📚 Year 10 OCR Maths: Quick Reference Formula & Theorem Handbook | Year 10 OCR 数学：公式定理速查手册
This handbook provides a concise summary of all the essential formulas and theorems you need for the Year 10 OCR Mathematics course. Each section is presented with clear explanations and paired Chinese translations to help you master key concepts quickly. Use it for revision, homework, or exam preparation.
本手册简明扼要地总结了 Year 10 OCR 数学课程所需的所有核心公式与定理。每个部分都配有清晰的中英文对照解释，帮助你快速掌握关键概念，适合用于复习、作业或备考冲刺。
1. Algebraic Laws and Indices | 代数定律与指数
When multiplying powers with the same base, add the exponents. When dividing, subtract the exponents.
同底数幂相乘，指数相加；同底数幂相除，指数相减。
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Raising a power to another power means multiplying the indices. Any non‑zero number raised to the power of zero equals 1.
幂的乘方，指数相乘。任何非零数的零次幂等于1。
(aᵐ)ⁿ = aᵐⁿ
a⁰ = 1 (a ≠ 0)
Negative indices represent reciprocals. Fractional indices correspond to roots.
负指数表示倒数，分数指数对应根式。
a⁻ⁿ = 1 / aⁿ
a1/n = ⁿ√a
Expanding binomials follows standard patterns. The square of a binomial is a perfect square trinomial.
二项式展开遵循固定模式。完全平方公式是常用的恒等式。
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
The difference of two squares factorises into a product of a sum and a difference.
平方差公式可分解为和与差的乘积。
a² − b² = (a − b)(a + b)
2. Linear Equations and Graphs | 线性方程与图像
The equation of a straight line is usually written in slope‑intercept form, where m is the gradient and c is the y‑intercept.
直线方程通常写成斜截式，其中 m 是斜率（梯度），c 是 y 轴截距。
y = mx + c
Gradient is calculated as the change in y divided by the change in x between two points.
斜率通过两点间纵坐标的差除以横坐标的差来计算。
m = (y₂ − y₁) / (x₂ − x₁)
The midpoint of two points is found by averaging their x‑coordinates and y‑coordinates separately.
两点间的中点坐标是横纵坐标分别取平均值。
M = ( (x₁+x₂)/2 , (y₁+y₂)/2 )
The distance between two points is derived from Pythagoras’ theorem.
两点之间的距离通过勾股定理求得。
d = √[ (x₂−x₁)² + (y₂−y₁)² ]
Parallel lines share the same gradient. Perpendicular lines have gradients whose product is −1.
平行线斜率相等；垂直线斜率乘积为 −1。
m₁ = m₂ (parallel)
m₁ × m₂ = −1 (perpendicular)
3. Quadratic Equations | 二次方程
A quadratic equation in standard form has the coefficient a ≠ 0. It can be solved by factorising, completing the square, or using the quadratic formula.
标准形式的二次方程中 a ≠ 0。可以通过因式分解、配方法或求根公式求解。
ax² + bx + c = 0
The quadratic formula gives the solutions directly using the coefficients. The discriminant determines the nature of the roots.
求根公式直接利用系数给出解。判别式决定了根的性质。
x = [ −b ± √b² − 4ac ] / (2a)
Δ = b² − 4ac
If Δ > 0, two distinct real roots.
Δ > 0，两个不等实根。
If Δ = 0, one repeated real root.
Δ = 0，一个重根。
If Δ < 0, no real roots.
Δ < 0，无实根。
Factorising a quadratic relies on finding two numbers that multiply to ac and add to b.
因式分解二次式需要找出两个数，使其乘积为 ac，和为 b。
Completing the square rewrites the expression as a perfect square plus a constant.
配方法将表达式改写为一个完全平方加上常数。
x² + bx = (x + b/2)² − (b/2)²
4. Pythagoras’ Theorem and Trigonometry | 勾股定理与三角学
In a right‑angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
在直角三角形中，斜边的平方等于两直角边的平方和。
a² + b² = c²
The three trigonometric ratios link angles to side lengths in a right‑angled triangle.
三个基本三角比将直角三角形中的角与边长联系起来。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
For non‑right‑angled triangles, the sine rule and cosine rule are used to find missing sides or angles.
对于非直角三角形，可使用正弦定理和余弦定理求解未知边或角。
Sine rule: a / sin A = b / sin B = c / sin C
Cosine rule: a² = b² + c² − 2bc cos A
The area of any triangle can be computed when two sides and the included angle are known.
若已知两边及其夹角，可用公式计算任意三角形的面积。
Area = ½ ab sin C
5. Area and Volume | 面积与体积
Key area formulas for 2D shapes are essential for solving geometry problems.
掌握二维图形的基本面积公式是解决几何问题的关键。
| Shape 图形 | Area 面积 |
| Rectangle 矩形 | A = l × w (length × width) |
| Triangle 三角形 | A = ½ × base × height |
| Parallelogram 平行四边形 | A = b × h (base × perpendicular height) |
| Trapezium 梯形 | A = ½ (a + b) h |
| Circle 圆 | A = π r² (r = radius) |
| Circle circumference 圆的周长 | C = 2π r or π d |
For 3D shapes, volume measures the space occupied. Surface area is the total area of all faces.
对于三维立体，体积度量所占空间，表面积是所有面的总面积。
| Solid 立体 | Volume 体积 | Surface area 表面积 |
| Cuboid 长方体 | V = l w h | SA = 2(lw + lh + wh) |
| Prism 棱柱 | V = area of cross‑section × length | Sum of areas of all faces |
| Cylinder 圆柱 | V = π r² h | SA = 2π r h + 2π r² |
| Cone 圆锥 | V = ⅓ π r² h | SA = π r l + π r² (l = slant height) |
| Sphere 球 | V = 4/3 π r³ | SA = 4π r² |
| Pyramid 棱锥 | V = ⅓ × base area × height | Base area + sum of triangular faces |
6. Circle Theorems and Sector Formulae | 圆定理与扇形公式
Several fundamental circle theorems help determine unknown angles without measuring.
若干基本圆定理可以帮助不通过测量而求出未知角。
Angle in a semicircle is a right angle (90°).
半圆上的圆周角是直角（90°）。
Angles in the same segment are equal.
同弧上的圆周角相等。
The angle at the centre is twice the angle at the circumference.
圆心角等于同弧上圆周角的两倍。
Opposite angles in a cyclic quadrilateral sum to 180°.
圆内接四边形对角互补，和为180°。
A tangent is perpendicular to the radius at the point of contact.
切线与过切点的半径垂直。
Sectors and arcs are proportional to the central angle θ (in degrees).
扇形与弧长与圆心角θ（度数）成比例。
Arc length = (θ/360) × 2π r
Sector area = (θ/360) × π r²
The perimeter of a sector includes two radii plus the arc length.
扇形周长包括两条半径加上弧长。
7. Measures of Central Tendency and Spread | 集中趋势与离散度量
Mean, median and mode summarise a data set with a single representative value.
平均数、中位数和众数用单一代表值概括数据集。
Mean = (sum of all values) / (number of values)
For grouped data, we estimate the mean using midpoints and frequencies.
对于分组数据，用组中值与频数估计平均数。
Estimated mean = Σ(f × x) / Σf
The median is the middle value when data is ordered; the mode is the most frequent value.
中位数是排序后中间的值，众数是出现最频繁的值。
Range gives a simple measure of spread. Interquartile range (IQR) focuses on the middle 50%.
极差给出简单的离散度量；四分位距 (IQR) 关注中间50%的数据。
Range = highest value − lowest value
IQR = upper quartile − lower quartile
Cumulative frequency graphs and box plots are powerful tools for visualising spread.
累计频率图与箱线图是可视化离散程度的有效工具。
8. Probability | 概率
Probability measures how likely an event is to happen, expressed as a number between 0 and 1.
概率衡量事件发生的可能性，用0到1之间的数表示。
P(event) = number of favourable outcomes / total number of possible outcomes
The addition rule handles ‘or’ probabilities, remembering to subtract overlap.
加法法则处理‘或’事件的概率，需减去重叠部分。
P(A or B) = P(A) + P(B) − P(A and B)
For independent events, multiply probabilities to find ‘and’ scenarios.
对于独立事件，‘且’的概率通过相乘计算。
P(A and B) = P(A) × P(B)
Conditional probability considers how the probability of an event changes given another has occurred.
条件概率考虑在另一事件已发生的情况下某事件的概率变化。
P(A|B) = P(A and B) / P(B)
Tree diagrams are a systematic way to multiply along branches and add paths.
树形图是一种系统方法，需要沿分支相乘并将各路径概率相加。
9. Sequences | 数列
An arithmetic sequence has a constant difference d between consecutive terms.
等差数列相邻两项的差 d 为常数。
nth term = a + (n−1)d
Here a is the first term, n is the position, and d is the common difference. You can generate any term without listing all previous ones.
其中 a 为首项，n 为项数，d 为公差。无需逐一列出即可求出任意项。
A geometric sequence has a common ratio r, and its nth term formula uses powers.
等比数列有公比 r，其第 n 项公式用到乘方。
nth term = a × rn−1
Recognising special sequences such as square numbers (n²), cube numbers (n³), and triangular numbers ( n(n+1)/2 ) is also useful.
识别特殊数列，如平方数 n²，立方数 n³，三角形数 n(n+1)/2，也很有帮助。
10. Angles and Polygons | 角与多边形
Angles on a straight line sum to 180°, and angles around a point sum to 360°.
直线上的角之和为180°，点周围的角度和为360°。
The interior angles of a triangle always add up to 180°. The exterior angle of a triangle equals the sum of the two opposite interior angles.
三角形内角和恒为180°。三角形的一个外角等于不相邻的两个内角之和。
For any polygon with n sides, the sum of interior angles and exterior angles are fixed.
对于任意 n 边形，内角和与外角和是确定值。
Sum of interior angles = (n − 2) × 180°
Sum of exterior angles = 360°
In a regular polygon, each interior angle equals the total sum divided by n.
在正多边形中，每个内角等于总和除以 n。
When parallel lines are cut by a transversal, special angle pairs appear:
平行线被截线所截时，出现特殊角对：
Corresponding angles are equal. | 同位角相等。
Alternate interior angles are equal. | 内错角相等。
Co‑interior (allied) angles sum to 180°. | 同旁内角互补，和为180°。
Vertically opposite angles are always equal.
对顶角始终相等。
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