📚 GCSE AQA Maths: Formula Summary Handbook | GCSE AQA 数学：公式汇总手册
This handbook brings together all the essential formulas you need to memorise and apply in your GCSE AQA Mathematics exams. Each section groups related formulas with clear explanations, helping you revise effectively and tackle both calculator and non-calculator papers with confidence.
本手册汇总了GCSE AQA数学考试中需要记忆并运用的所有核心公式。每个小节将相关公式归类并附上清晰说明，帮助你高效复习，自信应对计算器与非计算器试卷。
1. Arithmetic and Percentages | 算术与百分比
Percentage work appears in both calculator and non-calculator papers, often inside money, interest, depreciation and growth problems. The key idea is to convert every percentage into a decimal multiplier before you calculate.
百分比计算在计算器卷与非计算器卷中都会出现，常见于金钱、利息、折旧和增长类问题。核心思路是先把它化成小数乘数，再进行运算。
Standard percentage change compares the difference with the original value, so always divide by the original amount, not the new one.
标准百分比变化把差值与原始值比较，所以除数永远是原值，而不是新值。
Percentage Change = (New Value − Original Value) / Original Value × 100%
百分比变化 = (新值 − 原值) / 原值 × 100%
Simple interest grows by the same fixed amount each year, because interest is always calculated on the original principal.
单利每年增长相同的固定金额，因为利息始终以原始本金计算。
Simple Interest I = P × r × t (r as a decimal, t in years)
单利 I = 本金 × 年利率(小数) × 年数
Compound interest adds interest back to the principal each period, so the multiplier is raised to the power of the number of periods.
复利把每期利息并入本金，因此乘数要取期数的次方。
Compound Amount A = P (1 + r/100)ⁿ (r% per period, n periods)
复利本息和 A = 本金 × (1 + 期利率%)ⁿ（n为期数）
For a percentage increase use a multiplier greater than 1, and for a percentage decrease use a multiplier less than 1.
增长使用大于1的乘数，减少使用小于1的乘数。
Increase: × (1 + r/100) Decrease: × (1 − r/100)
增长乘数：× (1 + 增长率%/100) 减少乘数：× (1 − 减少率%/100)
Reverse percentage problems give you the final value and ask for the original, so divide by the multiplier instead of multiplying.
逆向百分比问题给出最终值、要求原值，因此要除以乘数而不是乘以乘数。
Original Value = Final Value ÷ Multiplier
原值 = 最终值 ÷ 乘数
Depreciation is repeated percentage decrease, so the same compound formula is used with a minus sign inside the bracket.
折旧属于重复百分比减少，公式结构与复利相同，只是括号内改为减号。
Depreciated Value = P (1 − r/100)ⁿ
折旧后价值 = 本金 × (1 − 折旧率%/100)ⁿ
2. Algebraic Formulas and Sequences | 代数公式与数列
Algebraic identities and the quadratic formula run through equations, inequalities, graphs and sequences. Examiners expect you to recognise these forms instantly rather than derive them each time.
代数恒等式与二次公式贯穿方程、不等式、图像与数列。考官希望你一眼识别这些形式，而不是每次重新推导。
solves any equation written in the standard form ax² + bx + c = 0, provided a is not zero.
二次公式可解任何写成标准形式 ax² + bx + c = 0 的方程，其中 a 不为零。
Quadratic Formula: x = (−b ± √b² − 4ac) / (2a)
二次公式：x = (−b ± √b² − 4ac) / (2a)
The discriminant is the expression under the square root, and its sign tells you how many real solutions exist.
判别式是根号下的表达式，它的符号决定实数解的个数。
Discriminant Δ = b² − 4ac
判别式 Δ = b² − 4ac
real roots, if Δ = 0 there is one repeated root, and if Δ < 0 there are no real roots.
当 Δ > 0 时有两个不同实根，Δ = 0 时有一个重根，Δ < 0 时没有实根。
The difference of two squares factorises any expression where one square is subtracted from another.
平方差公式可分解任意两个平方相减的表达式。
Difference of Two Squares: a² − b² = (a − b)(a + b)
平方差公式：a² − b² = (a − b)(a + b)
Perfect squares expand quickly once you remember the middle term is always twice the product of the two terms.
完全平方展开的关键是中间项始终等于两项乘积的两倍。
Perfect Square: (a ± b)² = a² ± 2ab + b²
完全平方公式：(a ± b)² = a² ± 2ab + b²
Completing the square rewrites a quadratic in vertex form, which helps you find the turning point of a graph.
配方法把二次式改写为顶点形式，便于求图像的转折点。
x² + bx + c = (x + b/2)² + c − (b/2)²
x² + bx + c = (x + b/2)² + c − (b/2)²
An arithmetic sequence changes by the same amount each term, and the nth term formula uses the first term a and the common difference d.
等差数列每一项的变化量相同，通项公式使用首项 a 与公差 d。
nth Term of an Arithmetic Sequence: aₙ = a + (n − 1)d
等差数列通项：aₙ = a + (n − 1)d
A geometric sequence multiplies by the same ratio each term, and the nth term formula uses the common ratio r.
等比数列每一项乘以相同比率，通项公式使用公比 r。
nth Term of a Geometric Sequence: aₙ = a × r⁽ⁿ⁻¹⁾
等比数列通项：aₙ = a × r⁽ⁿ⁻¹⁾
3. Angle Properties and Polygons | 角的性质与多边形
Angle facts support geometry, problems and reasoning and multi-step problems. You should quote the reason for every angle you find, since method marks are awarded for correct reasoning.
角的性质支撑几何、推理与多步问题。每求出一个角都应写出理由，因为过程分来自正确的推理表述。
Angles on a straight line add to 180°, and angles around a point add to 360°.
同一直线上的邻补角和为180°，围绕一点的所有角和为360°。
Angles on a Straight Line = 180° Angles Around a Point = 360°
直线上的角和 = 180° 围绕一点的角和 = 360°
The interior angles of any triangle add to 180°, which extends to the sum of interior angles of any polygon.
任意三角形的内角和为180°，并可推广到任意多边形的内角和。
Sum of Interior Angles = (n − 2) × 180°
内角和 = (n − 2) × 180°
The exterior angles of any convex polygon always add to 360°, no matter how many sides it has.
任意凸多边形的外角和恒为360°，与边数无关。
Sum of Exterior Angles = 360° (for any convex polygon)
外角和 = 360°（任意凸多边形）
In a regular polygon all interior angles are equal, so divide the interior angle sum by the number of sides.
正多边形各内角相等，因此把内角和除以边数。
One Interior Angle of a Regular Polygon = (n − 2) × 180° / n
正多边形一个内角 = (n − 2) × 180° / n
One exterior angle of a regular polygon is simply 360° divided by the number of sides, which is often the faster route to the interior angle.
正多边形的一个外角等于360°除以边数，这往往是求内角更快的途径。
One Exterior Angle of a Regular Polygon = 360° / n
正多边形一个外角 = 360° / n
Parallel line rules give you alternate, corresponding and co-interior angles, and these are frequently combined with triangle angle sums.
平行线性质给出内错角、同位角与同旁内角，常与三角形内角和结合使用。
4. Area and Perimeter of Common Shapes | 常见图形的面积与周长
These area formulas cover rectilinear shapes, triangles, and parallelograms and trapezia - must-knows for mensuration questions. Write the formula, substitute the values and keep units consistent.
以下面积公式涵盖直线图形、三角形、平行四边形和梯形——是计量问题的必会内容。先写公式、再代入数值，并保持单位一致。
The rectangle is the simplest case, and it also underpins the parallelogram and triangle formulas.
矩形是最简单的情形，同时也是平行四边形与三角形公式的基础。
Rectangle: A = l × w
矩形：A = 长 × 宽
A triangle is half of a parallelogram with the same base and perpendicular height, so the factor ½ appears.
三角形是同底等高平行四边形的一半，因此出现系数 ½。
Triangle: A = ½ × b × h
三角形：A = ½ × 底 × 高
A parallelogram has the same area as a rectangle with the same base and perpendicular height.
平行四边形的面积等于同底等高矩形的面积。
Parallelogram: A = b × h
平行四边形：A = 底 × 高
A trapezium has one pair of parallel sides, and the formula averages those two lengths before multiplying by the height.
梯形有一组平行边，公式先取两条平行边的平均值再乘以高。
Trapezium: A = ½ (a + b) × h
梯形：A = ½ (a + b) × h
Compound shapes should be split into rectangles, triangles or trapezia, and the separate areas added or subtracted.
组合图形应拆分为矩形、三角形或梯形，再把各部分面积相加或相减。
The circumference and area of a circle use π, and you should use the π button on your calculator unless the question says otherwise.
圆的周长与面积都要用 π，除非题目另有说明，应使用计算器上的 π 键。
Circumference: C = πd = 2πr Area: A = πr²
圆周长：C = πd = 2πr 圆面积：A = πr²
A semicircle is half a circle, so halve both the area and the arc length, and remember to add the straight diameter when finding the perimeter.
半圆是圆的一半，面积与弧长都要除以2；求周长时别忘了加上直径这条直边。
Semicircle: Area = ½πr² Perimeter = πr + 2r
半圆：面积 = ½πr² 周长 = πr + 2r
5. Circles: Arcs, Sectors, Segments and Circle Theorems | 圆：弧、扇形、弓形与圆定理
Arc and sector questions are common in Higher tier, and circle theorems appear in reasoning papers where each statement must be justified.
弧与扇形问题在Higher卷中很常见，圆定理则常出现在推理题中，每条结论都必须给出理由。
An arc is a fraction of the circumference, where the fraction is the angle at the centre divided by 360°.
弧是圆周的一部分，这个分数由圆心角除以360°得到。
Arc Length = θ/360 × 2πr
弧长 = θ/360 × 2πr
A sector is a fraction of the whole circle area, using the same angle fraction.
扇形是圆面积的一部分，所用分数与圆心角一致。
Sector Area = θ/360 × πr²
扇形面积 = θ/360 × πr²
A segment is the region between a chord and an arc, found by subtracting the triangle from the sector.
弓形是弦与弧之间的区域，用扇形面积减去三角形面积得到。
Segment Area = Sector Area − ½r² sin θ
弓形面积 = 扇形面积 − ½r² sin θ
The angle in a semicircle is always 90°, which is one of the most frequently tested circle theorems.
半圆所对的圆周角恒为90°，这是最常考的圆定理之一。
Angle in a Semicircle = 90°
半圆所对的圆周角 = 90°
The angle at the centre is twice the angle at the circumference when both stand on the same arc.
同弧所对的圆心角是圆周角的两倍。
Angle at Centre = 2 × Angle at Circumference
圆心角 = 2 × 圆周角
Angles in the same segment are equal, and opposite angles of a cyclic quadrilateral add to 180°.
同弧上的圆周角相等，圆内接四边形对角互补，和为180°。
es in the Same Segment are Equal Cyclic Quadrilateral: opposite angles = 180°
同弧圆周角相等 圆内接四边形：对角和 = 180°
A tangent meets a radius at 90°, and the alternate segment theorem links a tangent-chord angle with the angle in the opposite segment.
切线与半径垂直，交角为90°；弦切角定理把切线与弦的夹角与另一侧的圆周角联系起来。
Tangent ⊥ Radius Alternate Segment: tangent-chord angle = angle in opposite segment
切线 ⊥ 半径 弦切角 = 另一侧圆周角
6. Volume and Surface Area of 3D Shapes | 立体图形的体积与表面积
Volume and surface area questions reward careful identification of the shape and consistent use of units. Volume uses cubic units, while surface area uses square units.
体积与表面积题要求准确识别图形并保持单位一致。体积用立方单位，表面积用平方单位。
A cuboid volume is the product of its three dimensions, and a prism is the cross-sectional area multiplied by its length.
长方体体积是三边之积，棱柱体积是横截面积乘以长度。
Cuboid: V = l × w × h Prism: V = area of cross-section × length
长方体：V = 长 × 宽 × 高 棱柱：V = 横截面积 × 长度
A cylinder is a prism with a circular cross-section, so its volume combines πr² with the height.
圆柱是横截面为圆的棱柱，体积由 πr² 与高相乘得到。
πr²h Curved Surface Area = 2πrh Total Surface Area = 2πr² + 2πrh
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