📚 KS3 Maths: Formula Summary Handbook | KS3 数学：公式汇总手册
This handbook provides a quick reference to essential formulas covered in KS3 Mathematics. Understanding these formulas, knowing when to apply them, and using the correct units will help you solve problems confidently across topics including geometry, algebra, ratio, statistics and probability.
本手册提供 KS3 数学核心公式的快速参考。理解这些公式、清楚它们的使用场景，并正确书写单位，将帮助你自信地解答几何、代数、比与比例、统计与概率等各类题目。
Each section below pairs the English explanation with the Chinese version, so you can revise the mathematics and the language of the exam at the same time. Work through the formulas actively: cover the page, write each one from memory, then check.
下文每一节都给出英文说明与中文对照，方便你在复习数学的同时熟悉考试的英文表述。请主动使用这些公式：遮住页面，先凭记忆写出，再对照检查。
1. Core Area and Perimeter Formulas | 核心面积与周长公式
The area of a rectangle is found by multiplying length and width. The perimeter is the sum of all sides.
矩形的面积等于长乘以宽。周长是所有边长的总和。
A = l × w P = 2(l + w)
For a triangle, area is half the base times the height. Add the three sides for perimeter. The height must be measured perpendicular to the base.
三角形的面积等于底乘高的一半。周长等于三边之和。高必须从底边垂直量出。
A = ½ × b × h P = a + b + c
A parallelogram' (not the slant side). A square is a special rectangle, so its area is s² and its perimeter is 4s.
平行四边形的面积等于底乘垂直高（不是斜边长度）。正方形是特殊的矩形，面积是 s²，周长是 4s。
A = b × h A = s² P = 4s
Always check your units: area is given in square units (cm², m²) and perimeter in linear units (cm, m). Writing cm instead of cm² is one of the most common lost marks in KS3 exams.
务必检查单位：面积用平方单位（cm²、m²），周长用长度单位（cm、m）。把 cm² 写成 cm 是 KS3 考试中最常见的失分点之一。
| Shape 图形 | Area 面积 | Perimeter 周长 |
| Rectangle 矩形 | l × w | 2(l + w) |
| Square 正方形 | s² | 4s |
| Triangle 三角形 | ½ × b × h | a + b + c |
| Parallelogram 平行四边形 | b × h | 2(a + b) |
2. Trapezium and Compound Shapes | 梯形与复合图形
The area of a trapezium is found by adding the two parallel sides, dividing by 2, and multiplying by the perpendicular height between them.
梯形的面积求法：把两条平行边相加，除以 2，再乘以它们之间的垂直高。
A = ½ × (a + b) × h
Here a and b are the parallel sides and h is the perpendicular distance between them, not the length of a sloping edge.
其中 a 与 b 是两条平行边，h 是它们之间的垂直距离，而不是斜边长度。
Compound shapes are handled in one of two ways: split them into rectangles, triangles and trapezia and add the areas, or subtract the missing part from a larger shape.
复合图形的处理方法有两种：把它拆成矩形、三角形和梯形后把面积相加，或者用大图形减去缺少的部分。
For the perimeter of a compound shape, add only the outer edges. Interior dividing lines must not be counted.
求复合图形的周长时，只加外侧的边，内部的分割线不能计入。
formula for the arc length ½ × π × d for the curved part and add it to the straight edges.
常见陷阱是图形一端附有半圆：曲线部分用弧长公式 ½ × π × d，再与各条直边相加。
3. Circles, Arcs and Sectors | 圆、弧与扇形
The circumference of a circle is π times the diameter, which equals 2π times the radius. The area is π times the radius squared.
圆的周长等于 π 乘直径，也等于 2π 乘半径。面积等于 π 乘半径的平方。
C = πd = 2πr A = πr²
ulator. Never round the value of π in the middle of a calculation; round only at the final answer.
π 可取 3.14 或使用计算器上的 π 键。计算过程中不要提前约简 π，只在最终答案处取近似值。
An arc is a fraction of the circumference, and a sector is a fraction of the whole area. The fraction is the angle at the centre divided by 360°.
弧是圆周长的一部分，扇形是整个圆面积的一部分。这个分数由圆心角除以 360° 决定。
Arc length = (θ ÷ 360) × πd Sector area = (θ ÷ 360) × πr²
°, and a quadrant has θ = 90°. The perimeter of a sector is the arc length plus two radii.
半圆的 θ = 180°，四分之一圆的 θ = 90°。扇形的周长等于弧长加上两条半径。
4. Angles, Parallel Lines and Polygons | 角、平行线与多边形
Angles on a straight line add to 180°, and angles around a point add to 360°. Vertically opposite angles are equal.
同一直线上的角之和为 180°，围绕一点的角之和为 360°。对顶角相等。
The interior angles of a triangle sum to 180° and those of a quadrilateral sum to 360°.
三角形的内角和为 180°，四边形的内角和为 360°。
For any polygon with n sides, the interior angle sum is (n − 2) × 180°.
对于有 n 条边的任意多边形，内角和为 (n − 2) × 180°。
Interior angle sum = (n − 2) × 180°
The exterior angles of any polygon always add to 360°, so each exterior angle of a regular polygon is 360° ÷ n.
任意多边形的外角和恒为 360°，因此正多边形每个外角等于 360° ÷ n。
Exterior angle = 360° ÷ n Interior angle = 180° − exterior angle
With parallel lines and a transversal, alternate angles (the Z shape) and corresponding angles (the F shape) are equal, while co-interior angles (the C shape) add to 180°.
在平行线与截线中，内错角（Z 形）与同位角（F 形）相等，而同旁内角（C 形）之和为 180°。
5. Surface Area and Volume | 表面积与体积
Volume measures the space inside a solid and is written in cubic units (cm³, m³). Surface area measures the total area of all faces and is written in square units.
体积衡量立体内部的空间，用立方单位表示（cm³、m³）。表面积衡量所有面的总面积，用平方单位表示。
nd total surface area 6a². A cuboid has volume l × w × h and surface area 2(lw + lh + wh).
棱长为 a 的正方体体积为 a³，表面积为 6a²。长方体的体积为 l × w × h，表面积为 2(lw + lh + wh)。
V = l × w × h SA = 2(lw + lh + wh)
For any prism, the volume is the area of the cross-section multiplied by the length of the prism.
对于任意棱柱，体积等于横截面积乘以棱柱长度。
V = area of cross-section × length
A cylinder is a prism with a circular cross-section, so its volume is πr²h and its curved surface area is 2πrh.
圆柱是横截面为圆的棱柱，所以体积是 πr²h，侧面积是 2πrh。
V = πr²h Curved SA = 2πrh
conversions: 1 m² = 10 000 cm², 1 m³ = 1 000 000 cm³, 1 litre = 1000 cm³ and 1 ml = 1 cm³.
记住换算关系：1 m² = 10 000 cm²，1 m³ = 1 000 000 cm³，1 升 = 1000 cm³，1 毫升 = 1 cm³。
6. Right-Angled Triangles: Pythagoras and Trigonometry | 直角三角形：勾股定理与三角函数
Pythagoras' theorem states that in a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
勾股定理指出：在直角三角形中，斜边的平方等于另外两条边的平方之和。
a² + b² = c²
To find the hypotenuse, add the squares and take the square root. To find a shorter side, subtract the squares and take the square root.
求斜边时，把平方相加后开平方。求直角边时，把平方相减后开平方。
c = √a² + b² a = √c² − b²
17 and 7, 24, 25. The converse also works: if a² + b² = c², the triangle is right-angled.
值得记住的勾股数组有 3、4、5；5、12、13；8、15、17 以及 7、24、25。逆定理同样成立：若 a² + b² =
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