公式
A quadratic equation has the general form ax² + bx + c = 0 (a ≠ 0). Its solutions can be found directly using the quadratic formula.
二次方程的一般形式为 ax² + bx + c = 0（a ≠ 0）。其解可直接由求根公式得出。
x = (−b ± √b² − 4ac) / (2a)
The expression under the square root, Δ = b² − 4ac, is called the discriminant. It tells you the nature of the roots.
根号下的式子 Δ = b² − 4ac 称为判别式，可判断根的性质。
- If Δ > 0, there are two distinct real roots. 若 Δ > 0，则有两个不相等的实根。
- If Δ = 0, there is one repeated real root (or one real solution). 若 Δ = 0，则有一个重根（或唯一实解）。
- If Δ < 0, there are no real roots. 若 Δ < 0，则无实数根。
Always rearrange the equation into standard form before identifying a, b and c.
使用公式前务必将方程化为标准形式，确定 a、b、c。
2. Straight Line Coordinate Geometry | 直线坐标几何
The gradient (slope) of a line through two points (x₁, y₁) and (x₂, y₂) is given by:
过两点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率（梯度）公式：
m = (y₂ − y₁) / (x₂ − x₁)
The midpoint has coordinates:
中点坐标为：
M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
The distance between the two points is:
两点间的距离为：
d = √( (x₂ − x₁)² + (y₂ − y₁)² )
The equation of a straight line is commonly written as y = mx + c, where m is the gradient and c is the y-intercept. An alternative form is y − y₁ = m(x − x₁).
直线方程通常写为 y = mx + c，其中 m 为梯度，c 为 y 轴截距。也可用点斜式 y − y₁ = m(x − x₁)。
ual gradients. Perpendicular lines have gradients whose product is −1 (i.e. m₁ × m₂ = −1).
平行线梯度相等；垂直线梯度乘积为 −1（即 m₁ × m₂ = −1）。
3. Area of 2D Shapes | 平面图形的面积
Learn these core area formulas — many exam questions require you to combine them or work backwards to find a missing side.
牢记以下基本面积公式——考题常需组合使用或反求未知边长。
- Rectangle: A = length × width 长方形：A = 长 × 宽
- Triangle: A = ½ × base × height 三角形：A = ½ × 底 × 高
- Parallelogram: A = base × perpendicular height 平行四边形：A = 底 × 垂直高
- Trapezium: A = ½(a + b)h, where a and b are the parallel sides 梯形：A = ½(a + b)h，其中 a、b 为平行边
- Circle: A = πr², Circumference = 2πr = πd 圆：A = πr²，周长 = 2πr = πd
- Sector: A = (θ/360) × πr², 弧长 = (θ/360) × 2πr 扇形：A = (θ/360) × πr²，弧长 = (θ/360) × 2πr
For composite shapes, split the figure into standard shapes, find the area of each, and sum them.
对组合图形，可分解为标准图形，分别求面积再相加。
4. Volume and Surface Area of 3D Solids | 立体图形的体积和表面积
Volume is the space a solid occupies. Surface area is the total area of all its faces.
体积指立体图形所占空间；表面积为所有面的总面积。
- Cuboid: V = l × w × h, TSA = 2(lw + lh + wh) 长方体：V = 长×宽×高，TSA = 2(长×宽 + 长×高 + 宽×高)
- Prism: V = area of cross-section × length 棱柱：V = 横截面积 × 长度
- Cylinder: V = πr²h, Curved SA = 2πrh, Total SA = 2πr² + 2πrh 圆柱：V = πr²h，侧面积 = 2πrh，总表面积 = 2πr² + 2πrh
- Pyramid / Cone: V = ⅓ × area of base × height 棱锥/圆锥：V = ⅓ × 底面积 × 高
- Sphere: V = (4/3)πr³, SA = 4πr² 球：V = (4/3)πr³，SA = 4πr²
For a cone, the curved surface area is πrl, where l is the slant height. Remember l = √r² + h².
圆锥的侧面积为 πrl，其中 l 为斜高，满足 l = √r² + h²。
5. Pythagoras and Trigonometry | 勾股定理与三角学
In a right‑angled triangle with hypotenuse c, c² = a² + b². This is Pythagoras’ theorem.
在直角三角形中，斜边为 c，则 c² = a² + b²。这就是勾股定理。
The three trigonometric ratios (SOH CAH TOA) are:
三个基本三角比（SOH CAH TOA）为：
sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent
sin θ = 对边/斜边, cos θ = 邻边/斜边, tan θ = 对边/邻边
For any triangle (not just right‑angled), the sine rule holds:
对任意三角形（不限于直角三角形），正弦定理成立：
a / sin A = b / sin B = c / sin C （或 sin A / a = sin B / b = sin C / c）
The cosine rule connects a side with the opposite angle:
余弦定理联系边与对角：
a² = b² + c² − 2bc cos A
The area of any triangle can be calculated using two sides and the included angle:
任意三角形的面积可用两边及其夹角求出：
Area = ½ ab sin C
Always label triangles carefully: side a is opposite angle A, etc.
务必正确标记三角形：边 a 对应对角 A，以此类推。
6. Probability | 概率
Probability measures how likely an event is. It is always between 0 and 1 (or 0% and 100%).
概率度量事件发生的可能性，其值介于 0 到 1 之间（或 0% 至 100%）。
For mutually exclusive events (cannot happen together): P(A or B) = P(A) + P(B).
互斥事件（不能同时发生）：P(A 或 B) = P(A) + P(B)。
For non‑mutually exclusive events: P(A or B) = P(A) + P(B) − P(A and B).
非互斥事件：P(A 或 B) = P(A) + P(B) − P(A 且 B)。
For independent events (one does not affect the other): P(A and B) = P(A) × P(B).
独立事件（一事件不影响另一事件）：P(A 且 B) = P(A) × P(B)。
With a tree diagram, multiply along branches for combined events
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