GCSE Maths Formulas

\( d=\sqrt{a^2+b^2+c^2} \)

\( |x| = x \text{ if } x \ge 0,\; -x \text{ if } x < 0 \)

\( \frac{A}{B}=\frac{C}{D}\;(BD\ne0)\;\Rightarrow\; AD=BC \)

\( \angle\text{ between tangent and chord }=\angle\text{ in opposite arc} \)

\( \angle AOB = 2\,\angle ACB \)

\( \angle\text{ in a semicircle }=90^{\circ} \)

\( A+B+C+D=360^{\circ} \)

\( A+B+C=180^{\circ} \)

\( \text{Angles subtended by the same chord are equal} \)

\( L = \frac{\theta}{360^{\circ}}\, 2\pi r \)

\( A = \frac{\theta}{360^{\circ}}\, \pi r^2 \)

\( A = \tfrac{1}{2}(a+b)h \)

\( A = \pi r^2 \)

\( A=b\,h \)

\( A=\ell\,w \)

\( A=\tfrac{1}{2} d_1 d_2 \)

\( A=\tfrac{1}{2}\,\big|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\big| \)

\( A = \tfrac{1}{2}ab\sin C \)

\( A = \tfrac{1}{2} b h \)

\( A=\tfrac{1}{2}\,a\,P \)

\( A=\tfrac{n}{2}\,r^{2}\sin\!\left(\tfrac{2\pi}{n}\right) \)

\( 1\,\text{m}^2=10^4\,\text{cm}^2,\quad 1\,\text{m}^3=10^6\,\text{cm}^3 \)

\( u_n = a + (n-1)d \)

\( P(\text{at least one success})=1-(1-p)^{n} \)

\( \text{Average rate on }[x_1,x_2]=\frac{f(x_2)-f(x_1)}{x_2-x_1} \)

\( \text{Average speed}=\tfrac{\text{total distance}}{\text{total time}} \)

\( \text{Bearings are measured clockwise from North, as 3-digit angles} \)

\( (a+b)^2=a^2+2ab+b^2,\qquad (a-b)^2=a^2-2ab+b^2 \)

\( r\in[ L_r, U_r ]:\ C=2\pi r\Rightarrow [\,2\pi L_r,\ 2\pi U_r\,];\ A=\pi r^2\Rightarrow [\,\pi L_r^2,\ \pi U_r^2\,] \)

\( x \text{ rounded to unit } u \;\Rightarrow\; x\in\big[\,x_{\text{stated}}-\tfrac{u}{2},\;x_{\text{stated}}+\tfrac{u}{2}\,\big) \)

\( L_{x^n}=(L_x)^n,\ U_{x^n}=(U_x)^n;\quad L_{\sqrt[n]{x}}=\sqrt[n]{L_x},\ U_{\sqrt[n]{x}}=\sqrt[n]{U_x} \)

\( L_{xy}\approx L_x L_y,\;U_{xy}\approx U_x U_y;\quad L_{x/y}\approx \tfrac{L_x}{U_y},\;U_{x/y}\approx \tfrac{U_x}{L_y} \)

\( P=2(\ell+w)\Rightarrow [\,2(L_\ell+L_w),\ 2(U_\ell+U_w)\,];\quad A=\ell w\Rightarrow [\,L_\ell L_w,\ U_\ell U_w\,] \)

\( L_{x+y}=L_x+L_y,\ U_{x+y}=U_x+U_y;\quad L_{x-y}=L_x-U_y,\ U_{x-y}=U_x-L_y \)

\( \text{Chords at equal distances from the centre are equal (and converse)} \)

\( (x-h)^2+(y-k)^2=r^2 \)

\( x^2+y^2+gx+fy+c=0\;\Rightarrow\; \text{centre }(-\tfrac{g}{2},-\tfrac{f}{2}),\ r=\sqrt{\left(\tfrac{g}{2}\right)^2+\left(\tfrac{f}{2}\right)^2-c} \)

\( C = 2\pi r \;\;\text{ or }\; C = \pi d \)

\( \binom{n}{r}=\tfrac{n!}{r!(n-r)!} \)

\( \bar{x}=\tfrac{n_1\bar{x}_1+n_2\bar{x}_2}{n_1+n_2} \)

\( x^2+bx+c=\left(x+\tfrac{b}{2}\right)^2-\left(\tfrac{b^2}{4}-c\right) \)

\( A=P\left(1+\tfrac{r}{100}\right)^{n} \)

\( \text{Speed} = \frac{\text{Distance}}{\text{Time}},\quad \text{Density} = \frac{\text{Mass}}{\text{Volume}},\quad \text{Pressure} = \frac{\text{Force}}{\text{Area}} \)

\( P(A\mid B)=\tfrac{P(A\cap B)}{P(B)}\;(P(B)>0) \)

\( y\propto x\Rightarrow y=kx\;(k=\tfrac{y}{x});\qquad y\propto \tfrac{1}{x}\Rightarrow y=\tfrac{k}{x}\;(k=xy) \)

\( a^2=b^2+c^2-2bc\cos A \)

\( A+C=180^{\circ},\quad B+D=180^{\circ} \)

\( \rho=\tfrac{m}{V},\quad m=\rho V,\quad V=\tfrac{m}{\rho} \)

\( d=\sqrt{\ell^2+w^2} \)

\( a^2-b^2=(a-b)(a+b) \)

\( y\propto x \;\Rightarrow\; y=kx \)

\( y\propto x^{k}\;\Rightarrow\; y=kx^{k} \)

\( \Delta = b^2 - 4ac \)

\( d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} \)

\( s=\tfrac{d}{t},\qquad d=s\,t,\qquad t=\tfrac{d}{s} \)

\( \text{Speed}=\text{gradient} = \frac{\Delta\text{distance}}{\Delta\text{time}} \)

\( (x,y)\mapsto (\,h+k(x-h),\ k(y-k)+k\,) \)

\( \text{Equal chords} \iff \text{equal arcs} \iff \text{equal angles at centre} \)

\( PA=PB \)

\( x^2+y^2=r^2 \)

\( y = mx + c \)

\( y-y_1=m(x-x_1) \)

\( \text{Abs. error}=|x-\hat{x}|,\;\text{Rel. error}=\tfrac{|x-\hat{x}|}{|x|},\;\%\text{ error}=\tfrac{|x-\hat{x}|}{|x|}\times100\% \)

\( \text{Median}=L+\frac{\tfrac{N}{2}-\text{CF}}{f}\times w \)

\( Q_k\ \text{or}\ P_p= L+\frac{kN-\text{CF}}{f}\times w \)

\( E= n\,p \)

\( E(X)=\sum x\,P(X=x) \)

\( y=k\,a^{x}\;(a>0, a\ne1)\;\text{has asymptote }y=0 \)

\( \text{Exterior angle}=\frac{360^{\circ}}{n} \)

\( \text{Exterior angle}=\text{sum of opposite interior angles} \)

\( n!=n\times(n-1)\times\cdots\times2\times1,\;0!=1 \)

\( x^2+bx+c=(x+p)(x+q),\;p+q=b,\;pq=c \)

\( a^{1/n}=\sqrt[n]{a},\qquad a^{m/n}=\sqrt[n]{a^{m}}\;(a>0) \)

\( \text{Frequency Density}=\frac{\text{Frequency}}{\text{Class Width}} \)

\( u_n = ar^{\,n-1} \)

\( m = \frac{y_2 - y_1}{x_2 - x_1} \)

\( y=-f(x) \;\text{(reflect in x-axis)},\quad y=f(-x) \;\text{(reflect in y-axis)} \)

\( y=f(x)+a \;\text{(up by a)},\quad y=f(x-a) \;\text{(right by a)} \)

\( y=af(x) \;\text{(stretch scale }a\text{ in y)},\quad y=f(ax) \;\text{(scale }\tfrac{1}{a}\text{ in x)} \)

\( ab=\operatorname{HCF}(a,b)\times\operatorname{LCM}(a,b) \)

\( y\text{-intercept}:(0,c),\quad x\text{-intercept}:(-\tfrac{c}{m},0)\ (m\ne0) \)

\( IQR=Q_3-Q_1 \)

\( (AP)(PB)=(CP)(PD) \)

\( y\propto \tfrac{1}{x} \;\Rightarrow\; y=\tfrac{k}{x} \)

\( y\propto \tfrac{1}{x^{k}}\;\Rightarrow\; y=\tfrac{k}{x^{k}} \)

\( a^m a^n = a^{m+n},\quad \frac{a^m}{a^n}=a^{m-n},\quad (a^m)^n=a^{mn},\quad a^{-n}=\frac{1}{a^n},\quad a^0=1 \)

\( \frac{\text{drawing}}{\text{actual}}=\frac{1}{k},\qquad \text{actual}=k\times\text{drawing} \)

\( \bar{x}=\frac{\sum f x}{\sum f} \)

\( 1\,\text{cm}^3=1\,\text{mL},\quad 1000\,\text{cm}^3=1\,\text{L} \)

\( x_{\text{mid}}=\tfrac{\text{lower boundary}+\text{upper boundary}}{2} \)

\( \left( \tfrac{x_1+x_2}{2},\; \tfrac{y_1+y_2}{2} \right) \)

\( a\,\tfrac{b}{c}=\tfrac{ac+b}{c};\quad \tfrac{p}{q}=\left\lfloor\tfrac{p}{q}\right\rfloor\,\tfrac{p- q\left\lfloor\tfrac{p}{q}\right\rfloor}{q} \)

\( P(A\cap B)=0,\quad P(A\cup B)=P(A)+P(B) \)

\( \;\vec{a}\;\text{and}\;\vec{b}\;\text{parallel} \iff \vec{a}=k\,\vec{b}\;(k\in\mathbb{R}) \)

\( \text{Corresponding }\angle\text{s equal},\; \text{Alternate }\angle\text{s equal},\; \text{Co-interior }\angle\text{s sum }180^{\circ} \)

\( m_1=m_2 \)

\( \%\,\text{change}=\frac{\text{new}-\text{old}}{\text{old}}\times100\% \)

\( \text{Increase by }r\%:\;\times\,(1+\tfrac{r}{100});\qquad \text{Decrease by }r\%:\;\times\,(1-\tfrac{r}{100}) \)

\( P' = kP,\qquad A' = k^{2}A \)

\( P=\frac{\theta}{360^{\circ}}\,2\pi r+2r \)

\( P=2(\ell+w) \)

\( P=4s \)

\( _{n}P_{r}=\tfrac{n!}{(n-r)!} \)

\( m_{\perp}=-\tfrac{1}{m},\;\text{passes through }\left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2}\right) \)

\( \text{A radius/diameter perpendicular to a chord bisects the chord} \)

\( m_1\,m_2=-1 \)

\( p=\tfrac{F}{A},\quad F=pA,\quad A=\tfrac{F}{p} \)

\( P(A\cup B)=P(A)+P(B)-P(A\cap B) \)

\( P(A')=1-P(A) \)

\( P(A)=\tfrac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}} \)

\( P(A\cap B)=P(A)\,P(B) \)

\( a^2 + b^2 = c^2 \)

\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

\( u_n=an^2+bn+c,\quad a=\tfrac{1}{2}\,\Delta^2,\quad b=\Delta u_1-3a,\quad c=u_1-a-b \)

\( x_{\text{vertex}}= -\tfrac{b}{2a},\quad y_{\text{vertex}}= f\!\left(-\tfrac{b}{2a}\right),\quad \text{axis}:\;x=-\tfrac{b}{2a} \)

\( y=a(x-h)^2+k \)

\( y=a(x-r_1)(x-r_2)\;\Rightarrow\; r_1+r_2=-\tfrac{b}{a},\; r_1r_2=\tfrac{c}{a} \)

\( Q_1\text{ near }\tfrac{N+1}{4},\quad \text{Median near }\tfrac{N+1}{2},\quad Q_3\text{ near }\tfrac{3(N+1)}{4} \)

\( \text{radians}=\text{degrees}\times\tfrac{\pi}{180},\qquad \text{degrees}=\text{radians}\times\tfrac{180}{\pi} \)

\( \frac{1}{\sqrt{a}+\sqrt{b}}\cdot\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{\sqrt{a}-\sqrt{b}}{a-b} \)

\( \frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a}\;(a>0) \)

\( \frac{1}{a+\sqrt{b}}\times\frac{a-\sqrt{b}}{a-\sqrt{b}}=\frac{a-\sqrt{b}}{a^{2}-b} \)

\( y=\tfrac{k}{x}\;(k\ne0)\;\text{ has asymptotes }x=0,\;y=0 \)

\( d=\sqrt{\left(\tfrac{P}{2}\right)^2-2A} \)

\( \text{Exterior}=\tfrac{360^{\circ}}{n},\qquad \text{Interior}=180^{\circ}-\tfrac{360^{\circ}}{n} \)

\( x=\text{0.nonrep}\,\overline{\text{rep}}\;\Rightarrow\; x=\tfrac{\text{all digits}-\text{nonrep}}{\underbrace{99\dots 9}_{\lvert\text{rep}\rvert}\underbrace{00\dots 0}_{\lvert\text{nonrep}\rvert}} \)

\( x=0.\overline{a_1a_2\dots a_k}\;\Rightarrow\;x=\tfrac{a_1a_2\dots a_k}{\underbrace{99\dots 9}_{k\text{ nines}}} \)

\( \text{Original}=\frac{\text{Final}}{1+\tfrac{r}{100}} \)

\( \sin\theta=\tfrac{\text{opp}}{\text{hyp}},\quad \cos\theta=\tfrac{\text{adj}}{\text{hyp}},\quad \tan\theta=\tfrac{\text{opp}}{\text{adj}} \)

\( P=\left(\tfrac{mx_2+nx_1}{m+n},\;\tfrac{my_2+ny_1}{m+n}\right) \)

\( L=r\theta,\qquad A=\tfrac{1}{2}r^2\theta \)

\( \text{Share for }a:b=\frac{a}{a+b}\times\text{total} \)

\( L'=kL,\quad A'=k^2A,\quad V'=k^3V \)

\( I=\tfrac{P r t}{100},\qquad A=P+I \)

\( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \)

\( ax+b=c\;\Rightarrow\;x=\tfrac{c-b}{a}\;(a\ne0) \)

\( \sin45^{\circ}=\cos45^{\circ}=\tfrac{\sqrt{2}}{2};\;\sin30^{\circ}=\tfrac{1}{2},\;\cos30^{\circ}=\tfrac{\sqrt{3}}{2};\;\sin60^{\circ}=\tfrac{\sqrt{3}}{2},\;\cos60^{\circ}=\tfrac{1}{2} \)

\( \text{m/s}=\tfrac{\text{km/h}}{3.6},\qquad \text{km/h}=3.6\times\text{m/s} \)

\( s=\tfrac{d}{t},\quad d=st,\quad t=\tfrac{d}{s} \)

\( N \times 10^{n}\;\text{with}\;1\le N<10,\;n\in\mathbb{Z} \)

\( (a\times10^{m})(b\times10^{n})=(ab)\times10^{m+n},\quad \frac{a\times10^{m}}{b\times10^{n}}=\left(\frac{a}{b}\right)\times10^{m-n} \)

\( S_n=\tfrac{n}{2}\,[2a+(n-1)d]=\tfrac{n}{2}(a_1+a_n) \)

\( S=(n-2)\times180^{\circ} \)

\( \sqrt{ab}=\sqrt{a}\,\sqrt{b},\quad \sqrt{\tfrac{a}{b}}=\tfrac{\sqrt{a}}{\sqrt{b}}\;(b>0),\quad (\sqrt{a})^2=a \)

\( a\sqrt{b}+c\sqrt{b}=(a+c)\sqrt{b} \)

\( S=\pi r\,\ell+\pi r^2 \)

\( S=2(\ell w+\ell h+wh) \)

\( S=2\pi r h+2\pi r^2 \)

\( S=\text{perimeter of cross-section}\times\text{length}+2\times\text{area of cross-section} \)

\( S=4\pi r^2 \)

\( \text{Tangent at a point is perpendicular to the radius through that point} \)

\( m_{\text{tan}}\,m_{\text{radius}}=-1 \)

\( t^{2}=e(\,e+\ell\,) \)

\( \text{Reflect x-axis}:(x,y)\to(x,-y);\;\text{Reflect y-axis}:(x,y)\to(-x,y);\;\text{Rotate }90^{\circ}:(x,y)\to(-y,x);\;180^{\circ}:(x,y)\to(-x,-y);\;270^{\circ}:(x,y)\to(y,-x) \)

\( (x,y)\mapsto(x+a,\ y+b) \)

\( A\approx \tfrac{h}{2}\,[y_0+y_n+2(y_1+y_2+\cdots+y_{n-1})] \)

\( \sin^2\theta+\cos^2\theta=1,\quad \tan\theta=\tfrac{\sin\theta}{\cos\theta} \)

\( P=\left(\tfrac{k}{n}\right)^2 \)

\( P=\tfrac{k}{n}\times\tfrac{k-1}{n-1} \)

\[ \hat{u}=\frac{1}{\sqrt{x^2+y^2}}\begin{pmatrix}x\y\end{pmatrix} \]

\[ \begin{pmatrix}x\y\end{pmatrix}+\begin{pmatrix}u\v\end{pmatrix}=\begin{pmatrix}x+u\y+v\end{pmatrix},\qquad k\begin{pmatrix}x\y\end{pmatrix}=\begin{pmatrix}kx\ky\end{pmatrix} \]

\[ \overrightarrow{AB}=\begin{pmatrix}x_2-x_1\ y_2-y_1\end{pmatrix} \]

\[ \left\lVert\begin{pmatrix}x\y\end{pmatrix}\right\rVert=\sqrt{x^{2}+y^{2}} \]

\( \text{Acceleration}=\text{gradient},\quad \text{Distance travelled}=\text{area under the graph} \)

\( \text{Opposite angles at an intersection are equal} \)

\( V=\text{area of cross-section}\times\text{length} \)

\( V=\tfrac{1}{3}\pi r^2 h \)

\( V=\ell\,w\,h \)

\( V=\pi r^2 h \)

\( V=\tfrac{1}{3}\pi h\,(R^2+r^2+Rr) \)

\( V=\tfrac{1}{3}\,\text{area of base}\times h \)

\( V=\tfrac{4}{3}\pi r^3 \)

\( V' = k^{3}V \)