📚 Year 9 SQA Advanced Mathematics: Formula & Theorem Quick Reference Handbook | 九年级SQA进阶数学：公式定理速查手册
Welcome to the ultimate quick reference handbook for Year 9 SQA Advanced Mathematics. This guide compiles the essential formulas, theorems and key concepts you need to master for your assessments. Keep it handy for revision sessions, homework tasks and quick fact-checking whenever you get stuck.
欢迎使用九年级SQA进阶数学的终极速查手册。本指南汇编了你需要掌握的重要公式、定理和核心概念，适用于评估备考。在复习、做作业或遇到难题时，随时打开这本手册查漏补缺。
1. Pythagoras’ Theorem and the Converse | 勾股定理及其逆定理
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 legs.
在任何直角三角形中，斜边（直角对边）的平方等于两条直角边的平方和。
c² = a² + b²
c² = a² + b²
atisfy c² = a² + b², then the triangle is right-angled. This is the converse of Pythagoras' theorem and helps verify right angles.
如果三角形的三边满足 c² = a² + b²，那么该三角形是直角三角形。这是勾股定理的逆定理，常用于验证直角。
Example: For a triangle with legs 6 cm and 8 cm, the hypotenuse is √6² + 8² = √36 + 64 = √100 = 10 cm.
例子：一直角边为6 cm和8 cm，斜边为 √6² + 8² = √36 + 64 = √100 = 10 cm。
2. Trigonometric Ratios in Right-Angled Triangles | 直角三角形的三角比
The three basic trigonometric ratios relate an acute angle in a right triangle to the ratios of side lengths. Remember the acronym SOH CAH TOA.
三个基本三角比将直角三角形中的一个锐角与边长比值联系起来。记住助记口诀 SOH CAH TOA。
sin θ = opposite / hypotenuse
sin θ = 对边 / 斜边
cos θ = adjacent / hypotenuse
cos θ = 邻边 / 斜边
tan θ = opposite / adjacent
tan θ = 对边 / 邻边
The exact values for the special angles 30°, 45° and 60° are commonly tested and must be memorised.
特殊角30°、45°和60°的精确值经常考查，必须牢记。
| 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 |
The exact trigonometric values table above shows the key ratios in simplest surd form. Use these to evaluate expressions without a calculator.
上表以最简根式给出了关键三角比。利用这些值可在没有计算器的情况下求值。
3. Area of Triangles and Circles | 三角形和圆的面积
The area of any triangle can be found by multiplying half the base by the perpendicular height. For a circle, the area depends on the square of the radius.
任何三角形的面积等于底乘以高的一半。圆的面积取决于半径的平方。
Area of triangle = ½ × base × height
三角形面积 = ½ × 底 × 高
Area of circle = π × radius², i.e. A = πr²
圆面积 = π × 半径²，即 A = πr²
If you are given the diameter of a circle, remember to halve it to obtain the radius before using the area formula.
如果已知圆的直径，在使用面积公式前记得先除以2得到半径。
6 cm has area ½ × 10 × 6 = 30 cm². A circle with radius 7 cm has area π × 7² ≈ 153.94 cm².
例子：底为10 cm、高为6 cm的三角形面积为 ½ × 10 × 6 = 30 cm²。半径为7 cm的圆面积为 π × 7² ≈ 153.94 cm²。
4. Circumference and Perimeter | 周长
The perimeter of any shape is the total distance around its outer edge. For circles, we use the term circumference.
任何图形的周长是指围绕其外边界的总长度。对于圆，我们使用“圆周”一词。
Circumference = 2 × π × radius, i.e. C = 2πr
圆周长 = 2 × π × 半径，即 C = 2πr
Alternatively, circumference = π × diameter, C = πd
或 圆周长 = π × 直径，C = πd
For composite shapes, perimeter is found by summing the lengths of all straight edges and arcs. Semicircle perimeter: half the circumference plus the diameter.
对于组合图形，周长是所有直边和弧长的总和。半圆周长等于圆周长的一半加上直径。
≈ 31.42 m. A rectangle with length 8 cm and width 3 cm has perimeter 2 × (8 + 3) = 22 cm.
例子：半径为5 m的圆，周长为 2π × 5 = 10π ≈ 31.42 m。长为8 cm、宽为3 cm的矩形，周长为 2 × (8 + 3) = 22 cm。
5. Straight Line Graphs: y = mx + c | 直线图像：y = mx + c
All straight lines (except vertical lines) can be written in the slope–intercept form y = mx + c. Here m represents the gradient (steepness) and c is the y‑intercept (where the line crosses the y‑axis).
所有直线（竖直线除外）都可以写成斜截式 y = mx + c。其中 m 表示斜率（倾斜程度），c 是 y 轴截距（直线与 y 轴的交点）。
y = mx + c
y = mx + c
Gradient m can be calculated from two points (x₁, y₁) and (x₂, y₂) using the rise over run formula:
斜率 m 可通过两点 (x₁, y₁) 和 (x₂, y₂) 用“纵差除以横差”公式计算：
m = (y₂ – y₁) / (x₂ – x₁)
m = (y₂ – y₁) / (x₂ – x₁)
Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to -1 (negative reciprocals).
平行线斜率相等。垂直线的斜率乘积为 -1（两者互为负倒数）。
Example: A line through (1, 3) and (4, 9) has gradient m = (9 – 3) / (4 – 1) = 6/3 = 2. Its equation can be written as y = 2x + 1.
例子：过点 (1, 3) 和 (4, 9) 的直线斜率为 m = (9 – 3) / (4 – 1) = 6/3 = 2，其方程可写为 y = 2x + 1。
6. Solving Quadratic Equations | 解二次方程
lutions can be found by factorising, completing the square or using the quadratic formula.
二次方程通常写成 ax² + bx + c = 0 的形式，且 a ≠ 0。可通过因式分解、配方法或使用求根公式求解。
Quadratic Formula: x = (-b ± √b² – 4ac) / (2a)
求根公式：x = (-b ± √b² – 4ac) / (2a)
The expression b² – wo distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 means no real roots.
表达式 b² – 4ac 称为判别式（Δ）。它表示根的性质：若 Δ > 0，有两个不等实根；Δ = 0，有一个重根；Δ < 0，无实数根。
ers that multiply to give ac and add to give b. For example, x² + 5x + 6 = (x + 2)(x + 3).
因式分解时，寻找两个数使其乘积为 ac、和为 b。例如 x² + 5x + 6 = (x + 2)(x + 3)。
7. Volume and Surface Area of 3D Shapes | 三维图形的体积与表面积
The volume of a prism is found by multiplying the area of its cross‑section by its length. For a cylinder, the cross‑section is a circle. Surface area is the total area of all faces.
棱柱的体积等于横截面积乘以其长度。对于圆柱，横截面为圆。表面积是所有面的总面积。
Volume of prism = area of cross‑section × length
棱柱体积 = 横截面积 × 长度
Volume of cylinder = πr²h
圆柱体积 = πr²h
Surface area of cylinder = 2πr² + 2πrh
圆柱表面积 = 2πr² + 2πrh
id with length l, width w and height h, volume V = lwh and surface area = 2(lw + lh + wh).
对于长为 l、宽为 w、高为 h 的长方体，体积 V = lwh，表面积 = 2(lw + lh + wh)。
10 = 90π ≈ 282.74 cm³ and surface area 2π(3)² + 2π(3)(10) = 18π + 60π = 78π ≈ 245.04 cm².
例子：半径为3 cm、高为10 cm的圆柱，体积为 π × 3² × 10 = 90π ≈ 282.74 cm³，表面积为 18π + 60π = 78π ≈ 245.04 cm²。
8. Averages and Measures of Spread | 平均数与离散程度
The three common averages are mean, median and mode. The range measures how spread out the data are.
三种常见的平均数是均值、中位数和众数。极差衡量数据的分散程度。
Mean = sum of all data values / number of values
均值 = 所有数据之和 / 数据个数
Median: middle value when data are ordered
中位数：将数据排序后的中间值
Mode: most frequently occurring value
众数：出现次数最多的值
Range = largest value – smallest value
极差 = 最大值 – 最小值
If there is an even number of data items, the median is the mean of the two
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