📚 Cambridge Year 10 Maths: Formula & Theorem Quick Reference | 剑桥十年级数学：公式定理速查手册
This quick reference guide gathers the most essential formulas and theorems you will meet in the Cambridge Year 10 mathematics course. Use it as a handy companion for revision, homework, and exam preparation.
本速查手册汇集了剑桥十年级数学课程中最重要的公式与定理，可用作复习、作业和考试准备的便捷参考资料。
1. Algebraic Identities | 代数恒等式
Key algebraic identities used in expanding and factorising expressions include the perfect squares, difference of two squares, and cubes.
用于展开与因式分解的重要代数恒等式包括完全平方、平方差及立方公式。
Perfect square: (a + b)² = a² + 2ab + b²
完全平方： (a + b)² = a² + 2ab + b²
Perfect square: (a − b)² = a² − 2ab + b²
完全平方： (a − b)² = a² − 2ab + b²
Difference of two squares: a² − b² = (a − b)(a + b)
平方差： a² − b² = (a − b)(a + b)
Cube of a sum: (a + b)³ = a³ + 3a²b + 3ab² + b³
和的立方： (a + b)³ = a³ + 3a²b + 3ab² + b³
Cube of a difference: (a − b)³ = a³ − 3a²b + 3ab² − b³
差的立方： (a − b)³ = a³ − 3a²b + 3ab² − b³
d difference of cubes: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²)
立方和与立方差： a³ + b³ = (a + b)(a² − ab + b²)，a³ − b³ = (a − b)(a² + ab + b²)
2. Linear Equations & Inequalities | 线性方程与不等式
e variable have the general form ax + b = 0, and their solutions are found by isolating x.
一元一次方程的一般形式为 ax + b = 0，通过移项求解 x。
Solution: x = −b/a, provided a ≠ 0.
解为： x = −b/a，其中 a ≠ 0。
When solving simultaneous linear equations in two variables, you can use substitution or elimination. The point of intersection of the two lines gives the solution (x, y).
解二元一次方程组时，可使用代入法或消元法。两直线的交点坐标即为解 (x, y)。
Linear inequalities behave much like equations, except that multiplying or dividing by a negative number reverses the inequality sign.
线性不等式的解法与方程相似，但乘以或除以负数时，不等号方向要改变。
Example: If −2x > 6, then x < −3.
示例：若 −2x > 6，则 x < −3。
expressed on a number line, with an open circle for < or > and a closed circle for ≤ or ≥.
不等式的解集可以用数轴表示，< 或 > 用空心圆点，≤ 或 ≥ 用实心圆点。
3. Quadratic Expressions & Equations | 二次式与二次方程
0. Solutions can be found by factorising, completing the square, or the quadratic formula.
二次方程的标准形式为 ax² + bx + c = 0，且 a ≠ 0。求解方法包括因式分解、配方法和求根公式。
x = (−b ± √b² − 4ac) / (2a)
x = (−b ± √b² − 4ac) / (2a)
Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.
判别式 Δ = b² − 4ac 决定了根的性质：Δ > 0 时有两个相异实根；Δ = 0 时有一个重根；Δ < 0 时无实根。
Factorising quadratics often relies on finding two numbers that multiply to give ac and add to give b.
因式分解二次式常需找到两个数，其积为 ac，和为 b。
Completing the square rewrites the expression as a(x + p)² + q, revealing the vertex of the parabola.
配方法将二次式化为 a(x + p)² + q 的形式，从而展示抛物线的顶点。
4. Sequences & Series | 数列与级数
An arithmetic sequence has a constant difference d between consecutive terms. The nth term is given by:
等差数列中相邻两项的差为常数 d，其第 n 项公式为：
aₙ = a₁ + (n − 1)d
aₙ = a₁ + (n − 1)d
irst n terms of an arithmetic sequence is Sₙ = n/2 [2a₁ + (n − 1)d] or Sₙ = n/2 (a₁ + aₙ).
等差数列前 n 项和为 Sₙ = n/2 [2a₁ + (n − 1)d] 或 Sₙ = n/2 (a₁ + aₙ)。
A geometric sequence has a constant ratio r between consecutive terms. The nth term is:
等比数列中相邻两项的比值为常数 r，其第 n 项公式为：
aₙ = a₁ rⁿ⁻¹
aₙ = a₁ rⁿ⁻¹
The sum of the first n terms of a geometric sequence (r ≠ 1) is Sₙ = a₁(1 − rⁿ)/(1 − r).
等比数列前 n 项和（r ≠ 1）为 Sₙ = a₁(1 − rⁿ)/(1 − r)。
Other common sequences include square numbers (n²), cube numbers (n³), and triangular numbers (n(n+1)/2).
其他常见数列包括平方数 (n²)、立方数 (n³) 和三角形数 (n(n+1)/2)。
5. Coordinate Geometry | 坐标几何
The distance between two points A(x₁, y₁) and B(x₂, y₂) is calculated using the formula derived from Pythagoras' theorem.
两点 A(x₁, y₁) 与 B(x₂, y₂) 之间的距离公式由勾股定理导出。
AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
AB = √[(x₂ − x₁)² + (y₂ − y₁)²]
The midpoint M of AB is:
线段 AB 的中点 M 坐标为：
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
The gradient (slope) m of the line through A and B is given by:
过点 A 和 B 的直线的斜率 m 为：
m = (y₂ − y₁)/(x₂ − x₁)
m = (y₂ − y₁)/(x₂ − x₁)
mx + c, where c is the y-intercept. An alternative point-slope form is y − y₁ = m(x − x₁).
直线方程通常写作 y = mx + c，其中 c 为 y 轴截距；点斜式方程为 y − y₁ = m(x − x₁)。
llel lines have equal gradients (m₁ = m₂), while perpendicular lines satisfy m₁ × m₂ = −1.
平行线斜率相等 (m₁ = m₂)，垂直线斜率之积为 −1 (m₁ × m₂ = −1)。
6. Geometry: Angles & Polygons | 几何：角与多边形
When two parallel lines are cut by a transversal, several equal angle pairs appear: corresponding angles are equal, alternate interior angles are equal, and co-interior angles sum to 180°.
一组平行线被一条横切线所截，会出现多组相等角：同位角相等、内错角相等、同旁内角互补（和为 180°）。
Vertically opposite angles are always equal.
对顶角总是相等。
The sum of interior angles in a triangle is 180°.
三角形内角和为 180°。
les is (n − 2) × 180°, and the sum of exterior angles (one at each vertex) is always 360°.
对于任意 n 边形，内角和为 (n − 2) × 180°，外角和（每个顶点一个外角）恒为 360°。
ngles equal; each interior angle is [(n − 2) × 180°]/n, and each exterior angle is 360°/n.
正多边形的各边各角均相等，每个内角为 [(n − 2) × 180°]/n，每个外角为 360°/n。
Circle angle facts: the angle at the centre is twice the angle at the circumference subtended by the same arc. Angles in the same segment are equal.
圆中角度定理：同弧所对的圆心角是圆周角的 2 倍；同弦上的圆周角相等。
The angle in a semicircle is 90°.
半圆上的圆周角为 90°。
7. Pythagorean Theorem & Trigonometry | 勾股定理与三角学
In a right-angled triangle, the square of the hypotenuse is equal to 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。
c² = a² + b²
c² = a² + b²
The three primary trigonometric ratios for an acute angle θ are defined by the sides of a right triangle:
锐角 θ 的三组基本三角比由直角三角形的边长定义：
sin θ = opposite / hypotenuse, cos θ = adjacent / hypotenuse, tan θ = opposite / adjacent.
sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，tan θ = 对边 / 邻边。
Exact trigonometric values for special angles should be memorised:
特殊角的精确三角函数值需熟记：
| 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 |
ts reverse. The cosine rule relates sides and an included angle: a² = b² + c² − 2bc cos A.
正弦定理适用于任意三角形：a / sin A = b / sin B = c / sin C。余弦定理将三边与一角关联：a² = b² + c² − 2bc cos A。
The area of a triangle can also be found using Area = 1/2 ab sin C.
三角形面积也可用公式 面积 = 1/2 ab sin C 求得。
8. Mensuration: Perimeter, Area & Volume | 度量：周长、面积与体积
Here are the essential formulas for common 2D shapes:
以下是常见二维图形的重要公式：
Rectangle: Area = l × w, Perimeter = 2(l + w)
矩形：面积 = 长 × 宽，周长 = 2(长 + 宽)
Triangle: Area = 1/2 × base × height
三角形：面积 = 1/2 × 底 × 高
Parallelogram: Area = base × height
平行四边形：面积 = 底 × 高
Trapezium: Area = 1/2 (a + b) × h, where a and b are the parallel sides
梯形：面积 = 1/2 (a + b) × h，其中 a、b 为平行边
Circle: Area = πr², Circumference = 2πr
圆：面积 = πr²，周长 = 2πr
Sector: Area = (θ/360) × πr², Arc length = (θ/360) × 2πr
扇形：面积 = (θ/360) × πr²，弧长 = (θ/360) × 2πr
Volume and surface area of 3D shapes:
三维图形的体积与表面积：
Cuboid: Volume = l × w × h, Surface area = 2(lw + lh + wh)
长方体：体积 = 长 × 宽 × 高，表面积 = 2(长×宽 + 长×高 + 宽×高)
Prism: Volume = area of cross-section × length
棱柱：体积 = 横截面积 × 长度
Cylinder: Volume = πr²h, Curved surface area = 2πrh, Total surface area = 2πr(r + h)
圆柱：体积 = πr²h，侧面积 = 2πrh，全面积 = 2πr(r + h)
Pyramid: Volume = 1/3 × base area × height
棱锥：体积 = 1/3 × 底面积 × 高
Cone: Volume = 1/3 πr²h, Curved surface area = πrl, Total surface area = πr(r + l)
圆锥：体积 = 1/3 πr²h，侧面积 = πrl，全面积 = πr(r + l)
Sphere: Volume = 4/3 πr³, Surface area = 4πr²
球体：体积 = 4/3 πr³，表面积 = 4πr²
9. Statistics Essentials | 统计基础
Measures of central tendency summarise data with a typical value:
集中量数用一个典型值概括数据：
Mean = (sum of all data values) / (number of data values)
平均数 = 所有数据值之和 / 数据个数
the middle value when data are ordered; if there are two middle numbers, take their mean.
中位数 = 排序后居中的值；如果有两个中间数，则取两者的平均。
hat appears most frequently. A data set can have one mode, more than one mode, or no mode.
众数 = 出现次数最多的数据值；一组数据可以有一个、多个或无众数。
Range = maximum value − minimum value, a measure of spread.
极差 = 最大值 − 最小值，用于度量离散程度。
pper quartile Q₃ is the median of the upper half. The interquartile range (IQR) = Q₃ − Q₁.
四分位数：下四分位数 Q₁ 是数据下半部分的中位数，上四分位数 Q₃ 是上半部分的中位数。四分位距 IQR = Q₃ − Q₁。
Frequency tables, bar charts, pie charts and cumulative frequency graphs are common ways to display and interpret data.
频数表、条形图、饼图和累积频数图是常见的展示与解读数据的方式。
10. Probability Rules | 概率规则
t A is P(A) = (number of favourable outcomes) / (total number of equally likely outcomes).
事件 A 的概率为 P(A) = 有利结果数 / 所有等可能结果的总数。
obabilities always lie between 0 and 1. P(A) = 0 means impossible; P(A) = 1 means certain.
概率值总介于 0 和 1 之间。P(A) = 0 表示不可能事件；P(A) = 1 表示必然事件。
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)。
For independent events, P(A and B) = P(A) × P(B).
对于独立事件，P(A 且 B) = P(A) × P(B)。
The complement rule: P(not A) = 1 − P(A).
互补事件：P(非 A) = 1 − P(A)。
Tree diagrams help to map out combined events and multiply along branches for 'and' probabilities; add at the ends for 'or' probabilities when appropriate.
树形图可帮助理清组合事件，沿分支相乘得到“且”的概率，必要时在末端相加得到“或”的概率。
11. Transformations & Vectors | 变换与向量
A translation moves every point of a shape by the same vector.
Published by TutorHao | Year 10 Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图