A list of formulas is provided in each paper, so check what it contains before the exam and learn what it does not give you. This sheet covers the formulas and facts that are best learned by heart, grouped by topic. Items marked Extended are for the Extended level only. Two of the four papers do not allow a calculator, so exact values matter.
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\text{percentage}=\dfrac{\text{part}}{\text{whole}}\times100\) | Part as a percentage of the whole | Finding percentages and proportions |
| \(\text{percentage change}=\dfrac{\text{change}}{\text{original}}\times100\) | Change divided by the starting value | Increase, decrease, profit and loss |
| \(\text{new}=\text{original}\times\left(1\pm\dfrac p{100}\right)\) | Multiplier method with percentage \(p\) | Quick percentage change |
| \(\text{original}=\dfrac{\text{new}}{\text{multiplier}}\) | Reverse percentage | Finding the amount before a change |
| \(I=\dfrac{PRT}{100}\) | Simple interest: principal \(P\), rate \(R\)% per year, time \(T\) years | Interest questions |
| \(A=P\left(1+\dfrac R{100}\right)^n\) | Compound interest over \(n\) years | Savings, growth and depreciation |
| \(a^m\times a^n=a^{m+n},\quad \dfrac{a^m}{a^n}=a^{m-n}\) | Index laws | Simplifying powers |
| \((a^m)^n=a^{mn},\quad a^0=1,\quad a^{-n}=\dfrac1{a^n}\) | Further index laws | Simplifying and evaluating |
| \(a^{1/n}=\sqrt[n]{a}\) | Fractional index means a root (Extended) | Evaluating roots |
| \(a\times10^n,\ 1\le a<10\) | Standard form with integer \(n\) | Large and small numbers |
| \(\sqrt a\times\sqrt b=\sqrt{ab}\) | Surd multiplication (Extended) | Simplifying surds |
| \(\text{ratio }a:b\Rightarrow\text{share }=\dfrac a{a+b}\text{ of the total}\) | Dividing a quantity in a ratio | Sharing problems |
| \(\text{speed}=\dfrac{\text{distance}}{\text{time}}\) | Metres per second or kilometres per hour | Journey problems |
| \(\text{density}=\dfrac{\text{mass}}{\text{volume}}\) | For example g/cm\(^3\) | Mass and volume problems |
| \(\text{average speed}=\dfrac{\text{total distance}}{\text{total time}}\) | Not the average of the speeds | Multi-stage journeys |
| \(\text{LB}=x-\tfrac12u,\quad \text{UB}=x+\tfrac12u\) | Bounds for a value \(x\) rounded to unit \(u\) (Extended) | Limits of accuracy |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\) | Solutions of \(ax^2+bx+c=0\) (Extended) | Quadratics that do not factorise |
| \(a^2-b^2=(a-b)(a+b)\) | Difference of two squares | Factorising |
| \((a+b)^2=a^2+2ab+b^2\) | Square of a sum | Expanding |
| \(y=mx+c\) | Gradient \(m\), \(y\)-intercept \(c\) | Equation of a line |
| \(m=\dfrac{y_2-y_1}{x_2-x_1}\) | Gradient between two points | Equation of a line |
| \(m_1m_2=-1\) | Gradients of perpendicular lines (Extended) | Perpendicular lines |
| \(\left(\dfrac{x_1+x_2}2,\dfrac{y_1+y_2}2\right)\) | Midpoint of a line segment | Coordinate geometry |
| \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) | Length of a line segment | Distance between points |
| \(u_n=a+(n-1)d\) | \(n\)th term of an arithmetic sequence | Finding terms and rules |
| \(u_n=ar^{n-1}\) | \(n\)th term of a geometric sequence (Extended) | Growth sequences |
| \(y\propto x\Rightarrow y=kx\) | Direct proportion with constant \(k\) | Proportion questions |
| \(y\propto\dfrac1x\Rightarrow y=\dfrac kx\) | Inverse proportion | Proportion questions |
| \(y\propto x^2\Rightarrow y=kx^2\) | Square proportion (Extended) | Area-style relationships |
| \(\dfrac{dy}{dx}=nax^{n-1}\ \text{for}\ y=ax^n\) | Differentiating a power (Extended) | Gradients and turning points |
| \(\text{distance}=\text{area under a speed-time graph}\) | Speed in m/s, time in s (Extended) | Motion graphs |
| \(\text{acceleration}=\text{gradient of a speed-time graph}\) | Change in speed over time (Extended) | Motion graphs |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\text{angle sum of a triangle}=180^\circ\) | Interior angles | Missing angles |
| \(\text{interior angle sum}=(n-2)\times180^\circ\) | Polygon with \(n\) sides | Polygon problems |
| \(\text{exterior angle}=\dfrac{360^\circ}n\) | Regular polygon | Finding the number of sides |
| \(A=\tfrac12bh\) | Triangle: base and perpendicular height | Area |
| \(A=\tfrac12(a+b)h\) | Trapezium with parallel sides \(a,b\) and height \(h\) | Area |
| \(A=\pi r^2,\quad C=2\pi r\) | Circle area and circumference | Circle problems |
| \(\text{arc}=\dfrac\theta{360}\times2\pi r\) | Arc length for angle \(\theta\) | Sectors |
| \(\text{sector area}=\dfrac\theta{360}\times\pi r^2\) | Area of a sector | Sectors |
| \(V=\text{area of cross-section}\times\text{length}\) | Volume of a prism | Prisms and cylinders |
| \(V=\pi r^2h\) | Cylinder volume | Cylinders |
| \(a^2+b^2=c^2\) | Pythagoras, hypotenuse \(c\) | Right-angled triangles |
| \(\text{area factor}=k^2,\quad \text{volume factor}=k^3\) | \(k\) is the length scale factor of similar shapes (Extended) | Similarity |
| \(\text{angle at centre}=2\times\text{angle at circumference}\) | Circle theorem (Extended) | Circle angle questions |
| \(\text{angle in a semicircle}=90^\circ\) | Circle theorem | Circle angle questions |
| \(\text{tangent}\perp\text{radius}\) | Circle theorem: they meet at \(90^\circ\) | Tangent problems |
| \(\text{opposite angles of a cyclic quadrilateral}=180^\circ\text{ together}\) | Circle theorem (Extended) | Circle angle questions |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}},\ \cos\theta=\dfrac{\text{adj}}{\text{hyp}},\ \tan\theta=\dfrac{\text{opp}}{\text{adj}}\) | Right-angled triangle ratios | Finding sides and angles |
| \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) | Link between the ratios | Simplifying (Extended) |
| \(\dfrac a{\sin A}=\dfrac b{\sin B}=\dfrac c{\sin C}\) | Sine rule (Extended) | Non-right-angled triangles |
| \(a^2=b^2+c^2-2bc\cos A\) | Cosine rule (Extended) | Two sides and included angle |
| \(\text{Area}=\tfrac12ab\sin C\) | Triangle area from two sides and the included angle (Extended) | Triangles without a known height |
| \(\sin30^\circ=\tfrac12,\ \cos60^\circ=\tfrac12\) | Exact values (Extended) | Non-calculator paper |
| \(\sin45^\circ=\cos45^\circ=\tfrac{\sqrt2}2,\ \tan45^\circ=1\) | Exact values (Extended) | Non-calculator paper |
| \(\cos30^\circ=\sin60^\circ=\tfrac{\sqrt3}2,\ \tan60^\circ=\sqrt3\) | Exact values (Extended) | Non-calculator paper |
| \(\text{bearing}=\text{clockwise angle from north, three figures}\) | For example \(045^\circ\) | Navigation problems |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\text{mean}=\dfrac{\sum fx}{\sum f}\) | Mean from a frequency table | Grouped and ungrouped data |
| \(\text{IQR}=Q_3-Q_1\) | Spread of the middle half | Comparing data (Extended) |
| \(P(\text{event})=\dfrac{\text{favourable}}{\text{total}}\) | Equally likely outcomes | Basic probability |
| \(P(A\text{ or }B)=P(A)+P(B)\) | Mutually exclusive events only | Either-or |
| \(P(A\text{ and }B)=P(A)\times P(B)\) | Independent events only | Tree diagrams |
| \(P(\text{not }A)=1-P(A)\) | Complement rule | "At least one" questions |
| \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\) | Number of elements in a union of sets | Venn diagrams |
| \(\text{frequency density}=\dfrac{\text{frequency}}{\text{class width}}\) | Height of a histogram bar (Extended) | Histograms |