Complete Mathematics Formulas for Class 6 to 10 – Free PDF Guide with Explanations

Complete Mathematics Formulas Guide for Class 6 to 10

Mathematics is built on fundamental formulas that serve as building blocks for advanced concepts. This comprehensive guide presents essential formulas from Class 6 through Class 10, organized systematically to help students master mathematical concepts with clarity and confidence.

Class 6 Mathematics Formulas

1. Number System

Concept	Formula	Explanation
Whole Numbers	0, 1, 2, 3, 4, …	Numbers starting from zero with no fractions
Natural Numbers	1, 2, 3, 4, 5, …	Counting numbers starting from 1
Integers	…, -3, -2, -1, 0, 1, 2, 3, …	Whole numbers including negative numbers
Even Numbers	2n (where n is any integer)	Numbers divisible by 2
Odd Numbers	2n + 1 (where n is any integer)	Numbers not divisible by 2

2. Basic Arithmetic Operations

Operation	Formula	Example
Addition	a + b = sum	5 + 3 = 8
Subtraction	a – b = difference	10 – 4 = 6
Multiplication	a × b = product	6 × 7 = 42
Division	a ÷ b = quotient	20 ÷ 4 = 5

3. Fractions

Concept	Formula	Explanation
Proper Fraction	Numerator < Denominator	Example: 3/5, 2/7
Improper Fraction	Numerator ≥ Denominator	Example: 7/3, 5/5
Mixed Fraction	Whole number + Proper fraction	Example: 2¹/₃
Addition of Fractions	a/b + c/d = (ad + bc)/bd	Cross multiply and add
Subtraction of Fractions	a/b – c/d = (ad – bc)/bd	Cross multiply and subtract
Multiplication of Fractions	a/b × c/d = (a × c)/(b × d)	Multiply numerators and denominators
Division of Fractions	a/b ÷ c/d = a/b × d/c	Multiply by reciprocal

4. Decimals

Concept	Formula/Rule	Example
Decimal to Fraction	Place value method	0.5 = 5/10 = 1/2
Addition of Decimals	Align decimal points	2.5 + 3.75 = 6.25
Multiplication of Decimals	Multiply, count decimal places	2.5 × 0.4 = 1.00

5. Ratio and Proportion

Concept	Formula	Explanation
Ratio	a : b or a/b	Comparison of two quantities
Proportion	a : b = c : d or a/b = c/d	Equality of two ratios
Mean Proportional	b² = ac (if a : b = b : c)	b is mean proportional between a and c

6. Percentage

Concept	Formula	Explanation
Percentage	(Value/Total) × 100%	Parts per hundred
Converting Fraction to %	(Fraction) × 100%	Multiply fraction by 100
Converting % to Fraction	%/100	Divide percentage by 100
Percentage Increase	[(New – Old)/Old] × 100%	Change relative to original
Percentage Decrease	[(Old – New)/Old] × 100%	Decrease relative to original

7. Geometry – Basic Shapes

Shape	Perimeter	Area
Square	4 × side = 4s	side² = s²
Rectangle	2(length + breadth) = 2(l + b)	length × breadth = l × b
Triangle	a + b + c (sum of all sides)	½ × base × height = ½ × b × h
Circle	2πr (Circumference)	πr²

8. Mensuration (Basic)

Concept	Formula	Units
Perimeter	Sum of all sides	cm, m, km
Area	Space inside a shape	cm², m², km²
Volume	Space inside 3D object	cm³, m³, l

Class 7 Mathematics Formulas

1. Integers

Operation	Formula/Rule	Example
Addition (same sign)	Add and keep sign	(-5) + (-3) = -8
Addition (different sign)	Subtract and use larger sign	(-5) + 3 = -2
Multiplication	(+)(+) = +, (-)(-) = +, (+)(-) = –	(-3) × (-4) = 12
Division	Same rules as multiplication	(-12) ÷ (-3) = 4

2. Exponents and Powers

Concept	Formula	Explanation
Power	aⁿ = a × a × a … (n times)	a is base, n is exponent
Product Rule	aᵐ × aⁿ = aᵐ⁺ⁿ	Add exponents with same base
Quotient Rule	aᵐ ÷ aⁿ = aᵐ⁻ⁿ	Subtract exponents with same base
Power of Power	(aᵐ)ⁿ = aᵐⁿ	Multiply exponents
Zero Exponent	a⁰ = 1	Any number to power zero is 1
Negative Exponent	a⁻ⁿ = 1/aⁿ	Negative exponent means reciprocal

3. Algebraic Expressions

Identity	Formula	Expansion
Identity 1	(a + b)²	a² + 2ab + b²
Identity 2	(a – b)²	a² – 2ab + b²
Identity 3	(a + b)(a – b)	a² – b²
Identity 4	(x + a)(x + b)	x² + (a + b)x + ab

4. Simple Equations

Type	Standard Form	Solution Method
Linear Equation	ax + b = c	x = (c – b)/a
Transposition	Move term across = sign	Change sign when moving

5. Geometry – Lines and Angles

Concept	Formula/Property	Value
Straight Angle	–	180°
Right Angle	–	90°
Acute Angle	–	Less than 90°
Obtuse Angle	–	Between 90° and 180°
Complementary Angles	a + b = 90°	Sum is 90°
Supplementary Angles	a + b = 180°	Sum is 180°
Vertically Opposite Angles	Equal	∠1 = ∠3, ∠2 = ∠4
Linear Pair	a + b = 180°	Adjacent angles on straight line

6. Triangle Properties

Property	Formula	Explanation
Sum of Angles	∠A + ∠B + ∠C = 180°	Interior angles of triangle
Exterior Angle	Exterior angle = Sum of opposite interior angles	∠ACD = ∠A + ∠B
Pythagorean Theorem	a² + b² = c²	For right-angled triangles

7. Perimeter and Area (Extended)

Shape	Perimeter	Area
Parallelogram	2(a + b)	base × height = b × h
Rhombus	4 × side	½ × d₁ × d₂ (diagonals)
Trapezium	a + b + c + d	½(a + b) × h

8. Simple Interest

Concept	Formula	Variables
Simple Interest	SI = (P × R × T)/100	P = Principal, R = Rate, T = Time
Amount	A = P + SI	Total amount after interest
Principal	P = (SI × 100)/(R × T)	Original amount
Rate	R = (SI × 100)/(P × T)	Interest rate percentage
Time	T = (SI × 100)/(P × R)	Time period

Class 8 Mathematics Formulas

1. Rational Numbers

Property	Formula	Explanation
Rational Number	p/q where q ≠ 0	Can be expressed as fraction
Addition	a/b + c/d = (ad + bc)/bd	Common denominator method
Multiplication	a/b × c/d = ac/bd	Multiply numerators and denominators
Additive Identity	a/b + 0 = a/b	Zero is additive identity
Multiplicative Identity	a/b × 1 = a/b	One is multiplicative identity

2. Squares and Square Roots

Concept	Formula	Example
Perfect Square	n²	5² = 25
Square Root	√n	√25 = 5
Pythagorean Triplet	a² + b² = c²	3, 4, 5 or 5, 12, 13
Property 1	√(a × b) = √a × √b	√(4 × 9) = 2 × 3 = 6
Property 2	√(a/b) = √a/√b	√(25/4) = 5/2

3. Cubes and Cube Roots

Concept	Formula	Example
Perfect Cube	n³	3³ = 27
Cube Root	∛n	∛27 = 3
Sum of Cubes	a³ + b³ = (a + b)(a² – ab + b²)	–
Difference of Cubes	a³ – b³ = (a – b)(a² + ab + b²)	–

4. Algebraic Identities (Extended)

Identity	Formula	Expansion
Identity 5	(a + b)³	a³ + b³ + 3ab(a + b)
Identity 6	(a – b)³	a³ – b³ – 3ab(a – b)
Identity 7	a³ + b³	(a + b)(a² – ab + b²)
Identity 8	a³ – b³	(a – b)(a² + ab + b²)
Identity 9	(a + b + c)²	a² + b² + c² + 2ab + 2bc + 2ca

5. Direct and Inverse Proportion

Type	Formula	Relationship
Direct Proportion	x₁/y₁ = x₂/y₂ or x/y = k	If x increases, y increases
Inverse Proportion	x₁y₁ = x₂y₂ or xy = k	If x increases, y decreases

6. Compound Interest

Concept	Formula	Explanation
Amount	A = P(1 + R/100)ⁿ	Compounded annually
Compound Interest	CI = A – P	Difference between amount and principal
Half-Yearly Compounding	A = P(1 + R/200)²ⁿ	Compounded twice a year
Quarterly Compounding	A = P(1 + R/400)⁴ⁿ	Compounded four times a year

7. Mensuration – Surface Areas and Volumes

Shape	Total Surface Area	Volume
Cube	6a²	a³
Cuboid	2(lb + bh + hl)	l × b × h
Cylinder	2πr(r + h)	πr²h
Cone	πr(r + l) where l = slant height	⅓πr²h
Sphere	4πr²	⁴⁄₃πr³
Hemisphere	3πr²	⅔πr³

8. Quadrilaterals

Shape	Area Formula	Special Properties
Square	side² = a²	All sides equal, all angles 90°
Rectangle	length × breadth = l × b	Opposite sides equal, all angles 90°
Parallelogram	base × height = b × h	Opposite sides parallel and equal
Rhombus	½ × d₁ × d₂	All sides equal, diagonals perpendicular
Trapezium	½(a + b) × h	One pair of parallel sides

Class 9 Mathematics Formulas

1. Number System (Real Numbers)

Type	Definition	Examples
Natural Numbers (N)	Counting numbers	1, 2, 3, 4, …
Whole Numbers (W)	Natural numbers + 0	0, 1, 2, 3, …
Integers (Z)	Whole numbers + negatives	…, -2, -1, 0, 1, 2, …
Rational Numbers (Q)	p/q form where q ≠ 0	2/3, -5/7, 0.5
Irrational Numbers	Non-terminating, non-repeating	√2, π, e
Real Numbers (R)	Rational + Irrational	All numbers on number line

2. Polynomials

Concept	Formula	Degree
Linear	ax + b	1
Quadratic	ax² + bx + c	2
Cubic	ax³ + bx² + cx + d	3
Remainder Theorem	p(a) = remainder when p(x) divided by (x – a)	–
Factor Theorem	If p(a) = 0, then (x – a) is a factor	–

3. Algebraic Identities (Complete Set)

Identity	Formula
1	(a + b)² = a² + 2ab + b²
2	(a – b)² = a² – 2ab + b²
3	a² – b² = (a + b)(a – b)
4	(a + b)³ = a³ + b³ + 3ab(a + b)
5	(a – b)³ = a³ – b³ – 3ab(a – b)
6	a³ + b³ = (a + b)(a² – ab + b²)
7	a³ – b³ = (a – b)(a² + ab + b²)
8	(x + a)(x + b) = x² + (a + b)x + ab
9	(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca

4. Linear Equations in Two Variables

Concept	Formula	Explanation
Standard Form	ax + by + c = 0	Linear equation in two variables
Slope-Intercept Form	y = mx + c	m is slope, c is y-intercept
Point-Slope Form	y – y₁ = m(x – x₁)	Line through (x₁, y₁) with slope m
Two-Point Form	(y – y₁)/(y₂ – y₁) = (x – x₁)/(x₂ – x₁)	Line through two points

5. Coordinate Geometry

Concept	Formula	Explanation
Distance Formula	d = √[(x₂ – x₁)² + (y₂ – y₁)²]	Distance between two points
Section Formula	x = (mx₂ + nx₁)/(m + n), y = (my₂ + ny₁)/(m + n)	Point dividing line in ratio m:n
Midpoint Formula	x = (x₁ + x₂)/2, y = (y₁ + y₂)/2	Midpoint of line segment
Area of Triangle	½	x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)

6. Euclid’s Geometry

Concept	Statement
Euclid’s Division Lemma	a = bq + r where 0 ≤ r < b
Euclid’s Algorithm	HCF(a, b) = HCF(b, r) where a = bq + r

7. Triangles

Concept	Formula/Theorem
Heron’s Formula	Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Congruency (SSS)	Three sides equal
Congruency (SAS)	Two sides and included angle equal
Congruency (ASA)	Two angles and included side equal
Congruency (RHS)	Right angle, hypotenuse, one side equal

8. Circles

Concept	Formula	Explanation
Circumference	2πr	Perimeter of circle
Area	πr²	Space inside circle
Length of Arc	(θ/360°) × 2πr	θ is central angle
Area of Sector	(θ/360°) × πr²	Pie-shaped region
Area of Segment	Area of sector – Area of triangle	Region between chord and arc

9. Surface Areas and Volumes (Advanced)

Shape	Curved Surface Area	Total Surface Area	Volume
Sphere	4πr²	4πr²	⁴⁄₃πr³
Hemisphere	2πr²	3πr²	⅔πr³
Cone	πrl	πr(l + r)	⅓πr²h
Cylinder	2πrh	2πr(r + h)	πr²h
Frustum of Cone	πl(r₁ + r₂)	π[r₁² + r₂² + l(r₁ + r₂)]	⅓πh(r₁² + r₂² + r₁r₂)

10. Statistics

Concept	Formula	Explanation
Mean	x̄ = Σx/n	Average of observations
Median	Middle value when arranged	For odd n: (n+1)/2th term
Mode	Most frequently occurring value	Value with highest frequency
Range	Maximum – Minimum	Spread of data

11. Probability

Concept	Formula	Range
Probability	P(E) = (Number of favorable outcomes)/(Total number of outcomes)	0 ≤ P(E) ≤ 1
Certain Event	P(E) = 1	Always occurs
Impossible Event	P(E) = 0	Never occurs
Complementary Events	P(E) + P(not E) = 1	Sum equals 1

Class 10 Mathematics Formulas

1. Real Numbers (Advanced)

Concept	Formula/Theorem	Application
Fundamental Theorem of Arithmetic	Every composite number can be expressed as product of primes uniquely	Prime factorization
HCF × LCM	HCF(a,b) × LCM(a,b) = a × b	For two numbers
Rational Number Property	p/q is terminating if q = 2ᵐ × 5ⁿ	Decimal expansion

2. Quadratic Equations

Concept	Formula	Explanation
Standard Form	ax² + bx + c = 0	a ≠ 0
Quadratic Formula	x = [-b ± √(b² – 4ac)]/2a	Solutions of quadratic equation
Discriminant	D = b² – 4ac	Determines nature of roots
Nature of Roots (D > 0)	Two distinct real roots	Unequal roots
Nature of Roots (D = 0)	Two equal real roots	Equal roots
Nature of Roots (D < 0)	No real roots	Complex roots
Sum of Roots	α + β = -b/a	Relationship between roots and coefficients
Product of Roots	αβ = c/a	Relationship between roots and coefficients
Forming Equation	x² – (sum of roots)x + (product of roots) = 0	When roots are given

3. Arithmetic Progression (AP)

Concept	Formula	Variables
nth Term	aₙ = a + (n-1)d	a = first term, d = common difference
Sum of n Terms	Sₙ = n/2[2a + (n-1)d]	When first term and common difference known
Sum of n Terms (alternate)	Sₙ = n/2(a + l)	When first and last terms known
Common Difference	d = a₂ – a₁	Difference between consecutive terms

4. Coordinate Geometry (Advanced)

Concept	Formula	Application
Distance Formula	d = √[(x₂-x₁)² + (y₂-y₁)²]	Distance between two points
Section Formula (Internal)	P(x,y) = [(mx₂+nx₁)/(m+n), (my₂+ny₁)/(m+n)]	Point dividing line internally in m:n
Section Formula (External)	P(x,y) = [(mx₂-nx₁)/(m-n), (my₂-ny₁)/(m-n)]	Point dividing line externally in m:n
Midpoint Formula	M = [(x₁+x₂)/2, (y₁+y₂)/2]	Middle point of line segment
Area of Triangle	A = ½	x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)
Collinearity Condition	Area = 0	Three points are collinear if area = 0

5. Triangles (Similarity)

Concept	Theorem/Criteria	Statement
Basic Proportionality Theorem (Thales)	If a line parallel to one side intersects other two sides	It divides them proportionally
AA Similarity	Two angles equal	Triangles are similar
SSS Similarity	Three sides proportional	Triangles are similar
SAS Similarity	Two sides proportional and included angle equal	Triangles are similar
Pythagoras Theorem	a² + b² = c²	For right-angled triangles
Converse of Pythagoras	If a² + b² = c², then triangle is right-angled	–

6. Circles (Advanced)

Concept	Formula/Theorem	Explanation
Length of Tangent	√(d² – r²)	d = distance from center to external point
Tangent Property	Tangent ⊥ Radius at point of contact	Perpendicular relationship
Two Tangents from External Point	PA = PB	Equal lengths from external point
Angle between Tangent and Chord	Equal to angle in alternate segment	Alternate segment theorem

7. Trigonometry

Ratio	Formula	Reciprocal
sine	sin θ = Perpendicular/Hypotenuse	cosec θ = 1/sin θ
cosine	cos θ = Base/Hypotenuse	sec θ = 1/cos θ
tangent	tan θ = Perpendicular/Base	cot θ = 1/tan θ

8. Trigonometric Identities

Identity	Formula
Fundamental Identity	sin²θ + cos²θ = 1
Identity 2	1 + tan²θ = sec²θ
Identity 3	1 + cot²θ = cosec²θ

9. Trigonometric Ratios of Standard Angles

Angle	0°	30°	45°	60°	90°
sin θ	0	1/2	1/√2	√3/2	1
cos θ	1	√3/2	1/√2	1/2	0
tan θ	0	1/√3	1	√3	∞
cot θ	∞	√3	1	1/√3	0
sec θ	1	2/√3	√2	2	∞
cosec θ	∞	2	√2	2/√3	1

10. Trigonometric Ratios of Complementary Angles

Identity	Formula
sin(90° – θ)	cos θ
cos(90° – θ)	sin θ
tan(90° – θ)	cot θ
cot(90° – θ)	tan θ
sec(90° – θ)	cosec θ
cosec(90° – θ)	sec θ

11. Heights and Distances

Concept	Definition	Formula
Angle of Elevation	Angle above horizontal	tan θ = height/distance
Angle of Depression	Angle below horizontal	tan θ = height/distance

12. Statistics (Advanced)

Concept	Formula	For Grouped Data
Mean	x̄ = Σfx/Σf	Direct method
Mean (Assumed Mean)	x̄ = a + Σfd/Σf	Where d = x – a
Mean (Step Deviation)	x̄ = a + (Σfu/Σf) × h	Where u = (x-a)/h
Median	l + [(n/2 – cf)/f] × h	l = lower boundary, cf = cumulative frequency
Mode	l + [(f₁-f₀)/(2f₁-f₀-f₂)] × h	l = lower boundary of modal class

13. Probability (Advanced)

Concept	Formula	Explanation
Theoretical Probability	P(E) = Number of favorable outcomes / Total outcomes	Based on theory
Experimental Probability	P(E) = Number of times event occurred / Total trials	Based on experiment
Sum of Probabilities	P(E) + P(Ē) = 1	Complementary events

14. Surface Areas and Volumes (Combinations)

Concept	Formula	Application
Combination of Solids	Total Surface Area = Sum – Common Area	When two solids joined
Volume of Combination	Total Volume = Sum of individual volumes	Adding volumes
Conversion	1 m³ = 1000 liters	Volume to capacity

Important Constants

Constant	Symbol	Value (approx.)
Pi	π	3.14159 or 22/7
Euler’s Number	e	2.71828
Golden Ratio	φ	1.61803

Units and Conversions

Quantity	SI Unit	Common Conversions
Length	meter (m)	1 km = 1000 m, 1 m = 100 cm
Area	square meter (m²)	1 hectare = 10,000 m²
Volume	cubic meter (m³)	1 m³ = 1000 liters
Time	second (s)	1 hour = 3600 s
Speed	meter/second (m/s)	1 km/h = 5/18 m/s

Conclusion

This comprehensive guide covers all essential mathematics formulas from Class 6 to Class 10, organized systematically for easy reference. Regular practice with these formulas, combined with conceptual understanding, will build a strong mathematical foundation. Remember that formulas are tools understanding when and how to apply them is as important as memorizing them.

For best results, students should practice problems from textbooks, solve worksheets, and attempt sample papers regularly. This guide serves as a quick reference during revision and exam preparation.

Note: While this guide is comprehensive, students should always refer to their NCERT textbooks and consult teachers for detailed explanations and additional practice problems.

FAQs on Maths Formulas for Class 6 to 10

Q. What are the most important maths formulas for Class 6?

The most important maths formulas for Class 6 include:

Basic Arithmetic & Number System:

• Even numbers: 2n (where n is any integer)
• Odd numbers: 2n + 1

Fractions & Decimals:

• Addition of fractions: a/b + c/d = (ad + bc)/bd
• Multiplication of fractions: a/b × c/d = (a × c)/(b × d)

Ratio & Percentage:

• Percentage = (Value/Total) × 100%
• Ratio: a : b or a/b

Geometry – Area & Perimeter:

• Square: Perimeter = 4s, Area = s²
• Rectangle: Perimeter = 2(l + b), Area = l × b
• Triangle: Area = ½ × base × height
• Circle: Circumference = 2πr, Area = πr²

Q. How can I remember all maths formulas easily?

Here are proven techniques to memorize maths formulas effectively:

1. Understanding over Memorization:

• Learn how formulas are derived rather than rote learning
• Understand the logic behind each formula

2. Practice Daily:

• Solve 5-10 problems daily using different formulas
• Application helps retention better than reading

3. Create Visual Aids:

• Make colorful formula charts for your study room
• Use flashcards for quick revision
• Draw diagrams for geometry formulas

4. Use Mnemonics:

• For trigonometry: “Some People Have Curly Brown Hair Through Proper Brushing” (sin = P/H, cos = B/H, tan = P/B)
• Create your own memory tricks for complex formulas

5. Group Similar Formulas:

• Organize by topic (algebra, geometry, mensuration)
• Study related formulas together

6. Regular Revision:

• Review formulas weekly, not just before exams
• Use spaced repetition technique

7. Teach Others:

• Explain formulas to classmates or family members
• Teaching reinforces your own understanding

8. Make Formula Sheets:

• Create separate sheets for each chapter
• Keep them handy during homework and revision

Q. Where can I download maths formulas PDF for Class 6 to 10?

You can access maths formulas PDFs from several reliable sources:

Official & Educational Resources:

1. NCERT Official Website (www.ncert.nic.in)
   • Free textbooks with all formulas
   • Chapter-wise PDFs available
   • Most authentic source for CBSE students
2. School Websites
   • Many schools provide formula sheets
   • Check your school’s academic resources section
3. Educational Platforms:
   • BYJU’S, Khan Academy, Vedantu
   • Offer free downloadable formula PDFs
   • Often include solved examples
4. Government Educational Portals:
   • DIKSHA platform
   • State education board websites

What to Look For in a Good Formula PDF:

• Organized by class and chapter
• Clear explanations with examples
• Includes all NCERT topics
• Easy-to-read formatting
• Printable format

Q. What is the difference between Class 9 and Class 10 maths formulas?

Class 9 and Class 10 maths formulas differ significantly in complexity and application:

Class 9 Maths Formulas:

Foundation Building:

• Introduction to coordinate geometry (distance formula, section formula)
• Basic polynomial identities
• Heron’s formula for triangle area
• Simple surface area and volume formulas
• Basic trigonometric ratios (sin, cos, tan)
• Introduction to statistics (mean, median, mode)
• Basic probability concepts

Class 10 Maths Formulas:

Advanced Applications:

• Quadratic Equations: Complete formula including discriminant (D = b² – 4ac)
• Arithmetic Progressions: nth term and sum formulas
• Advanced Coordinate Geometry: Section formula (internal & external division)
• Trigonometry: All six ratios, identities (sin²θ + cos²θ = 1), complementary angles
• Standard trigonometric values: For 0°, 30°, 45°, 60°, 90°
• Advanced Statistics: Formulas for grouped data, median, mode calculations
• Circle Theorems: Tangent properties, length of tangent formula
• Similar Triangles: Proportionality theorems

Differences: