📚 KS3 WJEC Chinese Mathematics: Formula and Theorem Quick Reference Handbook | KS3 WJEC 中文数学：公式定理速查手册
Mastering the key formulas and theorems is an essential step towards success in Key Stage 3 Mathematics. This bilingual quick reference handbook is designed to give you instant access to the most important facts you need to know, with clear explanations in both English and Chinese. Whether you are revising for a WJEC assessment, completing a homework task, or simply reinforcing your classroom learning, this resource will help you build confidence and accuracy.
掌握关键公式和定理是通往 KS3 数学成功的重要一步。这本双语速查手册旨在让你即时查看到必须掌握的最重要知识点，并以清晰的英文和中文进行解释。无论你是在为 WJEC 评估复习、完成家庭作业，还是仅仅巩固课堂所学，这个资源都将帮助你增强信心，提高准确性。
1. Perimeter and Area | 周长与面积
ctangle, the perimeter is given by P = 2(l + w), where l is the length and w is the width.
形状的周长是指围绕其外边缘的总长度。对于矩形，周长公式为 P = 2(长 + 宽)，其中 l 表示长度，w 表示宽度。
The area of a rectangle is found by multiplying its length by its width: A = l × w.
矩形的面积通过长乘以宽求得：A = 长 × 宽。
e area is A = ½ × base × height. The height must be measured at a right angle to the base.
对于三角形，面积公式为 A = ½ × 底 × 高。高必须是相对于底边的垂直距离。
nd b are the lengths of the parallel sides and h is the perpendicular height between them.
平行四边形的面积同样是底 × 高 (A = b × h)，而梯形的面积计算为 A = ½ (a + b) × 高，其中 a 和 b 是两条平行边的长度，h 是它们之间的垂直高度。
2. Volume and Surface Area | 体积与表面积
of a cuboid (rectangular prism) is V = length × width × height, often written as V = lwh.
体积衡量的是三维形状内部的空间。长方体（矩形棱柱）的体积是 V = 长 × 宽 × 高，通常写作 V = lwh。
ace area of a cuboid is the total area of all six rectangular faces: SA = 2(lw + lh + wh).
长方体的表面积是所有六个矩形面的总面积：SA = 2(lw + lh + wh)。
For a cube with side length s, the volume is V = s³ and the surface area is SA = 6s².
对于边长为 s 的立方体，体积为 V = s³，表面积为 SA = 6s²。
Prisms have a constant cross‑section; their volume is always V = area of cross‑section × length.
棱柱具有恒定的横截面；其体积总是 V = 横截面积 × 长度。
3. Pythagoras' Theorem | 勾股定理
In a right‑ ides. This is written as a² + b² = c², where c is the hypotenuse and a and b are the legs.
在直角三角形中，斜边（直角所对的边）的平方等于另外两条边的平方和。这写作 a² + b² = c²，其中 c 是斜边，a 和 b 是直角边。
You can use Pythagoras' theorem to find the length of an unknown side in a right‑angled triangle. To find the hypotenuse, use c = √a² + b². To find a shorter side, rearrange to a = √c² − b².
你可以利用勾股定理来求直角三角形中未知边的长度。求斜边用 c = √a² + b²。求一条直角边，则变形为 a = √c² − b²。
Pythagoras' theorem only works for right‑angled triangles. Always check that the triangle has a 90° angle before using it.
勾股定理仅适用于直角三角形。使用前务必确认三角形有一个 90° 的角。
4. Angle Facts | 角度基本事实
Angles on a straight line add up to 180°.
平角（直线上的邻角）之和为 180°。
Angles around a point sum to 360°.
围绕一个点的周角之和为 360°。
Vertically opposite angles are equal.
对顶角相等。
When two parallel lines are crossed by a transversal, corresponding angles are equal, alternate angles are equal, and co‑interior (allied) angles add up to 180°.
当两条平行线被一条截线所截时，同位角相等，内错角相等，同旁内角（互补）之和为 180°。
Angles in a triangle always total 180°. Angles in a quadrilateral total 360°.
三角形的内角和总是 180°。四边形的内角和总是 360°。
5. Properties of Triangles and Quadrilaterals | 三角形与四边形的性质
Triangles can be classified by their sides: equilateral (all sides equal, all angles 60°), isosceles (two equal sides, two equal base angles), and scalene (no equal sides, no equal angles).
三角形可按边分类：等边三角形（三边相等，三个角均为 60°），等腰三角形（两条边相等，两个底角相等），不等边三角形（无边相等，无角相等）。
Special quadrilaterals include squares (four equal sides, four 90° angles), rectangles (opposite sides equal, four 90° angles), parallelograms (opposite sides parallel and equal, opposite angles equal), rhombuses (four equal sides, opposite angles equal, diagonals bisect at 90°), trapeziums (one pair of parallel sides) and kites (two pairs of adjacent equal sides, one pair of opposite angles equal).
特殊的四边形包括正方形（四条边相等，四个 90° 角），矩形（对边相等，四个 90° 角），平行四边形（对边平行且相等，对角相等），菱形（四条边相等，对角相等，对角线互相垂直平分），梯形（一组对边平行）和风筝形（两组邻边相等，一组对角相等）。
The sum of the exterior angles of any convex polygon is always 360°, regardless of the number of sides.
任何凸多边形的外角和总是 360°，与边数无关。
6. Algebraic Expressions | 代数表达式
r subtracting their coefficients. For example, 3x + 5x simplifies to 8x, and 7y − 2y = 5y.
同类项可以通过加减其系数进行合并。例如，3x + 5x 简化为 8x，7y − 2y = 5y。
brackets, multiply each term inside the bracket by the factor outside: a(b + c) = ab + ac.
展开括号时，将括号外的因数乘以括号内的每一项：a(b + c) = ab + ac。
Factorisation is the reverse of expansion. For instance, 6x + 9 can be factorised to 3(2x + 3) by taking out the highest common factor.
因式分解是展开的逆过程。例如，6x + 9 可以提取最大公因数分解为 3(2x + 3)。
An expression like (x + a)(x + b) expands to x² + (a+b)x + ab. This pattern helps when expanding double brackets.
像 (x + a)(x + b) 这样的表达式展开为 x² + (a+b)x + ab。这个模式有助于双括号的展开。
7. Linear Equations | 线性方程
A linear equation is an equation where the unknown variable is raised to the power of 1. To solve it, perform inverse operations to isolate the variable on one side of the equals sign.
线性方程是未知变量的指数为 1 的方程。要解方程，可进行逆运算以将变量单独留在等号的一边。
rst subtract 5 from both sides to get 2x = 8, then divide both sides by 2 to obtain x = 4.
例如，解方程 2x + 5 = 13，首先两边减去 5 得到 2x = 8，然后两边除以 2 得到 x = 4。
Always check your solution by substituting it back into the original equation.
务必将所得的解代入原方程进行检验。
uld be solved by collecting the variable terms on one side and the constants on the other.
对于两边都有未知数的方程，如 3x − 2 = x + 6，应将含变量的项移到一边，常数项移到另一边来求解。
8. Ratio and Proportion | 比与比例
A ratio compares two or more quantities. Ratios can be simplified by dividing all parts by their greatest common divisor. For example, 12:8 simplified is 3:2.
比用于比较两个或多个量。比可以通过将所有部分除以它们的最大公约数来化简。例如，12:8 化简后为 3:2。
To divide a quantity in a given ratio, add the parts, then calculate the value of one share. For instance, dividing £70 in the ratio 2:5 gives shares of £20 and £50.
要按照给定比例分配一个量，先将比的各项相加，然后计算一份的值。例如，以 2:5 的比例分配 70 英镑，得到 20 英镑和 50 英镑。
Proportion describes a multiplicative relationship. Two quantities are in direct proportion if they increase or decrease at the same rate. They are in inverse proportion if one increases as the other decreases at a constant product.
比例描述一种乘法关系。如果两个量以相同的速率增加或减少，则它们成正比。如果一个量随着另一个量的增加而减少且乘积恒定，则它们成反比。
The unitary method is useful for solving proportion problems: find the value of one unit first, then multiply to find the required amount.
归一法对解决比例问题很有用：先计算一个单位的量，然后乘以需要的数量。
9. Percentages | 百分数
ator of 100. To convert a fraction to a percentage, multiply by 100. For example, 3/4 = 75%.
百分数是分母为 100 的分数。要将分数转换为百分数，乘以 100。例如，3/4 = 75%。
a percentage of an amount, change the percentage to a decimal or fraction and multiply. 20% of 80 is 0.20 × 80 = 16.
要求一个量的百分之几，先将百分数转化为小数或分数再相乘。80 的 20% 是 0.20 × 80 = 16。
Percentage increase or decrease is calculated using: change / original × 100%. A price rise from £50 to £60 is a 20% increase.
百分比增减的计算公式为：变化量 / 原值 × 100%。价格从 50 英镑涨到 60 英镑，涨幅为 20%。
verse a percentage change (find the original amount), divide by the multiplier. After a 15% decrease, an item costs £34; the original price is £34 ÷ 0.85 = £40.
要逆向计算百分比变化（求原值），可以除以乘数。一件商品降价 15% 后售价为 34 英镑，原价为 34 ÷ 0.85 = 40 英镑。
10. Averages and Range | 平均数与范围
The three common averages are the mean, the median and the mode. The mean is the sum of all values divided by the number of values.
三种常见的平均数是平均数、中位数和众数。平均数（算术平均）是所有数值之和除以数值的个数。
The median is the middle value when data are ordered. If there are two middle numbers, the median is their mean. The mode is the value that appears most often.
中位数是将数据排序后位于中间的值。如果有两个中间的数，中位数是这两个数的平均数。众数是出现次数最多的值。
The range measures how spread out the data are. Range = highest value − lowest value.
极差（范围）衡量数据的分散程度。极差 = 最大值 − 最小值。
An outlier can significantly affect the mean but has little impact on the median and mode.
异常值会对平均数产生较大影响，但对中位数和众数的影响很小。
11. Probability Basics | 概率基础
The probability of an event is a number between 0 and 1 that describes how likely it is to happen. It can be expressed as a fraction, decimal or percentage.
事件的概率是一个介于 0 和 1 之间的数，用来描述事件发生的可能性大小。它可以用分数、小数或百分数表示。
le outcomes / total number of possible outcomes, provided all outcomes are equally likely.
概率 = 有利结果的数量 / 所有可能结果的总数，前提是每个结果发生的可能性相同。
The probabilities of all possible outcomes add up to 1. So the probability of an event not occurring is 1 − P(event).
所有可能结果的概率之和为 1。因此，事件不发生的概率为 1 − P(事件发生)。
In KS3, you also work with sample space diagrams, two‑way tables and frequency trees to list outcomes and calculate probabilities.
在 KS3 阶段，你还会使用样本空间图表、双向表格和频率树来列出结果并计算概率。
12. Unit Conversions | 单位换算
00 m, 1 m = 100 cm, 1 cm = 10 mm, 1 kg = 1000 g, 1 litre = 1000 ml, and 1 tonne = 1000 kg.
必须掌握的常用公制换算包括：1 公里 = 1000 米，1 米 = 100 厘米，1 厘米 = 10 毫米，1 千克 = 1000 克，1 升 = 1000 毫升，1 吨 = 1000 千克。
10000 cm² (since 100 cm × 100 cm). For volume, the factor is cubed: 1 m³ = 1 000 000 cm³.
进行面积单位换算时，记住换算倍数是平方关系。1 平方米 = 10000 平方厘米（因为 100 cm × 100 cm）。对于体积，倍数是立方关系：1 立方米 = 1 000 000 立方厘米。
1 minute = 60 seconds. Speed can be calculated using the formula: speed = distance / time.
处理时间时，记住 1 小时 = 60 分钟，1 分钟 = 60 秒。速度可以用公式计算：速度 = 距离 / 时间。
Use proportion to convert between currencies. If the exchange rate is known, multiplying or dividing by that rate gives the converted amount.
可以用比例方法进行货币换算。知道汇率后，乘以或除以该汇率即可得到换算后的金额。
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