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Maths Formula Sheet

All important Maths formulas for Class 6–12, grouped chapter-wise for fast, exam-ready revision.

Numbers & Arithmetic

Order of operations
BODMAS — Brackets, Orders, Division, Multiplication, Addition, Subtraction
Dividend relation
Dividend = Divisor × Quotient + Remainder
HCF × LCM
HCF × LCM = product of the two numbers
Fraction of a number
(a/b) of N = (a × N) / b
Average
Average = sum of observations / number of observations

Ratio, Proportion & Percentage

Ratio
a : b = a / b
Proportion
a : b :: c : d ⇒ a × d = b × c
Percentage
x% = x / 100
Percentage of a number
x% of N = (x × N) / 100
Percentage change
% change = (change / original value) × 100

Commercial Maths

Profit
Profit = SP − CP
Loss
Loss = CP − SP
Profit / Loss %
% = (Profit or Loss / CP) × 100
Simple interest
SI = (P × R × T) / 100
Amount (SI)
A = P + SI
Compound interest
A = P(1 + R/100)n, CI = A − P

Exponents & Powers

Product law
am × an = a(m+n)
Quotient law
am ÷ an = a(m−n)
Power of a power
(am)n = amn
Zero exponent
a0 = 1, a ≠ 0
Negative exponent
a−n = 1 / an
Fractional exponent
a(1/n) = n√a

Algebraic Identities

Square of a sum
(a + b)2 = a2 + 2ab + b2
Square of a difference
(a − b)2 = a2 − 2ab + b2
Difference of squares
a2 − b2 = (a + b)(a − b)
Cube of a sum
(a + b)3 = a3 + b3 + 3ab(a + b)
Cube of a difference
(a − b)3 = a3 − b3 − 3ab(a − b)
Sum of cubes
a3 + b3 = (a + b)(a2 − ab + b2)
Difference of cubes
a3 − b3 = (a − b)(a2 + ab + b2)
Three-term square
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
Sum of three cubes
a3 + b3 + c3 − 3abc = (a+b+c)(a2+b2+c2−ab−bc−ca)

Linear & Quadratic Equations

Linear equation
ax + b = 0 ⇒ x = −b / a
Quadratic — standard form
ax2 + bx + c = 0
Quadratic formula
x = [ −b ± √(b2 − 4ac) ] / 2a
Discriminant
D = b2 − 4acD > 0 real & distinct, D = 0 equal, D < 0 imaginary
Sum of roots
α + β = −b / a
Product of roots
α × β = c / a

Sequences & Series

AP — nth term
aₙ = a + (n − 1)d
AP — sum of n terms
Sₙ = (n/2)[ 2a + (n − 1)d ] = (n/2)(a + l)
GP — nth term
aₙ = a·r(n−1)
GP — sum of n terms
Sₙ = a(rn − 1)/(r − 1), r ≠ 1
GP — infinite sum
S∞ = a / (1 − r), |r| < 1
Sum of first n naturals
Σn = n(n + 1) / 2
Sum of squares
Σn2 = n(n + 1)(2n + 1) / 6
Sum of cubes
Σn3 = [ n(n + 1) / 2 ]2

Permutations, Combinations & Binomial

Factorial
n! = n × (n − 1) × … × 2 × 1, 0! = 1
Permutations
nPr = n! / (n − r)!
Combinations
nCr = n! / [ r!(n − r)! ]
Relation
nPr = nCr × r!
Binomial theorem
(a + b)n = Σ nCr a(n−r) br
General term
Tr+1 = nCr a(n−r) br

Sets, Relations & Functions

Union of sets
n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Subsets
number of subsets of a set with n elements = 2n
Cartesian product
n(A × B) = n(A) × n(B)
De Morgan’s laws
(A ∪ B)′ = A′ ∩ B′, (A ∩ B)′ = A′ ∪ B′

Complex Numbers

Imaginary unit
i = √(−1), i2 = −1
Complex number
z = a + ib
Modulus
|z| = √(a2 + b2)
Conjugate
z̄ = a − ib
Polar form
z = r(cos θ + i sin θ)

Lines, Angles & Triangles

Angle sum of triangle
∠A + ∠B + ∠C = 180°
Exterior angle
exterior angle = sum of two interior opposite angles
Angle sum of polygon
(n − 2) × 180°
Pythagoras theorem
hypotenuse2 = base2 + height2
Sum of exterior angles
sum of exterior angles of any polygon = 360°

Mensuration (2D)

Area of rectangle
A = length × breadth
Area of square
A = side2
Area of triangle
A = ½ × base × height
Heron's formula
A = √[ s(s−a)(s−b)(s−c) ], s = (a+b+c)/2
Area of parallelogram
A = base × height
Area of trapezium
A = ½ × (sum of parallel sides) × height
Area of circle
A = πr2
Circumference
C = 2πr

Surface Area & Volume (3D)

Volume of cuboid
V = l × b × h
Surface area of cuboid
S = 2(lb + bh + hl)
Volume of cube
V = a3
Volume of cylinder
V = πr2h
CSA of cylinder
CSA = 2πrh
Volume of cone
V = (1/3)πr2h
CSA of cone
CSA = πrl, l = √(r2 + h2)
Volume of sphere
V = (4/3)πr3
Surface area of sphere
S = 4πr2

Coordinate Geometry & Straight Lines

Distance formula
d = √[ (x₂ − x₁)2 + (y₂ − y₁)2 ]
Section formula
P = ( (mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n) )
Midpoint
M = ( (x₁ + x₂)/2, (y₁ + y₂)/2 )
Slope of a line
m = (y₂ − y₁) / (x₂ − x₁)
Slope–intercept form
y = mx + c
Point–slope form
y − y₁ = m(x − x₁)
Area of triangle (coords)
A = ½ |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Conic Sections

Circle (centre origin)
x2 + y2 = r2
Circle (centre h,k)
(x − h)2 + (y − k)2 = r2
Parabola
y2 = 4ax
Ellipse
x2/a2 + y2/b2 = 1
Hyperbola
x2/a2 − y2/b2 = 1

Trigonometry — Ratios & Identities

Basic ratios
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
Reciprocal ratios
cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
Pythagorean identity
sin2θ + cos2θ = 1
Secant identity
1 + tan2θ = sec2θ
Cosecant identity
1 + cot2θ = cosec2θ
Sine rule
a/sin A = b/sin B = c/sin C
Cosine rule
c2 = a2 + b2 − 2ab·cos C

Trigonometry — Compound & Multiple Angles

Sum formula (sin)
sin(A ± B) = sin A cos B ± cos A sin B
Sum formula (cos)
cos(A ± B) = cos A cos B ∓ sin A sin B
Tangent of sum
tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
Double angle (sin)
sin 2A = 2 sin A cos A
Double angle (cos)
cos 2A = cos2A − sin2A = 1 − 2sin2A
Double angle (tan)
tan 2A = 2 tan A / (1 − tan2A)

Limits & Derivatives

Limit (sin x)/x
lim (x→0) sin x / x = 1
Power rule
d/dx (xn) = n·x(n−1)
Product rule
(uv)′ = u′v + uv′
Quotient rule
(u/v)′ = (u′v − uv′) / v2
Chain rule
dy/dx = (dy/du)·(du/dx)
Standard derivatives
d/dx(sin x) = cos x, d/dx(ex) = ex, d/dx(ln x) = 1/x

Integration

Power rule
∫ xn dx = x(n+1)/(n+1) + C, n ≠ −1
Integral of 1/x
∫ (1/x) dx = ln|x| + C
Exponential
∫ ex dx = ex + C
Trigonometric
∫ sin x dx = −cos x + C, ∫ cos x dx = sin x + C
By parts
∫ u·v dx = u∫v dx − ∫ (u′ ∫v dx) dx

Matrices, Determinants & Vectors

2×2 determinant
|A| = ad − bc for A = [[a, b], [c, d]]
Inverse of matrix
A−1 = (1/|A|)·adj A, |A| ≠ 0
Magnitude of vector
|a→| = √(x2 + y2 + z2)
Dot product
a→·b→ = |a||b| cos θ
Cross product
|a→ × b→| = |a||b| sin θ

Statistics & Probability

Mean
x̄ = Σx / n (or Σfx / Σf)
Median (odd n)
middle observation when data is arranged in order
Mode
most frequently occurring observation
Empirical relation
Mode = 3 Median − 2 Mean
Probability
P(E) = favourable outcomes / total outcomes
Complement
P(not E) = 1 − P(E)
Addition rule
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
About this sheet

About the Maths Formula Sheet

This Maths formula sheet brings together every important formula you need from Class 6 to Class 12 in a single, organised page. Whether you are revising arithmetic and percentages for middle school, algebraic identities and mensuration for Class 9 and 10, or trigonometry, calculus and coordinate geometry for Class 11 and 12, the formulas are grouped chapter-wise so you can find what you need in seconds.

All formulas follow the NCERT / CBSE syllabus and are equally useful for ICSE and State Board students, as well as for competitive exams like JEE, NTSE and Olympiads. Use it for last-minute revision before a test, to build your own formula chart, or simply to memorise the identities and theorems you keep forgetting.

  • Algebraic identities, polynomials and quadratic equations
  • Mensuration, surface areas and volumes of solids
  • Coordinate geometry, straight lines and conic sections
  • Trigonometric ratios, identities and compound-angle formulas
  • Sequences, series, permutations, combinations and binomial theorem
  • Limits, derivatives, integration, matrices, vectors and statistics
FAQ

Maths Formulas — FAQs

Quick answers about using this Maths formula sheet for Class 6–12 revision.

Yes. All key Class 10 chapters — real numbers, polynomials, quadratic equations, arithmetic progressions, coordinate geometry, trigonometry, areas, surface areas & volumes, statistics and probability — are covered with their important formulas.
Absolutely. The sheet includes sets, relations & functions, complex numbers, conic sections, limits & derivatives, integration, matrices, determinants, vectors and probability that form the core of the Class 11–12 CBSE Maths syllabus.
Yes. The formulas are foundational for JEE Main, NDA, NTSE and Olympiad preparation, though competitive exams also expect you to apply them to harder, multi-step problems.
Revise one chapter group at a time, write each formula out by hand once, then solve a few practice questions from that chapter. Print the sheet and keep it handy for quick revision the night before an exam.

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