📚 Year 8 WJEC Further Maths: Formula & Theorem Quick Reference Guide | Year 8 WJEC 进阶数学：公式定理速查手册
This quick reference guide brings together all the essential formulas, theorems, and rules you will meet in Year 8 WJEC Further Maths. Keep it handy when revising, completing homework, or preparing for assessments. Every entry includes a clear statement of the rule, an example, and paired bilingual explanations to support learners working in both English and Chinese.
这本速查手册汇总了 Year 8 WJEC 进阶数学中你将遇到的所有核心公式、定理和法则。复习、写作业或备考时可以随时翻看。每个条目都配有清晰的规则陈述、例子以及中英对照讲解，方便双语学习者查阅。
1. Pythagoras' Theorem | 毕达哥拉斯定理
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. Written as an equation: a² + b² = c², where c is the hypotenuse.
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。用等式表示为：a² + b² = c²，其中 c 是斜边。
For example, if a = 3 cm and b = 4 cm, then c² = 3² + 4² = 9 + 16 = 25, so c = √25 = 5 cm.
例如，若 a = 3 cm, b = 4 cm，则 c² = 3² + 4² = 9 + 16 = 25，因此 c = √25 = 5 cm。
You can also use it to find a shorter side: a² = c² − b².
你也可以用它求直角边：a² = c² − b²。
2. Area Formulas for 2D Shapes | 二维图形的面积公式
Rectangle: Area = length × width. Square: Area = side².
矩形：面积 = 长 × 宽。正方形：面积 = 边长²。
Triangle: Area = ½ × base × vertical height. Remember the height must be perpendicular to the base.
三角形：面积 = ½ × 底 × 垂直高。注意高必须与底边垂直。
Parallelogram: Area = base × vertical height.
平行四边形：面积 = 底 × 垂直高。
Trapezium: b are the lengths of the parallel sides and h is the perpendicular distance between them.
梯形：面积 = ½ × (a + b) × h，其中 a 和 b 是两条平行边的长度，h 是它们之间的垂直距离。
Circle: Area = π × r² (use π ≈ 3.14 or the π button on your calculator).
圆：面积 = π × r²（一般取 π ≈ 3.14 或使用计算器上的 π 键）。
3. Circumference of a Circle | 圆的周长
Circumference is the distance around the circle. The formula is C = π × d or C = 2 × π × r, where d is the diameter and r is the radius.
周长是围绕圆一圈的长度。公式为 C = π × d 或 C = 2 × π × r，其中 d 是直径，r 是半径。
If a circle has radius 7 cm, C = 2 × π × 7 ≈ 2 × 3.14 × 7 = 43.96 cm.
如果圆的半径为 7 cm，则 C = 2 × π × 7 ≈ 2 × 3.14 × 7 = 43.96 cm。
4. Volume of Prisms | 棱柱的体积
A prism is a 3D shape with a constant cross‑section. Its volume is found by multiplying the area of the cross‑section by the length (or height) of the prism: Volume = area of cross‑section × length.
棱柱是一种具有恒定横截面的三维图形。它的体积等于横截面面积乘以棱柱的长（或高）：体积 = 横截面面积 × 长。
For a cuboid, the cross‑section is a rectangle, so Volume = l × w × h. For a cylinder, the cross‑section is a circle, so Volume = π r² h.
对于长方体，横截面是矩形，因此体积 = 长 × 宽 × 高。对于圆柱体，横截面是圆，因此体积 = π r² h。
5. Laws of Indices | 指数律
When multiplying powers with the same base, add the exponents: aᵐ × aⁿ = aᵐ⁺ⁿ.
同底数幂相乘，指数相加：aᵐ × aⁿ = aᵐ⁺ⁿ。
When dividing, subtract the exponents: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.
同底数幂相除，指数相减：aᵐ ÷ aⁿ = aᵐ⁻ⁿ。
Raising a power to another power: (aᵐ)ⁿ = aᵐ×ⁿ.
幂的乘方：(aᵐ)ⁿ = aᵐ×ⁿ。
Any non‑zero number to the power of zero is 1: a⁰ = 1 (a ≠ 0).
任何非零数的零次幂等于 1：a⁰ = 1 (a ≠ 0)。
Negative exponents give reciprocals: a⁻ⁿ = 1 / aⁿ.
负指数表示倒数：a⁻ⁿ = 1 / aⁿ。
6. Expanding Brackets | 展开括号
Use the distributive law to multiply a term outside the bracket by every term inside: a(b + c) = ab + ac.
利用乘法分配律，将括号外的项乘以括号内的每一项：a(b + c) = ab + ac。
When expanding two binomials, multiply each term in the first bracket by each term in the second: (x + a)(x + b) = x² + ax + bx + ab, which simplifies to x² + (a + b)x + ab.
展开两个二项式时，将第一个括号中的每一项乘以第二个括号中的每一项：(x + a)(x + b) = x² + ax + bx + ab，化简为 x² + (a + b)x + ab。
Example: (x + 3)(x − 2) = x² + 3x − 2x − 6 = x² + x − 6.
例子：(x + 3)(x − 2) = x² + 3x − 2x − 6 = x² + x − 6。
7. Factorising Expressions | 因式分解
Factorising is the reverse of expanding. Look for the highest common factor (HCF) of all terms and place it outside the bracket: ab + ac = a(b + c).
因式分解是展开的逆运算。找出所有项的最高公因式 (HCF)，把它提到括号外面：ab + ac = a(b + c)。
For quadratics of the form x² + bx + c, find two numbers that multiply to c and add to b: x² + bx + c = (x + p)(x + q), where p + q = b and pq = c.
对于形如 x² + bx + c 的二次式，找出两个数，它们的乘积为 c，和为 b：x² + bx + c = (x + p)(x + q)，其中 p + q = b，pq = c。
Example: x² + 5x + 6 = (x + 2)(x + 3) because 2 + 3 = 5 and 2 × 3 = 6.
例子：x² + 5x + 6 = (x + 2)(x + 3)，因为 2 + 3 = 5 且 2 × 3 = 6。
8. Solving Linear Equations | 解一元一次方程
To solve an equation, perform the same operation on both sides to isolate the unknown. Always keep the equation balanced.
解方程时，在等号两边同时进行相同的运算，以便把未知数单独留在一边。始终保持等式平衡。
Step‑by‑step: simplify each side, move variable terms to one side and constants to the other, then divide by the coefficient.
步骤：先化简每一边，再把含有未知数的项移到一边，常数项移到另一边，最后除以未知数的系数。
Example: 3x + 5 = 20 → subtract 5: 3x = 15 → divide by 3: x = 5.
例子：3x + 5 = 20 → 两边减 5：3x = 15 → 两边除以 3：x = 5。
9. Angle Rules | 角度规则
Angles on a straight line add up to 180°. Angles around a point sum to 360°.
直线上的角之和为 180°。绕一点一周的角之和为 360°。
Vertically opposite angles are equal.
对顶角相等。
Angles in a triangle always total 180°. In a quadrilateral, the interior angles add up to 360°.
三角形内角和总是 180°。四边形的内角和为 360°。
On parallel lines, alternate angles are equal, corresponding angles are equal, and co‑interior (allied) angles sum to 180°.
在平行线中，内错角相等，同位角相等，同旁内角互补（和为 180°）。
10. Mean, Median, Mode and Range | 平均数、中位数、众数和极差
Mean: sum of all data values divided by the number of values.
平均数：所有数据值的总和除以数据的个数。
Median: the middle value when the data are arranged in order. If there are two middle values, take their average.
中位数：将数据按大小排列后最中间的那个数。如果有两个中间数，则取它们的平均值。
Mode: the value that appears most often. A set can have one mode, more than one mode, or no mode at all.
众数：出现次数最多的数据值。一组数据可以有一个众数、多个众数，也可能没有众数。
Range: the largest value minus the smallest value. It measures the spread of the data.
极差：最大值减去最小值。它用来衡量数据的分散程度。
11. Probability Basics | 概率基础
The probability of an event happening is given by: P(event) = number of favourable outcomes / total number of possible outcomes, provided all outcomes are equally likely.
事件发生的概率公式为：P(事件) = 有利结果的数量 / 所有可能结果的总数，前提是所有结果发生的可能性相等。
Probabilities lie between 0 (impossible) and 1 (certain). They can be written as fractions, decimals, or percentages.
概率值介于 0（不可能发生）和 1（必然发生）之间，可以用分数、小数或百分数表示。
The sum of probabilities of all possible outcomes is 1. The probability of an event not happening is 1 − P(event).
所有可能结果的概率之和为 1。某事件不发生的概率为 1 − P(事件)。
12. Equation of a Straight Line | 直线方程
The equation of a straight line graph is usually written as y = mx + c, where m is the gradient (slope) and c is the y‑intercept (where the line crosses the y‑axis).
直线图像的方程通常写作 y = mx + c，其中 m 是斜率（坡度），c 是 y 轴截距（直线与 y 轴的交点）。
m = change in y / change in x = (y₂ − y₁) / (x₂ − x₁) between any two points on the line.
斜率 m = y 的变化量 / x 的变化量 = (y₂ − y₁) / (x₂ − x₁)，取自直线上任意两点。
gative gradient means it slopes downwards. A horizontal line has m = 0 and equation y = c.
斜率为正，直线向上倾斜；斜率为负，直线向下倾斜。水平线的斜率 m = 0，方程为 y = c。
13. Ratio and Proportion | 比和比例
A ratio compares two or more quantities. It is written in the form a : b and should be simplified by dividing both sides by their highest common factor.
比用来比较两个或更多的数量，写作 a : b 的形式，并且应该通过同时除以它们的最大公因数来化简。
To divide a quantity in a given ratio, add the parts to find the total number of shares, then multiply the quantity by each part over the total.
按给定比例分配一个量时，先把比的各项加起来算出总份数，再把该量乘以每项占总份数的比例。
Example: Divide £60 in the ratio 2 : 3. Total parts = 5. First part = (2/5) × 60 = £24, second part = (3/5) × 60 = £36.
例子：按 2 : 3 分配 £60。总份数 = 5，第一份 = (2/5) × 60 = £24，第二份 = (3/5) × 60 = £36。
gether at the same rate. If y is directly proportional to x, then y = kx for a constant k.
正比例意味着两个量以相同的比率同时增加或减少。如果 y 与 x 成正比例，那么 y = kx，其中 k 为常数。
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