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Cambridge IGCSE Mathematics (0580)Core & Extended · one-page formula sheet · ★ = not on the paper's formula list
★ not on the 0580 list of formulas — learn it. Ext = Extended tier only (Papers 2 and 4). Everything else is on the list printed on the paper.
Number
★Indices: \(a^m\times a^n=a^{m+n},\) \(a^m\div a^n=a^{m-n},\) \((a^m)^n=a^{mn},\) \(a^0=1,\) \(a^{-n}=\tfrac1{a^n}\)
★Fractional indices (Extended only): \(a^{\frac1n}=\sqrt[n]a,\) \(a^{\frac mn}=\big(\sqrt[n]a\big)^m\)
★Standard form: \(A\times10^n,\) \(1\le A<10,\ n\in\mathbb Z\)
★Sets: \(n(A\cup B)=n(A)+n(B)-n(A\cap B)\)
★Percentage change: \(\frac{\text{new}-\text{original}}{\text{original}}\times100\%\)
★Reverse percentage (Extended only): \(\text{original}=\frac{\text{new}}{1\pm\frac r{100}}\)
★Simple interest (rate \(r\)% a year, \(n\) years): \(I=\frac{Prn}{100}\)
★Compound interest: \(A=P\left(1+\tfrac r{100}\right)^n\)
★Growth and decay (Extended only): \(y=a\times k^n:\) \(k>1\text{ growth},\) \(0<k<1\text{ decay}\)
★Surds (Extended only): \(\sqrt{ab}=\sqrt a\sqrt b,\) \(\frac a{\sqrt b}=\frac{a\sqrt b}b,\) \(\frac1{c+\sqrt b}=\frac{c-\sqrt b}{c^2-b}\)
★Recurring decimal: \(x=0.\dot4\dot5\), so \(100x-x=45\) and \(x=\tfrac{45}{99}=\tfrac5{11}\) (Extended only)
★Bounds: a value to the nearest \(u\) lies in \([x-\tfrac u2,\ x+\tfrac u2)\)
★Bounds in calculations: for the largest \(a-b\) or \(a\div b\) use the upper bound of \(a\) with the lower bound of \(b\) (Extended only)
★Rates: \(\text{speed}=\frac{\text{distance}}{\text{time}},\) \(\text{density}=\frac{\text{mass}}{\text{volume}},\) \(\text{pressure}=\frac{\text{force}}{\text{area}}\)
Algebra & graphs
Quadratic formula, \(ax^2+bx+c=0\) (Extended only): \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
★Completing the square (Extended only): \(x^2+bx+c=\left(x+\tfrac b2\right)^2\) \({}-\tfrac{b^2}4+c\)
★Difference of two squares (Extended only): \(a^2-b^2=(a+b)(a-b)\)
★Linear sequence, \(n\)th term: \(u_n=a+(n-1)d\)
★Exponential sequence (Extended only): \(u_n=ar^{n-1}\)
★Proportion (Extended only): \(y\propto x^n\Rightarrow y\) \({}=kx^n;\) \(y\propto\tfrac1{x^n}\Rightarrow y\) \({}=\tfrac k{x^n}\)
★Functions (Extended only): \(fg(x)=f\big(g(x)\big);\) \(f^{-1}\text{: swap }x,y\text{ and rearrange}\)
★Inequalities: multiplying or dividing by a negative number reverses the sign
★Straight line: \(y=mx+c\), gradient \(m\), \(y\)-intercept \(c\); vertical line \(x=k\)
★Gradient from two points (Extended only): \(m=\frac{y_2-y_1}{x_2-x_1}\)
★Midpoint; length (Extended only): \(\left(\tfrac{x_1+x_2}2,\tfrac{y_1+y_2}2\right),\) \(\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
★Parallel lines: \(m_1=m_2\)
★Perpendicular lines (Extended only): \(m_1\times m_2=-1\)
★Differentiation (Extended only): \(y=ax^n\Rightarrow\frac{dy}{dx}\) \({}=anx^{n-1}\)
★Turning points: \(\frac{dy}{dx}=0\); the gradient of the curve at \(x=a\) is the value of \(\frac{dy}{dx}\) at \(a\) (Extended only)
Geometry & mensuration
★Angles: on a straight line \(180^\circ\); around a point \(360^\circ\); in a triangle \(180^\circ\)
★Polygons (\(n\) sides): \(\text{interior sum}=(n-2)\times180^\circ,\) \(\text{exterior angle (regular)}\) \({}=\tfrac{360^\circ}n\)
★Circle theorems: angle in a semicircle = 90°; tangent ⟂ radius
★More circle theorems: angle at centre = 2 × angle at circumference; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to 180°; alternate segment theorem; tangents from a point are equal; the perpendicular bisector of a chord passes through the centre (Extended only)
Triangle: \(A=\tfrac12bh\)
★Parallelogram; trapezium: \(A=bh;\) \(A=\tfrac12(a+b)h\)
Circle: \(A=\pi r^2,\) \(C=2\pi r\)
★Arc; sector (angle \(\theta\)): \(\text{arc}=\frac{\theta}{360}\times2\pi r,\) \(\text{sector}=\frac{\theta}{360}\times\pi r^2\)
Curved surface: cylinder; cone (slant \(l\)): \(A=2\pi rh;\) \(A=\pi rl\)
Sphere, surface area: \(A=4\pi r^2\)
Prism; pyramid: \(V=Al;\) \(V=\tfrac13Ah\)
Cylinder; cone: \(V=\pi r^2h;\) \(V=\tfrac13\pi r^2h\)
Sphere, volume: \(V=\tfrac43\pi r^3\)
★Similar shapes, scale factor \(k\): lengths \(\times k\)
★Similar shapes: areas \(\times k^2\), volumes \(\times k^3\) (Extended only)
Trigonometry & vectors
★Pythagoras: \(a^2+b^2=c^2\)
★SOHCAHTOA: \(\sin\theta=\frac{\text{opp}}{\text{hyp}},\) \(\cos\theta=\frac{\text{adj}}{\text{hyp}},\) \(\tan\theta=\frac{\text{opp}}{\text{adj}}\)
★Bearings: measured clockwise from north, written with 3 figures
★Exact values (Extended only): \(\sin30^\circ=\tfrac12,\) \(\cos60^\circ=\tfrac12,\) \(\tan45^\circ=1,\) \(\sin60^\circ=\tfrac{\sqrt3}2,\) \(\sin45^\circ=\tfrac{\sqrt2}2\)
Sine rule (Extended only): \(\frac a{\sin A}=\frac b{\sin B}\) \({}=\frac c{\sin C}\)
Cosine rule (Extended only): \(a^2=b^2+c^2-2bc\cos A\)
Area of a triangle (Extended only): \(\tfrac12ab\sin C\)
★Obtuse angles (Extended only): \(\sin(180^\circ-x)=\sin x,\) \(\cos(180^\circ-x)=-\cos x\)
★Vectors: add and subtract column vectors component by component; \(k\binom xy=\binom{kx}{ky}\) (Extended only)
★Magnitude; position vectors (Extended only): \(\left|\binom xy\right|=\sqrt{x^2+y^2},\) \(\overrightarrow{AB}=\mathbf b-\mathbf a\)
Probability & statistics
★Probability: \(P(\text{not }A)=1-P(A);\) \(\text{expected frequency}\) \({}=n\times P(A)\)
★Relative frequency: \(\frac{\text{frequency of the outcome}}{\text{number of trials}}\)
★Mutually exclusive: \(P(A\text{ or }B)=P(A)+P(B)\)
★Independent: \(P(A\text{ and }B)=P(A)\times P(B)\)
★Conditional (Extended only): \(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\)
★Tree diagrams: multiply along branches, add the outcomes you want
★Without replacement: the second-branch probabilities change (one fewer item) (Extended only)
★Mean from a frequency table: \(\bar x=\frac{\sum fx}{\sum f}\)
★Estimated mean, grouped data (class midpoints \(m\)) (Extended only): \(\bar x\approx\frac{\sum fm}{\sum f}\)
★Median position (\(n\) values in order): \(\tfrac{n+1}2\text{th value}\)
★Range: \(\text{largest}-\text{smallest}\)
★Interquartile range (Extended only): \(Q_3-Q_1\)
★Histogram (Extended only): \(\text{frequency density}=\frac{\text{frequency}}{\text{class width}}\)