📚 Year 8 Edexcel Quick Formula & Theorem Reference | 八年级 Edexcel 公式定理速查手册
This bilingual handbook brings together the must-know formulas and theorems for Year 8 Edexcel Mathematics and Science. Each entry is presented with a clear English explanation immediately followed by its Chinese counterpart, making it perfect for revision, homework support and building academic language in both English and Chinese.
这本双语手册汇集了八年级 Edexcel 数学与科学必会的公式和定理。每个条目先用英文解释，紧接着给出中文对应，非常适合复习、辅助作业以及同时提升英汉学术语言能力。
1. Algebraic Expressions | 代数表达式
Mastering algebraic manipulation is the foundation for all higher-level mathematics. The following identities let you expand and simplify expressions quickly.
掌握代数运算是所有高阶数学的基础。以下恒等式可以帮助你快速展开和简化表达式。
a(b + c) = ab + ac
The distributive property shows how a single term multiplies each term inside parentheses.
分配律表明单项式如何乘以括号内的每一项。
(a + b)² = a² + 2ab + b²
This perfect square expansion is one of the most commonly used algebraic identities.
这个完全平方展开式是最常用的代数恒等式之一。
(a – b)² = a² – 2ab + b²
The square of a difference follows a similar pattern but with a negative middle term.
差的平方遵循类似模式，只是中间项为负。
2. Linear Equations | 一次方程
Solving linear equations is a key skill. The goal is to isolate the variable using inverse operations.
解一次方程是一项关键技能。目标是利用逆运算把未知数单独留在等式一边。
ax + b = c → x = (c – b)/a
Subtract b from both sides, then divide by a to find the solution of a one-step or two-step equation.
两边同时减去 b，再除以 a，即可求得一步或两步方程的解。
If a > b, then a + c > b + c
Adding or subtracting the same number from both sides of an inequality preserves the inequality sign.
在不等式两边加上或减去同一个数，不等号方向保持不变。
If a > b and c > 0, then ac > bc
Multiplying or dividing by a positive number leaves the inequality unchanged. If c is negative, the sign reverses.
乘以或除以正数时，不等号方向不变。若 c 为负数，则不等号方向要反过来。
3. Coordinates and Straight Lines | 坐标与直线
Working with points and lines on the coordinate plane requires knowing how to calculate midpoints, gradients and line equations.
在坐标平面中处理点与直线，需要掌握中点、斜率和直线方程的计算方法。
Midpoint: ( (x₁+x₂)/2 , (y₁+y₂)/2 )
The midpoint is the average of the x-coordinates and the average of the y-coordinates of two points.
中点坐标等于两点 x 坐标的平均值与 y 坐标的平均值。
Gradient: m = (y₂ – y₁) / (x₂ – x₁)
The gradient measures the steepness of a line; it is the change in y divided by the change in x.
斜率衡量直线的陡峭程度，等于 y 的变化量除以 x 的变化量。
Equation: y = mx + c
A linear equation in slope-intercept form. m is the gradient and c is the y-intercept.
斜截式直线方程。m 表示斜率，c 表示 y 轴截距。
4. Ratio and Proportion | 比与比例
Ratios compare quantities, while proportions state that two ratios are equal. These tools appear frequently in scaling and problem‑solving.
比用于比较数量，比例说明两个比相等。这些工具在缩放和解题中经常出现。
a : b = c : d ⇔ ad = bc
In a proportion, the cross products are equal. This property helps find missing values.
在比例中，交叉乘积相等。利用这一性质可以求出缺失的值。
Percentage change = (change / original) × 100%
To calculate a percentage increase or decrease, divide the amount of change by the original value and multiply by 100.
计算百分比增减时，用变化量除以原值再乘以 100%。
Direct proportion: y = kx
When y is directly proportional to x, the ratio y/x is constant (k). The graph is a straight line through the origin.
当 y 与 x 成正比例时，y/x 为常数 k，图像是一条经过原点的直线。
5. Perimeter and Area | 周长与面积
Knowing how to find perimeter and area is essential for geometry problems. Formulas vary by shape.
掌握周长和面积的求法对于几何问题至关重要。不同形状的公式各不相同。
Rectangle: P = 2(l + w), A = lw
Perimeter is the total distance around the shape; area is the space inside, found by multiplying length and width.
周长是图形一周的总长度；面积是内部空间大小，通过长乘以宽求得。
Triangle: A = ½ × base × height
The area of any triangle is half the product of its base and the perpendicular height.
任意三角形的面积等于底与垂直高度乘积的一半。
Circle: C = 2πr, A = πr²
For circles, the circumference uses the radius, and the area uses radius squared. π is approximately 3.14.
对于圆，周长用到半径，面积用到半径的平方。π 约等于 3.14。
6. Volume and Surface Area | 体积与表面积
Three-dimensional shapes have volume (space inside) and surface area (total area of all faces).
三维图形有体积（内部空间）和表面积（所有面的总面积）。
Cuboid: V = l × w × h
The volume of a rectangular prism is the product of its length, width and height.
长方体的体积等于长、宽、高的乘积。
Prism: V = area of cross‑section × length
For any prism, multiply the area of its uniform cross‑section by the length of the prism.
对于任何棱柱，用其均匀横截面的面积乘以棱柱的长度。
Cube surface area: SA = 6s²
Since a cube has six identical square faces, total surface area is six times the area of one face.
正方体有六个相同的正方形面，因此总表面积为单个面面积的六倍。
7. Angles and Polygons | 角与多边形
Angle rules help you find missing angles in triangles, quadrilaterals and other polygons without measuring.
角的定理可以帮助你无需测量就能求出三角形、四边形和其他多边形中的未知角度。
Sum of angles in a triangle = 180°
The three interior angles of any triangle always add up to 180 degrees.
任意三角形的三个内角之和恒为 180 度。
Sum of interior angles of an n‑sided polygon = (n – 2) × 180°
For a polygon with n sides, the total interior angle sum is given by this formula.
对于 n 边形的多边形，其内角和由该公式给出。
Sum of exterior angles = 360°
The exterior angles of any convex polygon, taken one at each vertex, always sum to 360°.
任何凸多边形的外角和（每个顶点取一个外角）恒为 360°。
8. Pythagoras’ Theorem | 勾股定理（毕达哥拉斯定理）
Pythagoras’ theorem relates the sides of a right‑angled triangle and is one of the most powerful tools in geometry.
勾股定理揭示了直角三角形三边之间的关系，是几何中最强大的工具之一。
a² + b² = c²
In a right‑angled triangle, the square of the hypotenuse (c, the longest side) equals the sum of the squares of the other two sides (a and b).
在直角三角形中，斜边（c，最长边）的平方等于另外两条直角边（a 和 b）的平方之和。
Converse: If a² + b² = c², the triangle is right‑angled.
If the side lengths of a triangle satisfy this relationship, the angle opposite the longest side must be a right angle.
如果三角形三条边的长度满足该关系，那么最长边所对的角必定是直角。
Pythagorean triples: 3‑4‑5, 5‑12‑13
Common sets of whole numbers that work in the theorem can save time in problem‑solving.
满足定理的常用整数组（如 3-4-5, 5-12-13）能为解题节省时间。
9. Statistics: Averages and Spread | 统计：平均数与离散程度
Statistical measures summarise data sets. The three averages and the range give a quick overview of the data.
统计量用于概括数据集。三种平均数和极差可以快速呈现数据的概况。
Mean = (sum of all values) ÷ (number of values)
The mean is the arithmetic average, found by adding all data items and dividing by the count.
平均数即算术平均值，用所有数据的总和除以数据的个数求得。
Median = middle value when ordered
Arrange the data in order; the median is the value in the middle position. If there are two middle values, take their mean.
将数据按大小排列后，位于中间位置的数值即为中位数。若有两个中间数，则取它们的平均值。
Mode = most frequent value
The mode is the data value that appears most often. A data set can have more than one mode or no mode at all.
众数是出现次数最多的数据值。一组数据可以有多个众数，也可以没有众数。
10. Probability Basics | 概率基础
Probability quantifies the chance of an event happening. It is always a number between 0 (impossible) and 1 (certain).
概率用数值表示事件发生的可能性，总是在 0（不可能）和 1（必然）之间。
P(Event) = Number of favourable outcomes / Total number of possible outcomes
To calculate theoretical probability, count how many ways the desired event can occur and divide by the total number of equally likely outcomes.
计算理论概率时，先数出事件可能发生的方式数，再除以等可能结果的总数。
P(not A) = 1 – P(A)
The probability of an event not occurring is one minus the probability that it does occur.
事件不发生的概率等于 1 减去事件发生的概率。
Sum of all mutually exclusive outcomes = 1
If all possible outcomes are listed, their probabilities must add up to 1.
若列出所有可能的结果，它们的概率之和必定为 1。
11. Physics: Motion, Density and Pressure | 物理：运动、密度与压强
These science formulas link physical quantities and are used regularly in Key Stage 3 Physics.
这些科学公式将物理量联系起来，在第三学段物理课中经常使用。
Speed = distance ÷ time (v = s/t)
Average speed is calculated by dividing the total distance travelled by the total time taken.
平均速度等于总路程除以所用的总时间。
Density = mass ÷ volume (ρ = m/V)
Density describes how much mass is packed into a given volume. The symbol ρ (rho) is often used.
密度表示单位体积内所含的质量，常用符号 ρ（rho）表示。
Pressure = force ÷ area (P = F/A)
Pressure measures how a force is spread over a surface. The smaller the area, the greater the pressure for a given force.
压强衡量力在表面上的分布情况。在力一定时，受力面积越小，压强越大。
12. Chemistry: Conservation of Mass | 化学：质量守恒
Chemical reactions rearrange atoms but never create or destroy them. This principle is key to balancing equations.
化学反应只是重新排列原子，既不会创造也不会消灭原子。这一原理是配平化学方程式的关键。
Mass of reactants = Mass of products
In a closed system, the total mass before a chemical reaction equals the total mass after the reaction.
在封闭体系中，化学反应前的总质量等于反应后的总质量。
Number of each type of atom is conserved
When writing a chemical equation, make sure the count of each element on the left (reactants) matches the count on the right (products) by adding coefficients.
书写化学方程式时，通过添加系数确保左边（反应物）每种元素的原子个数与右边（生成物）完全相同。
Example: 2H₂ + O₂ → 2H₂O
Here, four hydrogen atoms and two oxygen atoms appear on both sides, satisfying the conservation law.
在本例中，反应前后均有四个氢原子和两个氧原子，满足守恒定律。
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