📚 Year 8 AQA Maths: Formula & Theorem Quick Reference Handbook | Year 8 AQA 数学：公式定理速查手册
This quick reference handbook brings together all the essential formulas, rules and theorems you will meet in Year 8 AQA Mathematics. Keep it handy for homework, revision and end‑of‑topic tests. Each entry is paired with a clear explanation in both English and Chinese, so you can deepen your understanding while building bilingual mathematical fluency.
这本速查手册汇集了 Year 8 AQA 数学课程中所有关键的公式、法则和定理。无论是做作业、复习还是单元测验，都可以随时查阅。每个条目都配有中英文解释，帮助你在巩固知识的同时提升双语数学表达能力。
1. Number Properties and Laws | 数的性质与运算定律
thout changing the result. For addition: a + b = b + a; for multiplication: a × b = b × a.
交换律指出，加法和乘法中可以交换数字的顺序，结果不变。加法：a + b = b + a；乘法：a × b = b × a。
mbers when adding or multiplying: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
结合律允许你在加法或乘法中重新分组： (a + b) + c = a + (b + c) 以及 (a × b) × c = a × (b × c)。
= a × b + a × c. It is the foundation for expanding brackets and factorising expressions.
分配律把乘法和加法联系起来：a × (b + c) = a × b + a × c。这是去括号和因式分解的基础。
2. Fractions, Decimals and Percentages | 分数、小数与百分数
on to a decimal, divide the numerator by the denominator. For example, 3/4 = 3 ÷ 4 = 0.75.
将分数化成小数，用分子除以分母。例如 3/4 = 3 ÷ 4 = 0.75。
To change a decimal to a percentage, multiply by 100 and add the % sign: 0.6 = 0.6 × 100% = 60%.
把小数化成百分数，乘以 100 再加上百分号：0.6 × 100% = 60%。
To write a percentage as a fraction, put it over 100 and simplify: 25% = 25/100 = 1/4.
把百分数写成分数，将其放在 100 的分母上再约分：25% = 25/100 = 1/4。
Fraction → Decimal: numerator ÷ denominator
分数 → 小数：分子 ÷ 分母
Decimal → Percentage: × 100%
小数 → 百分数：× 100%
3. Area Formulas | 面积公式
The area of a rectangle (or square) is length × width. A = l × w.
矩形（或正方形）的面积等于长 × 宽。A = l × w。
A triangle's area is half the base times the vertical height. A = ½ × b × h.
三角形的面积等于底乘高的一半。A = ½ × b × h。
For a parallelogram, area = base × perpendicular height. A = b × h.
平行四边形的面积 = 底 × 垂直高。A = b × h。
rapezium is the average of the parallel sides multiplied by the height. A = ½ (a + b) × h.
梯形的面积等于两平行边之和的一半乘以高。A = ½ (a + b) × h。
A circle' is π × radius². A = π r². Use the π button on your calculator or 3.14 as an approximation.
圆的面积等于 π × 半径²。A = π r²。可以用计算器上的 π 键或近似值 3.14。
| Shape | Formula (Area) |
|---|---|
| Rectangle | A = l × w |
| Triangle | A = ½ × b × h |
| Parallelogram | A = b × h |
| Trapezium | A = ½ (a + b) × h |
| Circle | A = π r² |
4. Volume and Surface Area | 体积与表面积
Volume of a cuboid (rectangular prism) = length × width × height. V = l × w × h.
长方体的体积 = 长 × 宽 × 高。V = l × w × h。
me = area of cross-section × length. This works for triangular prisms, cylinders and more.
任何棱柱（包括圆柱）的体积 = 横截面积 × 长度。适用于三棱柱、圆柱等。
Surface area of a cuboid = 2(lw + lh + wh). Add the areas of all six rectangular faces.
长方体的表面积 = 2(lw + lh + wh)。即六个矩形面的面积之和。
The curved surface area of a cylinder is 2πrh; total surface area = 2πr² + 2πrh.
圆柱的侧面积为 2πrh，总表面积 = 2πr² + 2πrh。
- Cuboid volume: V = lwh
- Prism volume: V = cross‑section area × length
- Cylinder volume: V = πr²h
5. Angle Facts and Triangle Theorems | 角度常识与三角形定理
Angles on a straight line add up to 180°. Angles around a point sum to 360°.
平角（直线上的角）之和为 180°。周角（一点周围的角）之和为 360°。
Vertically opposite angles are equal. When two lines cross, opposite angles match.
对顶角相等。两条直线相交时，对顶角相等。
The sum of interior angles in a triangle is always 180°. In a quadrilateral it is 360°.
三角形的内角和恒为 180°。四边形的内角和为 360°。
For any polygon, the sum of interior angles = (n – 2) × 180°, where n is the number of sides.
任意多边形的内角和 = (n – 2) × 180°，其中 n 为边数。
The exterior angle of a polygon is 180° minus the interior angle. The sum of exterior angles (one at each vertex) is always 360°.
多边形的外角 = 180° – 相邻内角。所有外角之和（每个顶点取一个）恒为 360°。
6. Parallel Lines and Angle Rules | 平行线与角度规则
When a transversal crosses parallel lines, corresponding angles are equal. (They sit in the same position at each intersection.)
当一条截线与两条平行线相交时，同位角相等。（同位角位于相同的位置。）
Alternate angles are also equal. They are inside the parallel lines and on opposite sides of the transversal, forming a 'Z' shape.
内错角也相等。它们在平行线内侧，位于截线两侧，呈 'Z' 形。
Co‑interior (allied) angles add up to 180°. They lie inside the parallel lines on the same side of the transversal, looking like a 'C' shape.
同旁内角互补，和为 180°。它们位于平行线内侧，在截线的同一侧，呈 'C' 形。
7. Pythagoras' Theorem | 毕达哥拉斯定理（勾股定理）
In any right‑angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two legs. This helps you find a missing side length.
在任何一个直角三角形中，斜边（最长边）的平方等于两条直角边的平方和。这可以用来求未知边长。
a² + b² = c²
直角边² + 直角边² = 斜边²
To find the hypotenuse c, use c = √a² + b². To find a leg a, use a = √c² – b². Always check that the triangle is right‑angled first.
求斜边 c：c = √a² + b²。求直角边 a：a = √c² – b²。使用前务必确认三角形为直角三角形。
8. Algebraic Expansion and Factorisation | 代数式的展开与因式分解
ltiplying a term by a bracket: a(b + c) = ab + ac. This is the distributive law in action.
单项式乘多项式：a(b + c) = ab + ac。这是分配律的直接应用。
= x² + (a + b)x + ab. Multiply each term in the first bracket by each term in the second.
两个二项式相乘：(x + a)(x + b) = x² + (a + b)x + ab。把第一个括号中的每一项分别乘以第二个括号中的每一项。
Special product – difference of two squares: (a + b)(a – b) = a² – b².
特殊乘积——平方差公式：(a + b)(a – b) = a² – b²。
Perfect square trinomials: (a + b)² = a² + 2ab + b² and (a – b)² = a² – 2ab + b².
完全平方公式：(a + b)² = a² + 2ab + b² 以及 (a – b)² = a² – 2ab + b²。
Factorising is the reverse process. Always look for the highest common factor first, then check for special patterns.
因式分解是展开的逆运算。先提取最大公因式，再检查是否符合特殊模型。
9. Equations and Linear Graphs | 方程与线性图像
straight line is often written as y = mx + c. m is the gradient (steepness) and c is the y‑intercept (where the line crosses the y‑axis).
直线方程通常写作 y = mx + c。m 代表斜率（倾斜程度），c 代表 y 轴截距（图像与 y 轴的交点）。
Gradient between two points (x₁, y₁) and (x₂, y₂) is m = (y₂ – y₁) / (x₂ – x₁). Rise over run.
两点 (x₁, y₁) 和 (x₂, y₂) 之间的斜率 m = (y₂ – y₁) / (x₂ – x₁)，即纵向变化量除以横向变化量。
ration on both sides until the unknown is isolated. Example: 2x + 5 = 13 → 2x = 8 → x = 4.
解一元一次方程时，等式两边同时进行相同的运算，直到未知数单独出现在一侧。例：2x + 5 = 13 → 2x = 8 → x = 4。
10. Sequences – nth Term | 数列 – 第 n 项公式
An arithmetic sequence has a constant difference between consecutive terms. The nth term formula lets you find any term directly.
等差数列的相邻两项之差（公差）保持不变。第 n 项公式可以直接求出数列中的任意一项。
nth term = dn + (a – d)
第 n 项 = 公差 × n + (首项 – 公差)
Here d is the common difference and a is the first term. For example, in the sequence 3, 7, 11, 15, … the difference is 4 and the first term is 3, so the nth term is 4n – 1.
其中 d 是公差，a 是首项。例如数列 3, 7, 11, 15, … 的公差是 4，首项是 3，所以第 n 项公式为 4n – 1。
To check a formula, substitute n = 1, 2, 3… and see if you get the original terms.
检验公式时，将 n = 1, 2, 3… 代入，看是否得到原数列的项。
11. Speed, Distance and Time | 速度、路程与时间
The three quantities are linked by the compound measure formula. If you know two, you can calculate the third.
这三个量通过复合单位公式紧密相连。已知其中两个，就能求出第三个。
Speed = Distance ÷ Time
速度 = 路程 ÷ 时间
tance ÷ Speed. Always make sure units are consistent (e.g. km with hours, m with seconds).
路程 = 速度 × 时间，时间 = 路程 ÷ 速度。要注意单位统一（例如公里与小时搭配，米与秒搭配）。
A useful triangle to remember: place D at the top, S and T at the bottom corners. Cover the one you want to find.
记忆技巧：将 D 放在三角形顶端，S 和 T 放在底部两角。遮住想求的量，由剩下的两个量决定运算。
12. Statistics – Mean, Median, Mode and Range | 统计 – 平均数、中位数、众数和极差
The mean is the total of all values divided by the number of values. Mean = Σx / n.
平均数（均值）等于所有数值之和除以数据个数。平均数 = Σx / n。
The median is the middle value when data is ordered. With an even number of values, take the average of the two middle numbers.
中位数是将数据排序后处于中间位置的数值。当数据个数为偶数时，取中间两个数的平均值。
The mode is the value that appears most often. A set can have one mode, more than one mode, or no mode at all.
众数是出现次数最多的数值。一组数据可能有一个众数、多个众数，也可能没有众数。
The range is the difference between the largest and smallest values. Range = max – min.
极差等于最大值与最小值之差。极差 = 最大值 – 最小值。
These four measures help you describe and compare data sets quickly.
这四个统计量可以帮助你快速描述和比较数据集。
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