KS3 OCR Further Mathematics: Formula and Theorem Quick Reference Handbook | KS3 OCR 进阶数学：公式定理速查手册

📚 KS3 OCR Further Mathematics: Formula and Theorem Quick Reference Handbook | KS3 OCR 进阶数学：公式定理速查手册

This quick reference handbook compiles all essential formulas, theorems and key facts you need to master in KS3 OCR Further Mathematics. From angles and shapes to algebra and data, you will find clear, concise summaries to boost your revision and problem-solving confidence. Every entry is presented with an English statement followed by its Chinese equivalent, helping you build a bilingual command of the subject.

本速查手册汇集了KS3 OCR进阶数学中所有必须掌握的公式、定理和关键知识点。从角、形状到代数与数据，简明扼要的总结将助你高效复习，提升解题信心。每条内容均先给出英文陈述，再配以中文对应，帮助你建立学科的双语能力。

1. Angle Facts | 角的基础定理

Mastering angle facts is essential for solving geometry problems. Angles are measured in degrees, and the relationships below appear frequently.

掌握角的基础定理是解决几何问题的关键。角以度为单位，下面的关系经常用到。

Angles on a straight line add up to 180 degrees.

平角（直线上相邻的角）之和为180°。

Angles around a point sum to 360 degrees.

一个点周围的角之和为360°。

Vertically opposite angles are equal.

对顶角相等。

Corresponding angles on parallel lines are equal.

平行线中的同位角相等。

Alternate interior angles on parallel lines are equal.

平行线中的内错角相等。

Co-interior (allied) angles on parallel lines add up to 180 degrees.

平行线中的同旁内角之和为180°。

In a triangle, the three interior angles always sum to 180 degrees.

三角形三个内角之和恒为180°。

The exterior angle of a triangle equals the sum of the two opposite interior angles.

三角形的一个外角等于与它不相邻的两个内角之和。

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2. Triangles: Types and Properties | 三角形的分类与性质

Triangles are the simplest polygons. Their names depend on side lengths or interior angles, and each type has special properties.

三角形是最简单的多边形。其名称取决于边长或内角，每种类型都有特殊的性质。

An equilateral triangle has three equal sides and three equal angles of 60 degrees each.

等边三角形三边相等，三个角均为60°。

An isosceles triangle has two equal sides and two equal base angles.

等腰三角形有两条相等的边和两个相等的底角。

A scalene triangle has no equal sides and no equal angles.

不等边三角形三边互不相等，三个角也互不相等。

In a right-angled triangle, one angle is exactly 90 degrees; the side opposite this angle is the hypotenuse.

直角三角形中有一个角恰好是90°；这个角所对的边称为斜边。

The perimeter of any triangle is the sum of its three side lengths.

任意三角形的周长等于三条边长之和。

Area of a triangle = ½ × base × height, where the height is perpendicular to the base.

三角形面积 = ½ × 底 × 高，其中高与底垂直。

Congruent triangles have exactly the same shape and size; they can be proved using SSS, SAS, ASA or RHS conditions.

全等三角形形状和大小完全相同；可通过SSS、SAS、ASA或RHS条件证明全等。

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3. Quadrilaterals and Other Polygons | 四边形及其他多边形

Quadrilaterals are four-sided polygons. Recognising their properties and angle sums is fundamental to geometry.

四边形是四条边的多边形。掌握其性质和内角和是几何学的基础。

The sum of interior angles of any quadrilateral is 360 degrees.

任意四边形的内角和为360°。

A square has four equal sides, four right angles, and both pairs of opposite sides are parallel.

正方形有四条等边、四个直角，且两组对边分别平行。

A rectangle has opposite sides equal and four right angles; its diagonals are equal and bisect each other.

矩形对边相等，四个角都是直角；对角线相等且互相平分。

A rhombus has four equal sides; opposite sides are parallel, opposite angles are equal, and diagonals bisect at right angles.

菱形四边相等；对边平行，对角相等，对角线互相垂直平分。

A parallelogram has both pairs of opposite sides parallel and equal; opposite angles are equal.

平行四边形两组对边分别平行且相等；对角相等。

A trapezium (UK definition) has at least one pair of parallel sides; an isosceles trapezium has non-parallel sides equal.

梯形（英国定义）至少有一组平行边；等腰梯形的两条非平行边相等。

For a regular polygon with n sides, each interior angle = (n – 2) × 180° / n.

对于有n条边的正多边形，每个内角 = (n – 2) × 180° / n。

The sum of exterior angles of any convex polygon is 360°.

任意凸多边形的外角和为360°。

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4. Circles: Key Definitions and Measurements | 圆：定义与度量

Circles have a unique set of vocabulary and formulas. Understanding parts of a circle and how to calculate circumference and area is crucial.

圆有一套独特的术语和公式。理解圆的各个部分以及如何计算周长和面积至关重要。

The circumference is the distance around the circle.

圆周是围绕圆一周的长度。

Circumference = π × diameter, or C = 2πr, where r is the radius.

周长 = π × 直径，即 C = 2πr，其中 r 为半径。

Area of a circle = π × radius², or A = πr².

圆的面积 = π × 半径²，即 A = πr²。

The radius is a line segment from the centre to any point on the circumference; the diameter is twice the radius.

半径是从圆心到圆周上任意一点的线段；直径是半径的两倍。

A chord is a line segment connecting two points on the circle; the longest chord is the diameter.

弦是连接圆上两点的线段；最长的弦是直径。

A tangent touches the circle at exactly one point and is perpendicular to the radius at that point.

切线仅在一点处接触圆，并在该点处与半径垂直。

An arc is a part of the circumference; a sector is the region enclosed by two radii and an arc.

弧是圆周的一部分；扇形是由两条半径和一段弧围成的区域。

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5. Perimeter, Area and Surface Area | 周长、面积与表面积

Calculating perimeters and areas of 2D shapes, as well as surface areas of 3D solids, relies on key formulas that must be memorised.

计算二维图形的周长、面积以及三维立体的表面积，需要熟记关键公式。

Perimeter of a rectangle = 2 × (length + width).

矩形周长 = 2 × (长 + 宽)。

Area of a rectangle = length × width.

矩形面积 = 长 × 宽。

Area of a parallelogram = base × perpendicular height.

平行四边形面积 = 底 × 垂直高。

Area of a trapezium = ½ × (sum of parallel sides) × height.

梯形面积 = ½ × (平行边之和) × 高。

Area of a triangle = ½ × base × height.

三角形面积 = ½ × 底 × 高。

Area of a kite or rhombus can be found using ½ × product of diagonals.

风筝形或菱形的面积可用 ½ × 两条对角线之积 求得。

Surface area of a cube = 6 × side².

立方体表面积 = 6 × 边长²。

Surface area of a cuboid = 2(lw + lh + wh), where l = length, w = width, h = height.

长方体表面积 = 2(lw + lh + wh)，其中 l 为长，w 为宽，h 为高。

Curved surface area of a cylinder = 2πrh; total surface area = 2πr(h + r).

圆柱的侧面积 = 2πrh；总表面积 = 2πr(h + r)。

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6. Volume of 3D Shapes | 立体图形的体积

Volume measures the space inside a 3D object. Units are typically cubic centimetres (cm³) or cubic metres (m³).

体积衡量三维物体内部的空间大小。单位常用立方厘米 (cm³) 或立方米 (m³)。

Volume of a cube = side³.

立方体体积 = 边长³。

Volume of a cuboid = length × width × height.

长方体体积 = 长 × 宽 × 高。

Volume of a prism = area of cross-section × length.

棱柱（角柱）体积 = 横截面积 × 长度。

Volume of a cylinder = π × radius² × height, i.e. V = πr²h.

圆柱体积 = π × 半径² × 高，即 V = πr²h。

Volume of a pyramid = ⅓ × base area × vertical height.

棱锥体积 = ⅓ × 底面积 × 垂直高度。

Volume of a cone = ⅓ × πr²h.

圆锥体积 = ⅓ × πr²h。

Volume of a sphere = ⁴⁄₃ × πr³.

球体体积 = ⁴⁄₃ × πr³。（这里使用分数符号⁴⁄₃表示4/3，但更稳妥用 4/3 × πr³，我会用 (4/3)πr³ 确保清晰。重新表达：Volume of a sphere = 4/3 × πr³. 用普通斜线。）更好：Volume of a sphere = (4/3)πr³.

Volume of a sphere = (4/3)πr³.

球体体积 = (4/3)πr³。

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7. Pythagoras’ Theorem | 勾股定理

Pythagoras’ theorem applies only to right-angled triangles and links the lengths of the three sides.

勾股定理仅适用于直角三角形，它建立了三边长度的联系。

In any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

在任意直角三角形中，斜边的平方等于另两条边的平方和。

a² + b² = c²

a² + b² = c² （c表示斜边）

To find the length of the hypotenuse, calculate c = √(a² + b²).

求斜边长用 c = √(a² + b²)。

To find a shorter side, rearrange: a = √(c² – b²).

求一条直角边用 a = √(c² – b²)。

The converse of Pythagoras’ theorem can be used to test whether a triangle is right-angled: if a² + b² = c², then the angle opposite side c is 90°.

勾股定理的逆定理可用于检验三角形是否为直角三角形：若 a² + b² = c²，则边c所对的角为90°。

Pythagorean triples are sets of three integers that satisfy the theorem, e.g. (3, 4, 5) and (5, 12, 13).

勾股数是满足该定理的整数三元组，例如 (3, 4, 5) 和 (5, 12, 13)。

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8. Algebraic Laws and Indices | 代数法则与指数

Algebraic manipulation is at the heart of Further Mathematics. The laws of indices simplify expressions involving powers.

代数运算是进阶数学的核心。指数法则可简化含有幂的表达式。

Multiplying powers with the same base: xᵐ × xⁿ = xᵐ⁺ⁿ.

同底数幂相乘：xᵐ × xⁿ = xᵐ⁺ⁿ。

Dividing powers: xᵐ / xⁿ = xᵐ⁻ⁿ.

同底数幂相除：xᵐ / xⁿ = xᵐ⁻ⁿ。

Raising a power to another power: (xᵐ)ⁿ = xᵐⁿ.

幂的乘方：(xᵐ)ⁿ = xᵐⁿ。

A term raised to the power 0 equals 1: x⁰ = 1, provided x ≠ 0.

任何非零数的零次幂等于1：x⁰ = 1（x ≠ 0）。

Negative indices indicate reciprocals: x⁻ⁿ = 1 / xⁿ.

负指数表示倒数：x⁻ⁿ = 1 / xⁿ。

Fractional indices represent roots: x^(½) = √x, and x^(1/n) = ⁿ√x.

分数指数表示方根：x^(½) = √x，x^(1/n) = ⁿ√x。 (用Unicode上标½表示½，但x^(½)可综合为x^{\frac{1}{2}}，此处用x½不够好。改为x^(1/2) = √x。使用普通形式：x^(1/2) = √x。不用上标½，直接用x^(1/2)=√x。对于n次根可以用ⁿ√x，上标n。) 我会改为：x^(1/2) = √x, and x^(1/n) = ⁿ√x.

The distributive law: a(b + c) = ab + ac.

乘法分配律：a(b + c) = ab + ac。

Expanding binomials: (a + b)(c + d) = ac + ad + bc + bd.

二项式展开：(a + b)(c + d) = ac + ad + bc + bd。

Special products: (a + b)² = a² + 2ab + b², and (a – b)² = a² – 2ab + b².

完全平方公式：(a + b)² = a² + 2ab + b²，(a – b)² = a² – 2ab + b²。

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9. Linear Equations and Straight-Line Graphs | 线性方程与直线图像

Linear relationships are fundamental in algebra. You need to be able to solve equations and interpret straight-line graphs.

线性关系是代数中的基础。你需要能够解方程并解读直线图像。

To solve a linear equation like 2x + 3 = 11, isolate x by performing inverse operations: subtract 3, then divide by 2, giving x = 4.

解线性方程如2x + 3 = 11，通过逆运算分离x：先减3，再除以2，得x = 4。

The general equation of a straight line is y = mx + c, where m is the gradient and c is the y-intercept.

直线的一般方程为 y = mx + c，其中m是斜率，c是y轴截距。

Gradient m = change in y / change in x = (y₂ – y₁) / (x₂ – x₁).

斜率 m = y的变化 / x的变化 = (y₂ – y₁) / (x₂ – x₁)。

Parallel lines have equal gradients.

平行线的斜率相等。

Perpendicular lines have gradients that multiply to –1, i.e. m₁ × m₂ = –1.

互相垂直的直线斜率乘积为–1，即 m₁ × m₂ = –1。

To find the x-intercept, set y = 0 and solve; to find the y-intercept, set x = 0.

求x轴截距，令 y = 0 解方程；求y轴截距，令 x = 0。

The midpoint of a line segment joining (x₁, y₁) and (x₂, y₂) is ((x₁+x₂)/2, (y₁+y₂)/2).

连接 (x₁, y₁) 与 (x₂, y₂) 的线段中点坐标为 ((x₁+x₂)/2, (y₁+y₂)/2)。

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10. Statistics and Probability | 统计与概率

Data handling and probability are key components of the OCR curriculum. Knowing how to calculate averages and probabilities is essential.

数据处理与概率是OCR课程的重要组成部分。掌握如何计算平均数和概率至关重要。

The mean of a set of numbers = (sum of all values) / (number of values), often denoted by x̄.

一组数的平均数 = 总和 / 个数，常记为 x̄。

The median is the middle value when data are ordered; if there are two middle numbers, take their mean.

中位数是将数据排序后处于中间的值；若中间有两个数，则取它们的平均数。

The mode is the value that appears most frequently.

众数是出现频率最高的值。

The range = maximum value – minimum value, measuring spread.

极差 = 最大值 – 最小值，用于衡量离散程度。

Probability of an event = (number of favourable outcomes) / (total number of possible outcomes), typically written as a fraction, decimal or percentage.

事件的概率 = 有利结果数 / 所有可能结果总数，通常用分数、小数或百分数表示。

The probability scale goes from 0 (impossible) to 1 (certain).

概率的取值范围从0（不可能）到1（必然）。

For mutually exclusive events, P(A or B) = P(A) + P(B).

对于互斥事件，P(A或B) = P(A) + P(B)。

Expected frequency = probability × number of trials.

期望频数 = 概率 × 试验次数。

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