A formulae sheet is printed in each paper, so check it before the exam and learn what it does and does not give you. This sheet covers the formulas and facts that are best learned by heart, grouped by topic. Items marked Higher are for the Higher tier only, and a calculator is allowed in both papers.
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\text{percentage change}=\dfrac{\text{change}}{\text{original}}\times100\) | Change divided by the starting value, as a percentage | Increase, decrease, profit and loss |
| \(\text{new}=\text{original}\times\left(1\pm\dfrac{p}{100}\right)\) | \(p\) is the percentage; use the multiplier for quick work | Percentage change in one step |
| \(A=P\left(1+\dfrac r{100}\right)^n\) | Compound interest: \(P\) start, \(r\)% per period, \(n\) periods | Savings, growth, depreciation (use minus for decrease) |
| \(\text{original}=\dfrac{\text{new}}{\text{multiplier}}\) | Reverse percentage | Finding the price before a change |
| \(a^m\times a^n=a^{m+n},\quad \dfrac{a^m}{a^n}=a^{m-n}\) | Index laws for a common base | Simplifying powers |
| \((a^m)^n=a^{mn},\quad a^0=1,\quad a^{-n}=\dfrac1{a^n}\) | Further index laws | Simplifying, standard form |
| \(a^{1/n}=\sqrt[n]{a},\quad a^{m/n}=(\sqrt[n]{a})^m\) | Fractional indices | Evaluating roots and powers together |
| \(a\times10^n,\ 1\le a<10\) | Standard form, \(n\) an integer | Very large and very small numbers |
| \(\sqrt a\times\sqrt b=\sqrt{ab},\quad \dfrac{1}{\sqrt a}=\dfrac{\sqrt a}{a}\) | Surd rules (Higher) | Simplifying and rationalising |
| \(\text{speed}=\dfrac{\text{distance}}{\text{time}}\) | Metres per second, or kilometres per hour | Journeys and rates |
| \(\text{density}=\dfrac{\text{mass}}{\text{volume}}\) | For example g/cm\(^3\) or kg/m\(^3\) | Mass and volume problems |
| \(\text{pressure}=\dfrac{\text{force}}{\text{area}}\) | Force in newtons, area in m\(^2\), pressure in pascals | Applied rate questions |
| \(\text{LB}=x-\tfrac12u,\quad \text{UB}=x+\tfrac12u\) | Bounds for a value \(x\) rounded to unit \(u\) (combining bounds is Higher) | Accuracy and limits |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \((a+b)^2=a^2+2ab+b^2\) | Square of a sum | Expanding and completing the square |
| \((a-b)^2=a^2-2ab+b^2\) | Square of a difference | Expanding |
| \(a^2-b^2=(a-b)(a+b)\) | Difference of two squares | Factorising, quick arithmetic |
| \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\) | Solutions of \(ax^2+bx+c=0\) (Higher) | Quadratics that do not factorise |
| \((x+p)^2+q\) | Completed-square form of a quadratic with vertex at \((-p,q)\) (Higher) | Turning points and solving |
| \(y=mx+c\) | Straight line: gradient \(m\), \(y\)-intercept \(c\) | Drawing and writing line equations |
| \(m=\dfrac{y_2-y_1}{x_2-x_1}\) | Gradient from two points | Finding line equations |
| \(m_1m_2=-1\) | Gradients of perpendicular lines (Higher) | Perpendicular lines and normals |
| \(\left(\dfrac{x_1+x_2}2,\dfrac{y_1+y_2}2\right)\) | Midpoint of two points | 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 a term or the rule |
| \(u_n=an^2+bn+c\) | General quadratic sequence; second difference is \(2a\) (Higher) | Quadratic sequences |
| \(\dfrac{dy}{dx}=nax^{n-1}\ \text{for}\ y=ax^n\) | Differentiation of a power (Higher) | Gradients and turning points |
| \(\text{gradient of tangent}=\dfrac{dy}{dx}\) | Rate of change at a point (Higher) | Tangents and optimisation |
| \(y=kx,\quad y=\dfrac kx,\quad y=kx^2\) | Direct, inverse and square proportion with constant \(k\) | Proportion problems |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\text{angle sum of a triangle}=180^\circ\) | Interior angles of a triangle | Finding missing angles |
| \(\text{interior angle sum}=(n-2)\times180^\circ\) | Polygon with \(n\) sides | Regular and irregular polygons |
| \(\text{exterior angle}=\dfrac{360^\circ}{n}\) | Regular polygon with \(n\) sides | Finding the number of sides |
| \(A=\tfrac12bh\) | Triangle: base \(b\), perpendicular height \(h\) | Areas of triangles |
| \(A=bh\) | Parallelogram: base \(b\), perpendicular height \(h\) | Areas of parallelograms |
| \(A=\pi r^2,\quad C=2\pi r=\pi d\) | Circle area and circumference, radius \(r\) | Circle problems |
| \(\text{arc}=\dfrac\theta{360}\times2\pi r\) | Arc length for angle \(\theta\) in degrees | Sectors and arcs |
| \(\text{sector area}=\dfrac\theta{360}\times\pi r^2\) | Area of a sector with angle \(\theta\) | Sectors |
| \(V=\text{area of cross-section}\times\text{length}\) | Volume of a prism | Prisms and cylinders |
| \(V=\pi r^2h,\quad \text{CSA}=2\pi rh\) | Cylinder volume and curved surface area | Cylinders |
| \(a^2+b^2=c^2\) | Pythagoras; \(c\) is the hypotenuse | Right-angled triangles |
| \(\text{area scale factor}=k^2,\quad \text{volume scale factor}=k^3\) | \(k\) is the length scale factor of similar shapes (Higher) | Similar shapes and solids |
| \(\text{angle at centre}=2\times\text{angle at circumference}\) | Circle theorem on the same arc (Higher) | Circle angle problems |
| \(\text{opposite angles of a cyclic quadrilateral sum to }180^\circ\) | Circle theorem (Higher) | Circle angle problems |
| \(\text{angle between tangent and radius}=90^\circ\) | Circle theorem | Tangent problems |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}},\ \cos\theta=\dfrac{\text{adj}}{\text{hyp}},\ \tan\theta=\dfrac{\text{opp}}{\text{adj}}\) | Ratios in a right-angled triangle | Finding sides and angles |
| \(\tan\theta=\dfrac{\sin\theta}{\cos\theta}\) | Link between the three ratios | Simplifying, solving equations |
| \(\sin^2\theta+\cos^2\theta=1\) | Identity true for every angle | Proof and simplification (Higher) |
| \(\dfrac a{\sin A}=\dfrac b{\sin B}=\dfrac c{\sin C}\) | Sine rule (Higher); side \(a\) is opposite angle \(A\) | Non-right-angled triangles |
| \(a^2=b^2+c^2-2bc\cos A\) | Cosine rule for a side (Higher) | Two sides and the angle between them |
| \(\cos A=\dfrac{b^2+c^2-a^2}{2bc}\) | Cosine rule for an angle (Higher) | Three sides known |
| \(\text{Area}=\tfrac12ab\sin C\) | Area of a triangle from two sides and the included angle (Higher) | Triangles without a known height |
| \(\sin30^\circ=\tfrac12,\ \cos30^\circ=\tfrac{\sqrt3}2,\ \tan30^\circ=\tfrac1{\sqrt3}\) | Exact values (Higher) | Exact trigonometry questions |
| \(\sin45^\circ=\cos45^\circ=\tfrac{\sqrt2}2,\ \tan45^\circ=1\) | Exact values (Higher) | Exact trigonometry questions |
| \(\sin60^\circ=\tfrac{\sqrt3}2,\ \cos60^\circ=\tfrac12,\ \tan60^\circ=\sqrt3\) | Exact values (Higher) | Exact trigonometry questions |
| Formula | Meaning, symbols and units | When you use it |
|---|
| \(\text{mean}=\dfrac{\sum fx}{\sum f}\) | Mean from a frequency table; \(f\) frequency, \(x\) value or midpoint | Grouped and ungrouped data |
| \(Q_2=\text{median},\quad \text{IQR}=Q_3-Q_1\) | Spread of the middle half of the data | Comparing distributions |
| \(P(\text{event})=\dfrac{\text{favourable outcomes}}{\text{total outcomes}}\) | Equally likely outcomes | Basic probability |
| \(P(A\ \text{or}\ B)=P(A)+P(B)\) | Mutually exclusive events only | Either-or questions |
| \(P(A\ \text{and}\ B)=P(A)\times P(B)\) | Independent events only | Tree diagrams |
| \(P(\text{not }A)=1-P(A)\) | Complement rule | "At least one" questions |
| \(\text{expected number}=n\times P(\text{event})\) | \(n\) trials | Predicting frequencies |
| \(\text{frequency density}=\dfrac{\text{frequency}}{\text{class width}}\) | Height of a histogram bar (Higher) | Drawing and reading histograms |