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Mathematics Formula Chart

Class 1 – 12 · 279+ formulas · All topics covered

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Class 9 – 10

63 formulas · 10 topics

Rational Number

p/q where p,q ∈ ℤ, q ≠ 0

Any number expressible as fraction

Eg: 3/4, −5/2, 7

Irrational Number

Cannot be written as p/q

Non-terminating, non-repeating decimals

Eg: √2, π, e

Laws of Radicals

√a × √b = √(ab) | √a / √b = √(a/b)

Product and quotient of surds

Eg: √2 × √8 = √16 = 4

Rationalising Factor

Multiply by conjugate: (a+√b)(a−√b) = a²−b

Remove surd from denominator

Eg: 1/(√2+1) × (√2−1)/(√2−1) = √2−1

Remainder Theorem

p(x) ÷ (x−a) → remainder = p(a)

Evaluate polynomial at x=a

Eg: p(x)=x²−3x+2, p(1)=0

Factor Theorem

(x−a) is factor of p(x) ↔ p(a) = 0

Zero of polynomial gives factor

Eg: p(2)=0 → (x−2) is factor

Quadratic Roots Product

α × β = c/a

Product of roots of ax²+bx+c=0

Eg: x²−5x+6=0 → αβ=6

Quadratic Roots Sum

α + β = −b/a

Sum of roots of ax²+bx+c=0

Eg: x²−5x+6=0 → α+β=5

Quadratic Formula

x = [−b ± √(b²−4ac)] / 2a

Roots of ax²+bx+c=0

Eg: x²−5x+6=0 → x=2 or x=3

Discriminant

D = b² − 4ac

D>0: two real roots, D=0: equal roots, D<0: no real roots

Eg: x²−4x+4=0 → D=0 (equal roots)

Nature of Roots

D > 0: distinct real | D = 0: equal real | D < 0: complex

Discriminant determines root type

Eg: x²+x+1=0 → D=−3 (complex)

Simultaneous Equations

a₁x+b₁y=c₁ and a₂x+b₂y=c₂

Solve by substitution or elimination

Eg: x+y=5, x−y=1 → x=3,y=2

Pythagoras Theorem

c² = a² + b²

Right triangle: hypotenuse² = sum of squares of legs

Eg: a=3,b=4 → c=5

Converse of Pythagoras

If c²=a²+b² then angle C = 90°

Test if triangle is right-angled

Eg: 5²=3²+4² → right triangle

Area of Triangle (Heron's)

A = √[s(s−a)(s−b)(s−c)], s=(a+b+c)/2

s = semi-perimeter

Eg: a=3,b=4,c=5 → s=6, A=6

Similarity Ratio

If △ABC ~ △DEF: AB/DE = BC/EF = CA/FD

Corresponding sides in same ratio

Eg: Scale factor k → areas in ratio k²

Basic Proportionality Theorem

DE ∥ BC → AD/DB = AE/EC

Line parallel to one side divides other two proportionally

Eg: AD=2,DB=4,AE=3 → EC=6

Tangent Length

PT² = PA × PB (secant-tangent)

P external point, T tangent point

Eg: PA=4,PB=9 → PT=6

Angle in Semicircle

∠ in semicircle = 90°

Angle subtended by diameter

Eg: Thales' theorem

Inscribed Angle Theorem

Inscribed angle = ½ × Central angle

Both subtend same arc

Eg: Central=80° → Inscribed=40°

Cyclic Quadrilateral

∠A + ∠C = 180°, ∠B + ∠D = 180°

Opposite angles are supplementary

Eg: If ∠A=110°, ∠C=70°

Length of Arc

l = (θ/360°) × 2πr

θ = central angle in degrees

Eg: θ=90°,r=7 → l=11 cm

Area of Sector

A = (θ/360°) × πr²

θ = central angle in degrees

Eg: θ=90°,r=7 → A=38.5 sq cm

sin θ

sin θ = Opposite / Hypotenuse

Ratio in right triangle

Eg: sin 30° = 1/2

cos θ

cos θ = Adjacent / Hypotenuse

Ratio in right triangle

Eg: cos 60° = 1/2

tan θ

tan θ = Opposite / Adjacent = sin θ / cos θ

Ratio in right triangle

Eg: tan 45° = 1

Reciprocal: cosec

cosec θ = 1/sin θ = H/O

Reciprocal of sine

Eg: cosec 30° = 2

Reciprocal: sec

sec θ = 1/cos θ = H/A

Reciprocal of cosine

Eg: sec 60° = 2

Reciprocal: cot

cot θ = 1/tan θ = cos θ/sin θ

Reciprocal of tangent

Eg: cot 45° = 1

Pythagorean Identity 1

sin²θ + cos²θ = 1

Fundamental trig identity

Eg: sin²30°+cos²30°=1/4+3/4=1

Pythagorean Identity 2

1 + tan²θ = sec²θ

Derived from identity 1

Eg: 1+tan²45°=1+1=2=sec²45°

Pythagorean Identity 3

1 + cot²θ = cosec²θ

Derived from identity 1

Eg: 1+cot²90°=1+0=1=cosec²90°

Standard Angles Table

sin: 0, 1/2, 1/√2, √3/2, 1 for 0°,30°,45°,60°,90°

Memorise these values

Note: cos is reverse order: 1, √3/2, 1/√2, 1/2, 0

Angle of Elevation

tan θ = height / horizontal distance

Looking up from ground

Eg: h=10m, d=10m → θ=45°

Mean (Ungrouped)

x̄ = Σxᵢ / n

Sum of all values divided by count

Eg: Data: 2,4,6 → Mean=4

Mean (Grouped)

x̄ = Σfᵢxᵢ / Σfᵢ

fᵢ = frequency, xᵢ = class mark

Eg: Weighted average of class marks

Median (Odd n)

Median = ((n+1)/2)th value

Middle value when sorted

Eg: n=5 → 3rd value

Median (Even n)

Median = average of (n/2)th and (n/2+1)th values

Average of two middle values

Eg: n=6 → avg of 3rd and 4th

Mode

Mode = value with highest frequency

Most frequently occurring value

Eg: Data: 1,2,2,3 → Mode=2

Empirical Relation

Mode = 3 Median − 2 Mean

Approximate relationship

Eg: Mean=4, Median=3 → Mode=1

Probability

P(E) = Number of favourable outcomes / Total outcomes

0 ≤ P(E) ≤ 1

Eg: P(Head) = 1/2

Complementary Events

P(E) + P(Ē) = 1

Event and its complement

Eg: P(not 6) = 1 − 1/6 = 5/6

Cumulative Frequency

CF = sum of all frequencies up to that class

Running total of frequencies

Eg: Used to find median graphically

Volume of Cone

V = ⅓πr²h

r = base radius, h = height

Eg: r=7,h=6 → V=308 cu cm

Curved SA of Cone

CSA = πrl

l = slant height = √(r²+h²)

Eg: r=3,l=5 → CSA=47.1

Total SA of Cone

TSA = πr(r + l)

Curved + base circle

Eg: r=3,l=5 → TSA=75.4

Volume of Sphere

V = (4/3)πr³

r = radius

Eg: r=3 → V=113.1 cu cm

Surface Area of Sphere

SA = 4πr²

Total surface area

Eg: r=7 → SA=616 sq cm

Volume of Hemisphere

V = (2/3)πr³

Half of sphere

Eg: r=3 → V=56.55 cu cm

CSA of Hemisphere

CSA = 2πr²

Curved part only

Eg: r=7 → CSA=308 sq cm

TSA of Hemisphere

TSA = 3πr²

Curved + flat circular base

Eg: r=7 → TSA=462 sq cm

Slant Height of Cone

l = √(r² + h²)

Pythagoras on cone

Eg: r=3,h=4 → l=5

Distance Formula

d = √[(x₂−x₁)² + (y₂−y₁)²]

Distance between two points

Eg: (0,0) to (3,4) → d=5

Midpoint Formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Midpoint of a line segment

Eg: (2,4) and (6,8) → M=(4,6)

Section Formula (Internal)

P = ((mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n))

Point dividing in ratio m:n internally

Eg: m=1,n=2 → P divides 1:2

Area of Triangle (Coordinate)

A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|

Using vertex coordinates

Eg: (0,0),(4,0),(0,3) → A=6

Slope of Line

m = (y₂−y₁)/(x₂−x₁) = tan θ

θ = angle with positive x-axis

Eg: (1,2) to (3,6) → m=2

Collinearity Condition

Area of triangle = 0

Three points are collinear if area=0

Eg: x₁(y₂−y₃)+x₂(y₃−y₁)+x₃(y₁−y₂)=0

nth Term of AP

aₙ = a + (n−1)d

a = first term, d = common difference

Eg: a=2,d=3,n=5 → a₅=14

Sum of n Terms of AP

Sₙ = n/2 × [2a + (n−1)d]

Sum of first n terms

Eg: a=1,d=1,n=10 → S₁₀=55

Sum using Last Term

Sₙ = n/2 × (a + l)

l = last term

Eg: a=1,l=10,n=10 → S=55

Common Difference

d = aₙ − aₙ₋₁

Difference between consecutive terms

Eg: 2,5,8,11 → d=3

Sum of First n Natural Numbers

Sₙ = n(n+1)/2

Special AP with a=1, d=1

Eg: n=100 → S=5050