📚 Year 10 CCEA Mathematics: Formula & Theorem Quick Reference Handbook | 10 年级 CCEA 数学：公式定理速查手册
This quick reference handbook brings together the essential formulas, theorems and key concepts you will meet in Year 10 CCEA Mathematics. Use it to check facts, consolidate your understanding and build speed when tackling problems – all the rules are presented side‑by‑side in English and Chinese so you can revise confidently in either language.
本速查手册汇集了 10 年级 CCEA 数学必学的公式、定理和核心概念。你可以用它核查要点、巩固理解并提升解题速度。所有规则均以中英双语并列呈现，方便你用任意一种语言自信地复习。
1. Number Operations & Properties | 数与运算性质
Indices (powers) follow a set of rules that simplify expressions. For any base a ≠ 0 and integers m, n the index laws are:
指数（幂）的运算遵循一套简化表达式的规则。对于任意底数 a ≠ 0 及整数 m, n，指数定律为：
am × an = am+n
am ÷ an = am−n
(am)n = amn
a0 = 1
a−n = 1 / an
Standard form writes very large or very small numbers as A × 10n where 1 ≤ A < 10 and n is an integer. When adding or subtracting in standard form, first make the powers of 10 equal.
科学记数法将极大或极小的数写成 A × 10n 的形式，其中 1 ≤ A < 10，n 为整数。进行科学记数法的加减运算时，须先将 10 的幂统一。
Percentage change = (change / original) × 100%. To increase by r%, multiply by (1 + r/100); to decrease, multiply by (1 − r/100). Reverse percentage problems require dividing by the appropriate multiplier.
百分比变化 = (变化量 / 原值) × 100%。增加 r%，乘以 (1 + r/100)；减少 r%，乘以 (1 − r/100)。逆向百分比问题需除以相应的乘数。
Upper and lower bounds: if a quantity is measured to a given accuracy, the true value lies within ± half of the smallest unit used. For example, 5.6 m correct to 1 decimal place lies between 5.55 m and 5.65 m.
上界与下界：若一个量被测量至给定精确度，真实值则在所用最小单位的 ±0.5 范围内。例如 5.6 m 精确到 1 位小数，则真实值介于 5.55 m 到 5.65 m 之间。
2. Algebra – Expressions & Equations | 代数 – 表达式与方程
Expanding brackets uses the distributive law: a(b + c) = ab + ac. For a pair of linear brackets, (x + a)(x + b) = x2 + (a + b)x + ab.
展开括号使用分配律：a(b + c) = ab + ac。对于两个一次括号，(x + a)(x + b) = x2 + (a + b)x + ab。
Factorising reverses expanding. Always look for the highest common factor first: e.g. 6x2 + 9x = 3x(2x + 3). A quadratic of the form x2 + bx + c can be factorised into (x + p)(x + q) if p + q = b and pq = c.
因式分解是展开的逆运算。务必先提取最大公因数，如 6x2 + 9x = 3x(2x + 3)。形如 x2 + bx + c 的二次式若能找到 p + q = b 且 pq = c，则可分解为 (x + p)(x + q)。
To solve a linear equation, isolate the variable by doing the same operation to both sides. If fractions appear, multiply every term by the common denominator first. Check your solution by substitution.
解线性方程时，两边同时进行相同的运算以分离变量。若出现分数，先将每一项乘以公分母。通过代入原方程来检验解答。
Changing the subject of a formula involves rearranging to isolate a different variable. Treat the desired variable as the unknown and apply inverse operations step by step, keeping the equation balanced.
变换公式的主项需要重新整理以分离另一个变量。把你要求的变量看作未知数，逐步应用逆运算，并保持等式平衡。
3. Ratio, Proportion & Percentages | 比、比例与百分比
Ratio compares the sizes of two or more parts. To divide a quantity in a given ratio, add the parts to find the total number of shares, then multiply the value of one share by each part.
比用于比较两个或多个部分的大小。按给定比例分配一个量时，先求出总份数，再以每份的值乘以各部分的份数。
Direct proportion means two quantities increase or decrease in the same ratio; their graph is a straight line through the origin. Inverse proportion means one quantity increases while the other decreases, so their product remains constant.
正比例表示两个量以相同比率增加或减少；其图像是一条过原点的直线。反比例意味着一个量增加时另一个减少，两者的乘积保持不变。
Compound interest is calculated using A = P(1 + r/100)n where P is the principal, r is the annual interest rate and n is the number of years. For depreciation (loss of value), replace the plus with a minus.
复利用公式 A = P(1 + r/100)n 计算，其中 P 为本金，r 为年利率，n 为年数。若是贬值，将加号改为减号即可。
A percentage error = |(measured − true) / true| × 100%. This helps to describe the accuracy of an approximation.
百分误差 = |(测量值 − 真实值) / 真实值| × 100%。这有助于描述近似值的准确性。
4. Linear Graphs | 线性图像
The equation of a straight line is usually written as y = mx + c, where m is the gradient and c is the y‑intercept (the point where the line crosses the y‑axis).
直线方程通常写作 y = mx + c，其中 m 是斜率，c 是 y 轴截距（直线与 y 轴交点的纵坐标）。
Gradient m = (change in y) / (change in x) = (y2 − y1) / (x2 − x1). A positive gradient slopes upward to the right; a negative gradient slopes downward.
斜率 m = (y 的变化量) / (x 的变化量) = (y2 − y1) / (x2 − x1)。正斜率向右上方倾斜；负斜率向右下方倾斜。
To find the equation of a line given two points, first calculate m, then substitute one point into y = mx + c to solve for c. If you know a point and the gradient, the line is y − y1 = m(x − x1).
已知两点求直线方程，先计算 m，再将一点代入 y = mx + c 求出 c。若已知一点和斜率，则直线为 y − y1 = m(x − x1)。
Parallel lines have equal gradients. Perpendicular lines satisfy m1 × m2 = −1 (provided neither is vertical). For example, if a line has gradient 2, a perpendicular line has gradient −½.
平行线的斜率相等。垂线的斜率满足 m1 × m2 = −1（前提是两条线都不是竖直的）。例如，若一条线斜率为 2，则与之垂直的线斜率为 −½。
5. Geometry – Angles & Shapes | 几何 – 角与形状
Angles on a straight line add up to 180°, angles around a point sum to 360°. Vertically opposite angles are equal. These facts are used repeatedly to find missing angles.
直线上的邻角之和为 180°，绕一点的周角之和为 360°。对顶角相等。这些事实经常用来求未知角。
When a transversal crosses parallel lines: corresponding angles are equal, alternate angles are equal, and interior (co‑interior) angles sum to 180°.
当一条截线与平行线相交时：同位角相等，内错角相等，同旁内角（同侧内角）之和为 180°。
The interior angle sum of a polygon with n sides is (n − 2) × 180°. For a regular polygon, each interior angle is [(n − 2) × 180°] / n. The sum of exterior angles of any convex polygon is always 360°.
n 边形的内角和为 (n − 2) × 180°。正多边形每个内角为 [(n − 2) × 180°] / n。任意凸多边形的外角和恒为 360°。
| Shape | Interior angle sum |
| Triangle | 180° |
| Quadrilateral | 360° |
| Pentagon | 540° |
| Hexagon | 720° |
In any triangle, an exterior angle equals the sum of the two opposite interior angles. This is particularly useful when working with proofs and angle chasing.
在任意三角形中，一个外角等于其两个不相邻内角之和。这在证明和导向求角时尤其有用。
6. Perimeter, Area & Volume | 周长、面积与体积
Perimeter is the distance around a shape. For common figures: rectangle P = 2(l + w), triangle P = a + b + c, circle circumference C = 2πr (or πd).
周长是图形边界的总长。常见的周长公式：矩形 P = 2(l + w)，三角形 P = a + b + c，圆周长 C = 2πr（或 πd）。
Area measures the surface covered. Key formulas:
- Rectangle: A = l × w
- Triangle: A = ½ × base × height
- Parallelogram: A = base × perpendicular height
- Trapezium: A = ½(a + b)h (a and b are parallel sides, h is perpendicular height)
- Circle: A = πr²
面积衡量一个表面覆盖的大小。关键公式：矩形 A = l × w；三角形 A = ½ × 底 × 高；平行四边形 A = 底 × 垂直高；梯形 A = ½(a + b)h（a,b 为平行边，h 为垂直高）；圆 A = πr²。
Volume of a prism = area of cross‑section × length. For a cylinder, V = πr²h. For a cuboid, V = l × w × h. Surface area is the total area of all faces; for a cylinder, SA = 2πr² + 2πrh.
棱柱体积 = 横截面积 × 长度。圆柱体积 V = πr²h，长方体体积 V = l × w × h。表面积是所有面的总面积；圆柱表面积 SA = 2πr² + 2πrh。
When finding the area of a composite shape, split it into simpler parts, calculate each area separately, then add or subtract as needed.
求复合图形的面积时，将其分割为简单图形，分别计算各部分面积，然后按需相加或相减。
7. Pythagoras’ Theorem | 勾股定理
In any right‑angled triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides: a² + b² = c², where c is the hypotenuse.
对于任意直角三角形，斜边（最长边）的平方等于两条直角边的平方和：a² + b² = c²，其中 c 为斜边。
To find the length of the hypotenuse, use c = √(a² + b²). To find a shorter side, rearrange: a = √(c² − b²) or b = √(c² − a²).
求斜边长度时用 c = √(a² + b²)。求一条直角边时则重排公式：a = √(c² − b²) 或 b = √(c² − a²)。
Pythagorean triples, such as (3, 4, 5) and (5, 12, 13), are sets of whole numbers that satisfy the theorem. Recognising them can save time.
勾股数（如 3, 4, 5 和 5, 12, 13）是满足定理的整数组合。识别它们能节省解题时间。
You can also use Pythagoras in 3D problems – find the diagonal of a cuboid by applying the theorem twice, first to a face and then to the space diagonal.
你也可以在三维问题中使用勾股定理——对长方体应用两次定理，先求面对角线，再求空间对角线。
8. Trigonometry Basics | 三角学基础
The three trigonometric ratios for a right‑angled triangle are defined as:
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
直角三角形中的三个三角比定义为：sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
To find a missing side, label the triangle (opposite, adjacent, hypotenuse relative to the given angle), choose the correct ratio, set up an equation and solve.
求未知边长时，标记三角形（相对于给定角的对边、邻边、斜边），选择合适的三角比，列出方程并求解。
To find an unknown angle, use the inverse functions: θ = sin⁻¹, cos⁻¹ or tan⁻¹ of the appropriate ratio. For example, if sin θ = 0.5, then θ = sin⁻¹(0.5) = 30°.
求未知角时使用反函数：θ = sin⁻¹、cos⁻¹ 或 tan⁻¹（相应比值）。例如若 sin θ = 0.5，则 θ = sin⁻¹(0.5) = 30°。
Exact values for key angles are worth memorising:
| θ | sin θ | cos θ | tan θ |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | 1/√2 | 1/√2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
These common values appear frequently in CCEA exam questions, especially when angles are 30°, 45° or 60°.
这些常用数值频繁出现在 CCEA 试题中，尤其是涉及 30°、45° 或 60° 角时。
9. Statistics – Averages & Spread | 统计 – 平均数与离散程度
Measures of central tendency summarise a data set. The mean is the sum of values divided by the number of values. The median is the middle value when data are ordered; for an even number of data, it is the average of the two middle numbers. The mode is the most frequent value.
集中量数概括数据集。平均数是所有数值之和除以数值的个数。中位数是排序后位于中间位置的数；数据个数为偶数时取中间两数的平均值。众数是出现频率最高的数值。
The range (max − min) gives a simple measure of spread. A large range indicates greater variability.
极差（最大值 − 最小值）是一个简单的离散度度量。极差大说明数据变动较大。
For grouped data, estimate the mean by multiplying each midpoint by its frequency, adding these products, and dividing by the total frequency.
对于分组数据，先用各组组中值乘以频数，再将这些乘积之和除以总频数，以估计平均数。
A box plot (box‑and‑whisker diagram) displays the minimum, lower quartile, median, upper quartile and maximum. The interquartile range (IQR = UQ − LQ) is a more robust measure of spread than the range.
箱线图展示最小值、下四分位数、中位数、上四分位数和最大值。四分位距 (IQR = UQ − LQ) 是比极差更稳健的离散度度量。
10. Probability | 概率
Probability of an event = number of favourable outcomes / total number of equally likely outcomes. It always lies between 0 (impossible) and 1 (certain).
事件的概率 = 有利结果数 / 所有等可能结果的总数。概率总是在 0（不可能）和 1（必然）之间。
For any event A, P(not A) = 1 − P(A). The probabilities of all mutually exclusive exhaustive events sum to 1.
对于任意事件 A，P(非 A) = 1 − P(A)。所有互斥且完备的事件的概率之和为 1。
Tree diagrams help show the outcomes of two or more successive events. Multiply along the branches to
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