Introduction
Algebraic Formulas list for Classes 6 to 12
Mathematics is one of the most important subjects for every student. Whether you are preparing for school examinations, Olympiads, scholarship exams, NTSE, Navodaya, Sainik School, or future competitive examinations like SSC, Banking, Railway, UPSC, JEE, and NEET, having a strong command over mathematical formulas is essential.
Many students lose marks not because they do not understand the concepts but because they forget the formulas during examinations. A well-organized formula sheet helps students revise quickly before exams and improves confidence while solving questions.
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Table of Contents
Fundamental Mathematics Formulas (Classes 6 & 7)
This comprehensive article has been prepared especially for students of Classes 6 to 12. Every important chapter is covered with easy-to-remember formulas arranged chapter-wise.
Chapter 1: Numbers and Number System
The Number System is the backbone of Mathematics. Every advanced topic begins with understanding numbers.
Even Numbers
Formula
2n
Example
2, 4, 6, 8, 10
Odd Numbers
Formula
2n + 1
Example
1, 3, 5, 7, 9
Sum of First n Natural Numbers
n(n+1)/2
Example
Sum of first 20 numbers
20 ร 21 รท 2 = 210
Sum of First n Even Numbers
n(n+1)
Example
First 5 even numbers
5 ร 6 = 30
Sum of First n Odd Numbers
nยฒ
Example
First 8 odd numbers
8ยฒ = 64
Square of a Number
Number ร Number
Example
18ยฒ = 324
Cube of a Number
Number ร Number ร Number
Example
7ยณ = 343
Prime Numbers
A prime number has exactly two factors:
- 1
- The number itself
Examples
2, 3, 5, 7, 11, 13
Composite Numbers
A composite number has more than two factors.
Examples
4, 6, 8, 10, 12
Chapter 2: Factors and Multiples
Highest Common Factor (HCF)
HCF = Greatest common factor of two or more numbers
Example
HCF (18, 24)
= 6
Least Common Multiple (LCM)
LCM = Smallest common multiple
Example
LCM (8,12)
=24
Relationship
LCM ร HCF = Product of the Numbers
Applicable for two numbers.
Divisibility Rules
Divisible by 2
Last digit is even.
Example
246
Divisible by 3
Sum of digits divisible by 3.
Example
327
3+2+7=12
12 divisible by 3
Divisible by 4
Last two digits divisible by 4.
Example
816
16 divisible by 4
Divisible by 5
Ends in
0 or 5
Divisible by 6
Divisible by both 2 and 3.
Divisible by 8
Last three digits divisible by 8.
Divisible by 9
Sum of digits divisible by 9.
Divisible by 10
Ends in 0.
Chapter 3: Fractions
A fraction represents a part of a whole.
General Form
Numerator / Denominator
Addition
a/b + c/d
= (ad + bc)/bd
Subtraction
a/b โ c/d
= (ad โ bc)/bd
Multiplication
a/b ร c/d
= ac/bd
Division
a/b รท c/d
= a/b ร d/c
Reciprocal
Reciprocal of
a/b
=
b/a
Equivalent Fractions
Multiply numerator and denominator by the same number.
Example
2/3
=
4/6
=
6/9
Chapter 4: Decimals
Fraction to Decimal
Divide numerator by denominator.
Example
3/4
=0.75
Decimal to Fraction
Example
0.25
=25/100
=1/4
Percentage to Decimal
Percentage รท100
Example
40%
=0.40
Decimal to Percentage
Decimal ร100
Example
0.75
=75%
Chapter 5: Ratio and Proportion
Ratio
Ratio
=a:b
=a/b
Equivalent Ratio
Multiply both numbers by the same value.
Example
2:5
=
4:10
=
6:15
Proportion
a:b = c:d
Cross multiplication
ad = bc
Unitary Method
Value of One Unit
=
Total Value รท Number of Units
Chapter 6: Percentage
Percentage
(Value / Total Value) ร100
Increase Percentage
(New โ Old)/Old ร100
Decrease Percentage
(Old โ New)/Old ร100
Percentage to Fraction
25%
=1/4
50%
=1/2
75%
=3/4
20%
=1/5
10%
=1/10
Chapter 7: Profit and Loss
Profit
Selling Price โ Cost Price
Loss
Cost Price โ Selling Price
Profit Percentage
Profit/Cost Price ร100
Loss Percentage
Loss/Cost Price ร100
Selling Price
Cost Price + Profit
or
Cost Price โ Loss
Chapter 8: Simple Interest
Simple Interest
SI
=(PรRรT)/100
Where
P = Principal
R = Rate
T = Time
Amount
Principal + Simple Interest
Chapter 9: Speed, Time and Distance
Speed
Distance รท Time
Distance
Speed ร Time
Time
Distance รท Speed
Unit Conversion
1 km
=1000 m
1 hour
=60 minutes
1 minute
=60 seconds
km/h to m/s
Multiply by
5/18
m/s to km/h
Multiply by
18/5
Chapter 10: Integers
Addition Rule
Same sign
Add numbers
Keep sign
Example
(+5)+(+3)=8
(-7)+(-4)=โ11
Different Signs
Subtract
Keep larger sign
Example
7+(โ5)=2
Multiplication Rules
Positive ร Positive = Positive
Negative ร Negative = Positive
Positive ร Negative = Negative
Negative ร Positive = Negative
Division Rules
Same as multiplication rules.
Chapter 11: Introduction to Algebra
Variable
Represented by letters
x, y, z
Constant
Fixed value
Example
10
25
100
Expression
3x+5
Equation
3x+5=20
Perimeter of Rectangle
2(l+b)
Perimeter of Square
4a
Algebraic Identity
(a+b)ยฒ
=aยฒ+2ab+bยฒ
Algebraic Identity
(aโb)ยฒ
=aยฒโ2ab+bยฒ
Difference of Squares
(a+b)(aโb)
=aยฒโbยฒ
Geometry, Mensuration, Algebra & Statistics Formulas (Classes 7โ9)
In this section, we cover the important formulas related to Geometry, Mensuration, Exponents, Algebra, Statistics, Coordinate Geometry, and Linear Equations. These topics form the backbone of Class 8 and Class 9 Mathematics and are frequently used in school examinations, Olympiads, NTSE, foundation courses, and competitive examinations.
Chapter 12: Lines and Angles
Understanding angles is essential for Geometry.
Types of Angles
- Acute Angle = Less than 90ยฐ
- Right Angle = 90ยฐ
- Obtuse Angle = Between 90ยฐ and 180ยฐ
- Straight Angle = 180ยฐ
- Reflex Angle = Between 180ยฐ and 360ยฐ
- Complete Angle = 360ยฐ
Complementary Angles
Sum = 90ยฐ
Example:
30ยฐ + 60ยฐ = 90ยฐ
Supplementary Angles
Sum = 180ยฐ
Example:
110ยฐ + 70ยฐ = 180ยฐ
Angles Around a Point
360ยฐ
Angles on a Straight Line
180ยฐ
Vertically Opposite Angles
Always Equal
Parallel Lines
When two parallel lines are cut by a transversal:
- Corresponding Angles = Equal
- Alternate Interior Angles = Equal
- Alternate Exterior Angles = Equal
- Co-interior Angles = 180ยฐ
Chapter 13: Triangles
Triangles are among the most important topics in Mathematics.
Sum of Interior Angles
180ยฐ
Exterior Angle Property
Exterior Angle = Sum of Opposite Interior Angles
Triangle Inequality
Sum of any two sides > Third side
Area of Triangle
Area = (1/2) ร Base ร Height
Equilateral Triangle
Perimeter = 3a
Area = (โ3/4)aยฒ
Isosceles Triangle
Two sides are equal.
Scalene Triangle
All sides are different.
Right Triangle (Pythagoras Theorem)
Hypotenuseยฒ = Baseยฒ + Perpendicularยฒ
or
cยฒ = aยฒ + bยฒ
Chapter 14: Quadrilaterals
Sum of Interior Angles
360ยฐ
Rectangle
Perimeter = 2(l + b)
Area = l ร b
Diagonal = โ(lยฒ + bยฒ)
Square
Perimeter = 4a
Area = aยฒ
Diagonal = aโ2
Parallelogram
Area = Base ร Height
Perimeter = 2(a + b)
Rhombus
Area = (1/2) ร dโ ร dโ
Perimeter = 4a
Trapezium
Area = (1/2)(a + b) ร h
Where
a and b are parallel sides.
Kite
Area = (1/2)dโdโ
Chapter 15: Circle
Circle formulas are frequently asked in examinations.
Circumference
2ฯr
Diameter
2r
Radius
Diameter รท 2
Area of Circle
ฯrยฒ
Semicircle
Area = (1/2)ฯrยฒ
Perimeter = ฯr + 2r
Arc Length
(ฮธ/360ยฐ) ร 2ฯr
Sector Area
(ฮธ/360ยฐ) ร ฯrยฒ
Chapter 16: Practical Geometry
Important properties:
- Radius of the same circle are equal.
- Diameter is the longest chord.
- Perpendicular from center bisects the chord.
- Equal chords subtend equal angles.
Chapter 17: Perimeter and Area
Rectangle
Area = Length ร Breadth
Perimeter = 2(l+b)
Square
Area = Sideยฒ
Perimeter = 4 ร Side
Triangle
Area = 1/2 ร Base ร Height
Parallelogram
Area = Base ร Height
Rhombus
Area = 1/2 ร dโ ร dโ
Trapezium
Area = 1/2 ร (Sum of Parallel Sides) ร Height
Chapter 18: Mensuration
Cube
Volume = aยณ
Total Surface Area = 6aยฒ
Lateral Surface Area = 4aยฒ
Diagonal = aโ3
Cuboid
Volume = l ร b ร h
Total Surface Area = 2(lb + bh + hl)
Lateral Surface Area = 2h(l+b)
Diagonal = โ(lยฒ+bยฒ+hยฒ)
Cylinder
Volume = ฯrยฒh
Curved Surface Area = 2ฯrh
Total Surface Area = 2ฯr(r+h)
Cone
Volume = (1/3)ฯrยฒh
Curved Surface Area = ฯrl
Total Surface Area = ฯr(r+l)
Slant Height
l = โ(rยฒ+hยฒ)
Sphere
Volume = (4/3)ฯrยณ
Surface Area = 4ฯrยฒ
Hemisphere
Volume = (2/3)ฯrยณ
Curved Surface Area = 2ฯrยฒ
Total Surface Area = 3ฯrยฒ
Chapter 19: Exponents and Powers
Product Rule
aแต ร aโฟ = aแตโบโฟ
Quotient Rule
aแต รท aโฟ = aแตโปโฟ
Power Rule
(aแต)โฟ = aแตโฟ
Product Power
(ab)โฟ = aโฟbโฟ
Quotient Power
(a/b)โฟ = aโฟ/bโฟ
Zero Exponent
aโฐ = 1
Negative Exponent
aโปโฟ = 1/aโฟ
Chapter 20: Squares and Square Roots
Square
nยฒ = n ร n
Square Root
โa ร โa = a
Perfect Squares
1ยฒ = 1
2ยฒ = 4
3ยฒ = 9
4ยฒ = 16
5ยฒ = 25
10ยฒ = 100
20ยฒ = 400
30ยฒ = 900
50ยฒ = 2500
100ยฒ = 10000
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Tips to Memorize Mathematics Formulas
- Understand the concept instead of memorizing formulas blindly.
- Prepare a chapter-wise formula notebook for quick revision.
- Revise formulas daily for 10โ15 minutes.
- Solve practice problems immediately after learning a formula.
- Create flashcards or sticky notes for frequently used formulas.
- Group similar formulas (such as area, volume, and trigonometric identities) to remember them more effectively.
- Practice previous yearsโ question papers to identify frequently tested formulas.
- Teach the formulas to someone else, as explaining concepts reinforces memory.
Recommended books for Students
Frequently Asked Questions (FAQs)
Q1. Why should students learn Mathematics formulas?
Mathematics formulas simplify calculations, improve problem-solving speed, and help students score higher in school and competitive examinations.
Q2. Which formulas are most important for board examinations?
Students should focus on:
- Algebraic Identities
- Quadratic Equations
- Trigonometric Ratios and Identities
- Coordinate Geometry
- Mensuration
- Calculus
- Probability
- Matrices and Determinants
Q3. Are these formulas useful for competitive examinations?
Yes. These formulas are useful for:
- JEE Main & Advanced
- CUET
- NDA
- SSC
- Banking Exams
- Railway Recruitment Exams
- Defence Exams
- State-level Entrance Tests
- Various Scholarship and Olympiad Examinations
Q4. How often should students revise formulas?
A quick revision of important formulas every day or at least 3โ4 times a week helps improve retention and boosts confidence before examinations.
Q5. Is understanding concepts more important than memorizing formulas?
Yes. Understanding how and why a formula works enables students to apply it correctly in different types of problems, making learning more effective than rote memorization.
Conclusion
Mathematics is built on concepts, and formulas are the tools that make solving problems efficient and accurate. A strong command of chapter-wise formulas from Classes 6 to 12 enables students to perform well in school examinations while also preparing them for higher education and competitive exams.
Instead of trying to memorize everything at once, revise formulas chapter by chapter, practice numerical problems regularly, and maintain your own personalized formula notebook. Consistent practice and conceptual understanding will significantly improve speed, accuracy, and confidence.
Keep this comprehensive formula guide handy for quick revision before tests and examinations. With regular practice, these formulas will become second nature and help you build a solid foundation for success in Mathematics.
