📚 Formula and Theorem Quick Reference Handbook | 公式定理速查手册
This quick reference handbook is designed for Year 8 CIE Chinese learners to master essential mathematical formulas and theorems with clear bilingual explanations. It covers key topics from geometry, algebra, statistics, and everyday problem-solving, allowing you to check rules instantly and build confidence in both Chinese and English mathematical language.
本速查手册专为 Year 8 CIE 中文学习者编写，用清晰的中英双语解释帮助掌握核心数学公式与定理。涵盖几何、代数、统计和日常应用中的关键知识点，让你随时查阅、快速巩固，同时提升用中文和英文表述数学概念的能力。
1. Area of Common Shapes | 常见图形面积
The area of a shape measures the surface it covers. Below are the most frequently used area formulas for quadrilaterals and triangles.
图形的面积衡量其所占平面的大小。以下是四边形和三角形最常用的面积公式。
Rectangle: A = l × w
Area equals length times width.
面积等于长乘以宽。
矩形面积：A = 长 × 宽
Triangle: A = ½ × b × h
Area is half of base times vertical height.
面积等于底乘以高的一半。
三角形面积：A = ½ × 底 × 高
Parallelogram: A = b × h
Area is base multiplied by the perpendicular height.
面积等于底乘以垂直高度。
平行四边形面积：A = 底 × 高
Trapezium: A = ½ × (a + b) × h
Area is half the sum of the parallel sides times the height.
面积等于上下底之和的一半再乘以高。
梯形面积：A = ½ × (上底 + 下底) × 高
2. Circle Formulas | 圆的相关公式
For any circle with radius r, the distance around and the space inside follow strict relationships involving the constant π.
对任意半径为 r 的圆，周长和面积都与常数 π 有固定的关系。
Circumference: C = 2πr or C = πd
The perimeter of a circle is called the circumference. d = 2r is the diameter.
圆的周长称为圆周，直径 d = 2r。
周长公式：C = 2πr 或 C = πd
Area: A = πr²
The area of a circle is pi times radius squared.
圆的面积等于 π 乘以半径的平方。
圆的面积：A = π × 半径²
3. Pythagorean Theorem | 勾股定理
The Pythagorean theorem describes the fundamental relationship among the three sides of a right-angled triangle. It is widely used in construction and navigation.
勾股定理描述了直角三角形三边之间的基本关系，广泛应用于建筑和导航中。
a² + b² = c²
In a right triangle, the square of the hypotenuse (c, the longest side) equals the sum of the squares of the other two legs (a and b).
在直角三角形中，斜边（c，最长边）的平方等于两直角边（a 和 b）的平方和。
To find the length of the hypotenuse, use c = √(a² + b²). To find a leg, use a = √(c² – b²).
求斜边：c = √(a² + b²)。求直角边：a = √(c² – b²)。
4. Fractions, Decimals and Percentages | 分数、小数与百分数
These three forms represent parts of a whole and can be converted into one another. Knowing the rules helps in shopping, statistics and probability.
这三种形式都表示整体的一部分，可以相互转换。掌握规则对购物、统计和概率问题很有帮助。
Addition and subtraction: Find a common denominator first. Example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12.
加减法：先找到公分母。例如：1/3 + 1/4 = 4/12 + 3/12 = 7/12。
Multiplication: Multiply numerators together and denominators together. a/b × c/d = (a×c)/(b×d).
乘法：分子相乘，分母相乘。a/b × c/d = (a×c)/(b×d)。
Division: Multiply by the reciprocal. a/b ÷ c/d = a/b × d/c.
除法：乘以倒数。a/b ÷ c/d = a/b × d/c。
Fraction to decimal: Divide numerator by denominator.
分数转小数：分子除以分母。
Percentage to fraction: Write over 100 and simplify. 25% = 25/100 = 1/4.
百分数转分数：写成以100为分母的分数并约分。25% = 25/100 = 1/4。
5. Solving Linear Equations | 解一元一次方程
A linear equation has one unknown, usually x. The goal is to isolate x on one side by performing inverse operations equally on both sides.
一元一次方程含有一个未知数，通常用 x 表示。目标是运用逆运算并在等式两边同时操作，使 x 独立出现在等式一边。
Example: 2x + 5 = 13
Subtract 5 from both sides: 2x = 8. Then divide both sides by 2: x = 4.
两边同时减5：2x = 8。然后两边同时除以2：x = 4。
Always check by substituting the solution back into the original equation.
一定要将解代入原方程进行检验。
Key rule: whatever you do to one side, you must do to the other to keep the balance.
关键规则：对等式一边进行的任何操作，必须对另一边进行相同操作以保持平衡。
6. Angle Rules | 角度规则
Angles are measured in degrees. Understanding angle facts helps solve problems involving parallel lines, triangles and quadrilaterals.
角度以度为单位。掌握角度关系有助于解决涉及平行线、三角形和四边形的问题。
Straight line: Angles on a straight line add up to 180°.
平角：平角等于180°，直线上相邻角之和为180°。
Vertically opposite angles: When two lines intersect, opposite angles are equal.
对顶角：两条直线相交时，对顶角相等。
Angles at a point: All angles around a point sum to 360°.
周角：绕一点一周的所有角之和为360°。
Triangle sum: The interior angles of a triangle always add up to 180°.
三角形内角和：三角形三个内角之和等于180°。
Quadrilateral sum: The interior angles of a quadrilateral add up to 360°.
四边形内角和：四边形四个内角之和为360°。
7. Perimeter and Circumference | 周长公式
Perimeter is the total distance around a 2D shape. For circles, the term circumference is used.
周长是二维图形一周的长度，对于圆则称为圆周。
Rectangle: P = 2l + 2w
Add the lengths of all four sides. For a rectangle, P = 2 × (length + width).
矩形四边相加。P = 2 × (长 + 宽)。
Square: P = 4s
All sides equal. P = 4 times side length.
正方形四边相等，周长 = 4 × 边长。
Triangle: P = a + b + c
Sum of the three side lengths.
三边长度之和。
Circle: C = 2πr or C = πd
As introduced in circle formulas.
如前圆公式所述。
8. Volume of 3D Shapes | 立体图形体积
Volume measures the space occupied by a solid. For prisms and cylinders, the general idea is base area times height.
体积表示立体所占空间的大小。棱柱和圆柱的体积通用思想是底面积乘以高。
Cube of side s: V = s³
Volume = side × side × side.
体积 = 棱长 × 棱长 × 棱长。
Cuboid: V = l × w × h
Length × width × height.
长 × 宽 × 高。
Prism: V = Abase × h
Volume equals area of cross-section multiplied by the length (or height).
体积等于截面面积乘以长度（或高度）。
Cylinder: V = πr²h
Base is a circle, so area × height gives πr² × h.
底面为圆，底面积乘以高得 πr² × h。
9. Mean, Median, Mode and Range | 平均数、中位数、众数和极差
These statistics summarise data sets. They help describe typical values and the spread of numbers.
这些统计量概括数据集，描述典型值和数据分散程度。
Mean (average): Sum of all values divided by the number of values.
平均数：所有数值之和除以数值个数。
Median: The middle value when data is ordered. If two middle numbers, find their mean.
中位数：数据排序后位于中间的值。如果有两个中间数，则取它们的平均数。
Mode: The most frequently occurring value. A set can have no mode, one mode or more.
众数：出现次数最多的数值。一组数据可能没有众数、有一个或多个众数。
Range: Largest value minus smallest value. Measures spread.
极差：最大值减去最小值，描述数据的离散程度。
10. Probability Basics | 概率基础
Probability measures how likely an event is to happen. It is expressed as a number between 0 and 1, or as a fraction, decimal or percentage.
概率衡量事件发生的可能性，用一个在0到1之间的数表示，也可用分数、小数或百分数表示。
Probability = Number of favourable outcomes / Total number of possible outcomes
The probability of an event equals the number of ways that event can occur divided by the total number of possible outcomes, provided all outcomes are equally likely.
事件发生的概率等于该事件可能出现的结果数除以所有等可能结果的总数。
Impossible event: probability = 0. Certain event: probability = 1.
不可能事件：概率为0。必然事件：概率为1。
The sum of probabilities of all possible outcomes is always 1.
所有可能结果的概率之和等于1。
11. Ratio and Proportion | 比与比例
A ratio compares the sizes of two or more quantities. Proportion states that two ratios are equal.
比用来比较两个或多个数量的大小，比例则表示两个比相等。
Simplifying ratios: Divide all parts by their greatest common factor. 12:8 = 3:2.
化简比：所有项除以最大公因数。12:8 = 3:2。
Sharing in a ratio: For a total T and ratio a:b, first add a+b parts, then one part = T/(a+b).
按比例分配：总数为 T，比为 a:b。先求总份数 a+b，每一份 = T ÷ (a+b)。
Proportion problems often involve finding an unknown quantity when four terms are in proportion. E.g., if a/b = c/d, then a × d = b × c (cross multiplication).
比例问题常涉及已知四项成比例求未知量。例如，若 a/b = c/d，则 a × d = b × c（交叉相乘）。
12. Speed, Distance and Time | 速度、距离与时间
The relationship between speed, distance and time is essential in everyday travel and in science. The formula links these three quantities.
速度、距离和时间的关系在日常出行和科学中都很重要，以下公式将三者联系起来。
Speed = Distance ÷ Time
To find average speed, divide the total distance travelled by the total time taken.
平均速度等于总距离除以总时间。
速度 = 距离 ÷ 时间
The same equation can be rearranged: Distance = Speed × Time, and Time = Distance ÷ Speed.
公式可变形为：距离 = 速度 × 时间；时间 = 距离 ÷ 速度。
Always ensure units match: if speed is in km/h, use distance in km and time in hours.
务必保持单位一致：如果速度单位是千米/小时，距离用千米，时间用小时。
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