Percentage
Express a part as a percentage of the whole.
\(\frac{\text{Part}}{\text{Whole}}\times100\%\)
Mathematics / Reference Library
Quick Formula Sheet.
Essential formulas, organized for quick revision. Search by a topic or keyword, filter by subject, and copy a formula's text when you need it.
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01. Arithmetic
Ratios, percentages, and rates
Percentage change
Compare a change with the original value.
\(\frac{\text{New}-\text{Original}}{\text{Original}}\times100\%\)
Proportion
Equivalent ratios have equal cross-products.
\(\frac{a}{b}=\frac{c}{d}\Rightarrow ad=bc\)
Speed
Average speed over a journey.
\(\text{Speed}=\frac{\text{Distance}}{\text{Time}}\)
02. Algebra
Equations and identities
Quadratic formula
Solve \(ax^2+bx+c=0\), where \(a\ne0\).
\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
Difference of squares
Factor a difference between two squares.
\(a^2-b^2=(a-b)(a+b)\)
Square of a sum
Expand a binomial square.
\((a+b)^2=a^2+2ab+b^2\)
Square of a difference
Expand a binomial difference.
\((a-b)^2=a^2-2ab+b^2\)
Product of powers
For the same nonzero base.
\(a^m a^n=a^{m+n}\)
Power of a power
Raise a power to another power.
\((a^m)^n=a^{mn}\)
03. Geometry
Perimeter, area, surface area, and volume
Rectangle
Length \(l\), width \(w\).
\(A=lw,\quad P=2(l+w)\)
Triangle
Base \(b\), perpendicular height \(h\).
\(A=\frac12 bh\)
Circle
Radius \(r\), diameter \(d=2r\).
\(A=\pi r^2,\quad C=2\pi r\)
Pythagorean theorem
For a right-angled triangle.
\(a^2+b^2=c^2\)
Cuboid
Length \(l\), width \(w\), height \(h\).
\(V=lwh\)
Sphere
Radius \(r\).
\(V=\frac43\pi r^3,\quad S=4\pi r^2\)
Cylinder
Radius \(r\), height \(h\).
\(V=\pi r^2h\)
Cone
Circular base radius \(r\), perpendicular height \(h\).
\(V=\frac13\pi r^2h\)
04. Trigonometry
Ratios and identities
Basic ratios
Relative to angle \(\theta\) in a right triangle.
\(\sin\theta=\frac{\text{opp}}{\text{hyp}}\)
\(\cos\theta=\frac{\text{adj}}{\text{hyp}}\)
\(\tan\theta=\frac{\text{opp}}{\text{adj}}\)
Pythagorean identity
A fundamental relationship between sine and cosine.
\(\sin^2\theta+\cos^2\theta=1\)
Sine of a sum
Expand the sine of two added angles.
\(\sin(A+B)=\sin A\cos B+\cos A\sin B\)
Cosine of a sum
Expand the cosine of two added angles.
\(\cos(A+B)=\cos A\cos B-\sin A\sin B\)
05. Coordinate geometry
Points, lines, and circles
Distance between points
Points \((x_1,y_1)\) and \((x_2,y_2)\).
\(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Midpoint
Find the centre of a line segment.
\(\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\)
Slope of a line
For two points with different \(x\)-coordinates.
\(m=\frac{y_2-y_1}{x_2-x_1}\)
Slope-intercept form
A straight line with slope \(m\) and intercept \(c\).
\(y=mx+c\)
Circle equation
Centre \((h,k)\), radius \(r\).
\((x-h)^2+(y-k)^2=r^2\)
06. Sequences and series
Arithmetic and geometric patterns
Arithmetic sequence
First term \(a\), common difference \(d\).
\(a_n=a+(n-1)d\)
Arithmetic series sum
Sum of the first \(n\) terms.
\(S_n=\frac n2[2a+(n-1)d]\)
Geometric sequence
First term \(a\), common ratio \(r\).
\(a_n=ar^{n-1}\)
Finite geometric sum
For \(r\ne1\).
\(S_n=a\frac{1-r^n}{1-r}\)
07. Calculus
Derivatives and integrals
Power rule
Differentiate a power of \(x\).
\(\frac{d}{dx}x^n=nx^{n-1}\)
Sum rule
Differentiate a sum of functions.
\((f+g)'=f'+g'\)
Product rule
Differentiate a product of functions.
\((fg)'=f'g+fg'\)
Chain rule
Differentiate a composite function.
\(\frac{d}{dx}f(g(x))=f'(g(x))g'(x)\)
Power rule for integration
For \(n\ne-1\).
\(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\)
Definite integral
If \(F'=f\), then:
\(\int_a^b f(x)\,dx=F(b)-F(a)\)
08. Probability
Events and counting
Classical probability
For equally likely outcomes.
\(P(A)=\frac{|A|}{|S|}\)
Complement rule
Probability that event \(A\) does not occur.
\(P(A^c)=1-P(A)\)
Addition rule
For any two events.
\(P(A\cup B)=P(A)+P(B)-P(A\cap B)\)
Conditional probability
When \(P(B)>0\).
\(P(A\mid B)=\frac{P(A\cap B)}{P(B)}\)
Permutations
Arrange \(r\) items selected from \(n\), where order matters.
\({}_nP_r=\frac{n!}{(n-r)!}\)
Combinations
Choose \(r\) items from \(n\), where order does not matter.
\({}_nC_r=\frac{n!}{r!(n-r)!}\)
09. Statistics
Central tendency and spread
Arithmetic mean
Average of \(n\) observations.
\(\bar{x}=\frac{\sum x_i}{n}\)
Range
Difference between the largest and smallest values.
\(\text{Range}=x_{\max}-x_{\min}\)
Population variance
Mean squared deviation from population mean \(\mu\).
\(\sigma^2=\frac{\sum(x_i-\mu)^2}{N}\)
Sample variance
Unbiased estimator of population variance.
\(s^2=\frac{\sum(x_i-\bar{x})^2}{n-1}\)
Standard deviation
Square root of the variance.
\(\sigma=\sqrt{\sigma^2}\)
Interquartile range
Spread of the middle half of the data.
\(\mathrm{IQR}=Q_3-Q_1\)
Important: Formulas may depend on their assumptions. For example, classical probability assumes equally likely outcomes, the sample variance formula uses \(n-1\), and trigonometric ratios depend on the angle and triangle context. Check definitions and conditions before applying a formula.