Full Math Cheat Sheet — Grades 1–10
One‑page Summary — Quick Reference (Grades 1–10)
Basic Arithmetic & Number Facts
Add/Subtract/Multiply/Divide properties: commutative, associative, distributive: a(b+c)=ab+ac
Sum of first n naturals: \(\displaystyle \sum_{k=1}^n k=\frac{n(n+1)}{2}\)
Sum of first n odd numbers: \(\displaystyle 1+3+\dots+(2n-1)=n^2\)
Sum of first n even numbers: \(\displaystyle 2+4+\dots+2n=n(n+1)\)
Fractions · Decimals · Percents
Convert: fraction → decimal: divide; decimal → percent: ×100; percent → fraction: ÷100 → simplify
Common: \(\tfrac12=0.5=50\%\)
\(\tfrac13\approx0.333=33.33\%\)
\(\tfrac14=0.25=25\%\)
Common fractions, decimals, and percentages:
- \(\tfrac{1}{2} = 0.5 = 50\%\)
- \(\tfrac{1}{3} \approx 0.333 = 33.33\%\)
- \(\tfrac{1}{4} = 0.25 = 25\%\)
- \(\tfrac{1}{5} = 0.2 = 20\%\)
- \(\tfrac{1}{8} = 0.125 = 12.5\%\)
Add/Subtract: common denominator; Multiply: multiply numerators/denominators; Divide: multiply by reciprocal.
Algebra & Polynomials
Expand: \((a+b)^2=a^2+2ab+b^2\), \((a-b)^2=a^2-2ab+b^2\)
Factor: \(a^2-b^2=(a-b)(a+b)\), \(ax+ay=a(x+y)\)
Quadratic: \(ax^2+bx+c=0\). Roots: \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\). Discriminant \(\Delta=b^2-4ac\)
Binomial theorem: \((x+y)^n=\sum_{k=0}^n {n\choose k}x^{n-k}y^k\)
Exponents, Roots & Logarithms
Exponent rules: \(a^m a^n=a^{m+n}\), \(\dfrac{a^m}{a^n}=a^{m-n}\), \((a^m)^n=a^{mn}\)
Negative/power: \(a^{-n}=\dfrac{1}{a^n}\), \(\sqrt[n]{a}=a^{1/n}\)
Log rules: \(\log_b(xy)=\log_b x+\log_b y\), \(\log_b(x^k)=k\log_b x\), change of base \(\log_b x=\dfrac{\log x}{\log b}\)
Sequences & Series
Arithmetic sequence: \(a_n=a_1+(n-1)d\). Sum: \(S_n=\dfrac{n}{2}(a_1+a_n)\)
Geometric sequence: \(a_n=a_1 r^{n-1}\). Sum (r≠1): \(S_n=a_1\dfrac{1-r^n}{1-r}\)
Perimeter, Area & Angles
Rectangle: \(P=2(l+w),\ A=lw\). Square: \(P=4a,\ A=a^2\).
Triangle: \(P=a+b+c,\ A=\tfrac12 bh\). Trapezium: \(A=\tfrac12(a+b)h\).
Circle: \(C=2\pi r,\ A=\pi r^2\). Sector: \(A=\tfrac12 r^2\theta\) (θ in radians). Arc length: \(s=r\theta\).
Polygon interior sum: \((n-2)\times180^\circ\). Interior angle (regular n-gon): \(\dfrac{(n-2)180^\circ}{n}\).
Surface Area & Volume
Cube: \(V=a^3,\ SA=6a^2\). Cuboid: \(V=lwh,\ SA=2(lb+bh+hl)\).
Cylinder: \(V=\pi r^2h,\ SA=2\pi r(h+r)\). Sphere: \(V=\tfrac{4}{3}\pi r^3,\ SA=4\pi r^2\).
Cone: \(V=\tfrac13\pi r^2 h,\ SA=\pi r(l+r)\) where l is slant height.
Coordinate Geometry & Conics
Distance: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\). Midpoint: \(M=(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2})\).
Line: \(y=mx+c\), point-slope: \(y-y_1=m(x-x_1)\). Perpendicular slopes: \(m_1 m_2=-1\).
Circle: \((x-h)^2+(y-k)^2=r^2\). Parabola: standard \(y=ax^2+bx+c\), vertex \(y=a(x-h)^2+k\).
Trigonometry — Identities & Rules
Definitions: \(\sin\theta=\tfrac{opp}{hyp},\ \cos\theta=\tfrac{adj}{hyp},\ \tan\theta=\tfrac{opp}{adj}\).
Reciprocals: \(\csc\theta=\tfrac{1}{\sin\theta},\ \sec\theta=\tfrac{1}{\cos\theta},\ \cot\theta=\tfrac{1}{\tan\theta}\).
Pythagorean: \(\sin^2\theta+\cos^2\theta=1\). Double-angle: \(\sin2\theta=2\sin\theta\cos\theta,\ \cos2\theta=\cos^2\theta-\sin^2\theta\).
Sum/difference: \(\sin(a\pm b)=\sin a\cos b\pm\cos a\sin b\), \(\cos(a\pm b)=\cos a\cos b\mp\sin a\sin b\).
Sine rule: \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\). Cosine rule: \(c^2=a^2+b^2-2ab\cos C\).
Basic solved forms: \(\sin^2\theta=\tfrac{1-\cos2\theta}{2},\ \cos^2\theta=\tfrac{1+\cos2\theta}{2}\).
Permutations · Combinations · Probability
Factorial: \(n!=n\times(n-1)\times\cdots\times1\).
Permutations: \(P(n,r)=\dfrac{n!}{(n-r)!}\). Combinations: \(C(n,r)=\dfrac{n!}{r!(n-r)!}\).
Probability: \(P(E)=\dfrac{\text{favourable}}{\text{total}}\). Conditional: \(P(A|B)=\dfrac{P(A\cap B)}{P(B)}\).
Independent: \(P(A\cap B)=P(A)P(B)\). Complement: \(P(\overline{E})=1-P(E)\).
Statistics & Data
Mean: \(\bar{x}=\dfrac{\sum x_i}{N}\). Median: middle value. Mode: most frequent.
Variance (population): \(\sigma^2=\dfrac{\sum (x_i-\bar{x})^2}{N}\). Std dev: \(\sigma=\sqrt{\sigma^2}\).
Grouped data: estimated mean using midpoints, cumulative frequency for median quartiles.
Finance & Growth
Simple interest: \(I=Prt\). Amount: \(A=P(1+rt)\).
Compound (annual): \(A=P(1+r)^t\). Compound (n times yearly): \(A=P\left(1+\dfrac{r}{n}\right)^{nt}\).
Continuous growth/decay: \(A=Pe^{kt}\).
Useful Identities & Constants
Euler: \(e^{i\pi}+1=0\). \(\pi\approx3.14159,\ e\approx2.71828\).
Factor & expansion shortcuts: \((x+y)^3=x^3+3x^2y+3xy^2+y^3\).
Angle conversion: \(1^\circ=\dfrac{\pi}{180}\) rad. Unit tips: 1 m = 100 cm = 1000 mm.
Grades 1–2 — Foundations
Counting & Place Value
Digits: 0–9. Place values: units, tens, hundreds.
Even / Odd: even ends 0,2,4,6,8. odd ends 1,3,5,7,9.
Addition & Subtraction
Commutative: a+b=b+a
Associative: (a+b)+c = a+(b+c)
Column addition/subtraction → carry / borrow rules.
Multiplication basics
Repeated addition: 3×4 = 3+3+3+3
Tables: learn ×1–×12 (practice)
Division basics
Sharing and grouping. Division fact: a ÷ b = c ⇔ b×c = a
Shapes
Square, rectangle, triangle, circle — name corners/edges.
Perimeter (simple): add sides.
Time & Money
Clock: 60 sec = 1 min, 60 min = 1 hr. Coins & notes: practice sums.
Grades 3–4 — Elementary
Fractions
Proper: numerator < denominator. Improper: numerator ≥ denominator.
Equivalent: \(\frac{a}{b}=\frac{ka}{kb}\)
Add/Subtract (common denom): \(\dfrac{a}{b}+\dfrac{c}{b}=\dfrac{a+c}{b}\)
Multiply: \(\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{ac}{bd}\)
Divide: \(\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}\)
Decimals & Percent
Decimals: place values tenths, hundredths, thousandths.
Convert: \(0.25=25\%= \tfrac{25}{100}=\tfrac{1}{4}\)
Percent of: \(p\%\text{ of }N = \dfrac{p}{100}\times N\)
Factors & Multiples
Factors: integers dividing a number. Multiples: product with integers.
GCF/LCM basics: list method or prime factors.
Perimeter & Area
Rectangle: \(P=2(l+w)\), \(A=l\times w\)
Square: \(P=4a\), \(A=a^2\)
Triangle: \(A=\tfrac{1}{2}bh\)
Circle: \(C=2\pi r\), \(A=\pi r^2\)
Angles & Polygons
Sum angles triangle = 180°
Regular polygon interior angle = \(\dfrac{(n-2)\times180^\circ}{n}\)
Basic Data
Mean (average): \(\bar{x}=\dfrac{\sum x_i}{N}\)
Mode: most frequent. Median: middle when ordered.
Grades 5–6 — Upper Elementary
Ratio & Proportion
Ratio: 3:2 means \(\dfrac{3}{2}\). Proportion: \(\dfrac{a}{b}=\dfrac{c}{d}\)
Cross-multiply: \(ad=bc\)
Exponents & Roots
\(a^m \times a^n = a^{m+n}\)
\(\dfrac{a^m}{a^n}=a^{m-n}\), \((a^m)^n=a^{mn}\)
Square root: \(\sqrt{a}\) such that \(\sqrt{a}^2=a\)
Algebra basics
Solve linear: ax+b=c → \(x=\dfrac{c-b}{a}\)
Simple identities: \((a+b)^2=a^2+2ab+b^2\)
Volume
Cube: \(V=a^3\)
Cuboid: \(V=lbh\)
Cylinder: \(V=\pi r^2 h\)
Prism: \(V=\text{base area}\times height\)
Percent & Interest (intro)
Percent change: \(\dfrac{\text{change}}{\text{original}}\times100\%\)
Simple interest: \(I=Prt\) (P principal, r rate, t time)
Statistics basics
Mean: \(\bar{x}=\dfrac{\sum x_i}{N}\)
Range = max − min
Frequency tables → mode, median estimation.
Grades 7–8 — Middle School
Linear equations
One variable: \(ax+b=0\Rightarrow x=-\dfrac{b}{a}\)
Two variables (slope form): \(y=mx+c\)
Slope between points: \(m=\dfrac{y_2-y_1}{x_2-x_1}\)
Coordinate Geometry
Distance: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)
Midpoint: \(M\!=\!\Big(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\Big)\)
Equation of line (point-slope): \(y-y_1=m(x-x_1)\)
Quadratic basics
Standard: \(ax^2+bx+c=0\)
Quadratic formula: \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\)
Discriminant \(\Delta=b^2-4ac\) (real roots if \(\Delta\ge0\))
Trigonometry (basic)
Right triangle definitions: \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}},\ \cos\theta=\dfrac{\text{adj}}{\text{hyp}},\ \tan\theta=\dfrac{\text{opp}}{\text{adj}}\)
Identity: \(\sin^2\theta+\cos^2\theta=1\)
Polynomials & Factorisation
Factor: \(a^2-b^2=(a-b)(a+b)\)
Common factor: \(ax+ay=a(x+y)\)
Probability
Simple probability: \(P(E)=\dfrac{\text{favourable}}{\text{total}}\)
Complement: \(P(\overline{E})=1-P(E)\)
Surface Area & Volume (refresher)
Cube: \(SA=6a^2, V=a^3\)
Cuboid: \(SA=2(lb+bh+hl), V=lbh\)
Sphere: \(SA=4\pi r^2, V=\tfrac{4}{3}\pi r^3\)
Cone: \(V=\tfrac13\pi r^2h\)
Sequences & Series (intro)
Arithmetic sequence: \(a_n=a_1+(n-1)d\)
Sum of n terms: \(S_n=\dfrac{n}{2}(a_1+a_n)\)
Geometric sum (r≠1): \(S_n=a_1\dfrac{1-r^n}{1-r}\)
Statistics (middle)
Mean: \(\bar{x}=\dfrac{\sum x_i}{N}\)
Median & Mode definitions. Variance (intro): \(\sigma^2=\dfrac{\sum (x_i-\bar{x})^2}{N}\)
Grade 8 — Key Formulas
Linear Algebra & Manipulation
Linear eqns: \(ax+b=0\Rightarrow x=-\dfrac{b}{a}\)
Simultaneous (2×2) solve (substitution/elimination).
Functions & Graphs
Function notation: \(f(x)=mx+c\). Evaluate: \(f(2)=2m+c\).
Trigonometry (right triangles)
\(\sin\theta=\dfrac{opp}{hyp}\), \(\cos\theta=\dfrac{adj}{hyp}\), \(\tan\theta=\dfrac{opp}{adj}\)
Use SOHCAHTOA to find missing sides/angles.
Geometry & Transformations
Scale factor area → \(k^2\), volume → \(k^3\)
Angle facts: parallel lines, alternate interior angles equal.
Data & Probability
Mean, median, mode, range. Simple probability and complements.
Number Theory
Prime, composite, HCF/GCF by prime factors, LCM from prime factors.
Grade 9 — Key Formulas
Polynomials & Factoring
Factorisation techniques: common factor, grouping, quadratic trinomials.
Remainder theorem: f(a) remainder when dividing by (x-a).
Quadratics
Vertex: \(x_v=-\dfrac{b}{2a}\). Axis of symmetry \(x=x_v\).
Roots via quadratic formula and discriminant tests.
Coordinate Geometry
Equation line: \(y=mx+c\). Perpendicular slopes: \(m_1m_2=-1\).
Circle eqn and distance/midpoint refresher.
Trigonometry (advanced basics)
Compound formulas, double-angle: \(\sin2\theta=2\sin\theta\cos\theta\).
Sine and cosine rule for any triangle (see summary).
Statistics
Grouped data: mean estimate, cumulative frequencies, median interpolation (concept).
Finance basics
Simple interest: \(I=Prt\). Compound annually: \(A=P(1+r)^t\)
Grade 10 — Key Formulas
Advanced Algebra
Quadratic forms, completing the square: \(ax^2+bx+c=a(x+\tfrac{b}{2a})^2+\dots\)
Binomial theorem and Pascal’s triangle applications.
Trigonometric Identities
Product-to-sum and sum-to-product (reference): \(\sin A\cos B=\tfrac12[\sin(A+B)+\sin(A-B)]\)
Triple-angle basics and solving trig equations (intro).
Logarithms & Exponentials
Exponential growth/decay models: \(A=Pe^{kt}\) (continuous).
Log rules and solving simple log equations (use change of base).
Probability & Combinatorics
Permutations & combinations (see summary). Conditional probability: \(P(A|B)=\dfrac{P(A\cap B)}{P(B)}\)
Independent events: \(P(A\cap B)=P(A)P(B)\)
Coordinate Geometry & Conics
Circle: \((x-h)^2+(y-k)^2=r^2\). Parabola: \(y=ax^2+bx+c\).
Focus/directrix form and basic locus concepts (intro).
Statistics (intro to distributions)
Mean, variance: \(\sigma^2=\dfrac{\sum (x_i-\bar{x})^2}{N}\). Std dev: \(\sigma=\sqrt{\sigma^2}\)
Normal approximation (concept) and uses of z-scores (intro).