Class 8 Maths Formulas 2026–27 – New NCERT Ganita Prakash

NEW NCERT 2026–27 GANITA PRAKASH

Class 8 Maths Formulas

All 14 Chapters • Important Rules • Examples • Practice

A colourful, chapter-wise mathematics revision guide from Maths Wallah.

Part IPart IIInteractive Learning

Welcome to Maths Wallah! This guide organises important formulas, mathematical properties, rules and worked examples around the 14 chapters of the new NCERT Class 8 Mathematics textbook series, Ganita Prakash.

How to use this guide: Read each chapter, understand the formula, follow the solved example and then try the practice questions. Mathematics is easier when you understand why a rule works instead of only memorising it.

Syllabus note: This article follows the new book's chapter structure. Some chapters focus on reasoning, patterns, diagrams and constructions rather than a list of formulas. A few additional revision rules are included for learning support; check your prescribed textbook and teacher's instructions for the exact examination syllabus.

📚 Table of Contents

1. A Square and A Cube

Squares and cubes are formed by multiplying a number by itself two or three times. These ideas are useful for number patterns, geometry and understanding powers.

Square of a Number

n² = n × n

Example: 12² = 12 × 12 = 144.

Cube of a Number

n³ = n × n × n

Example: 4³ = 4 × 4 × 4 = 64.

Square Numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ...

These are the squares of 1, 2, 3, 4, 5, and so on.

Cube Numbers

1, 8, 27, 64, 125, 216, ...

These are the cubes of positive integers.

Useful Square Identities

(a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b² (a + b)(a − b) = a² − b²

These are useful algebraic extensions for understanding square patterns and mental calculation.

Example: 21² = (20 + 1)² = 400 + 40 + 1 = 441.

Recognising Perfect Squares and Cubes

A perfect square is an integer multiplied by itself. A perfect cube is an integer multiplied by itself three times.

144 = 12² 343 = 7³

Prime factorisation can help determine whether a positive integer is a perfect square or perfect cube.

2. Power Play

Powers are a short way to write repeated multiplication. In aⁿ, a is the base and n is the exponent.

Meaning of a Power

aⁿ = a × a × ... × a (n factors)

Example: 2⁴ = 2 × 2 × 2 × 2 = 16.

Multiplying Powers with the Same Base

aᵐ × aⁿ = aᵐ⁺ⁿ

Example: 2³ × 2² = 2⁵ = 32.

Dividing Powers with the Same Base

aᵐ ÷ aⁿ = aᵐ⁻ⁿ, for a ≠ 0 and m ≥ n

Example: 5⁴ ÷ 5² = 5² = 25.

Power Raised to a Power

(aᵐ)ⁿ = aᵐⁿ

Example: (3²)³ = 3⁶.

Power of a Product

(ab)ⁿ = aⁿbⁿ

Example: (2 × 3)² = 2² × 3² = 36.

Special Powers

a¹ = a a⁰ = 1, for a ≠ 0 10ⁿ = 1 followed by n zeros, for whole-number n ≥ 0

Example: 10⁵ = 100,000.

Scientific notation: Very large or small numbers can be written as a × 10ⁿ, where 1 ≤ |a| < 10 and n is an integer. For example, 3,500,000 = 3.5 × 10⁶.

3. A Story of Numbers

Number systems can be studied through place value, patterns, factors, divisibility and the structure of integers.

Place Value and Expanded Form

Number = Sum of the place values of its digits

Example: 53,482 = 50,000 + 3,000 + 400 + 80 + 2.

Divisibility Rules

By 2: The last digit is 0, 2, 4, 6 or 8.
By 3: The sum of the digits is divisible by 3.
By 5: The last digit is 0 or 5.
By 9: The sum of the digits is divisible by 9.
By 10: The last digit is 0.
By 4: The number formed by the last two digits is divisible by 4.

Prime Factorisation

Express a positive integer as a product of prime numbers.

84 = 2 × 2 × 3 × 7 = 2² × 3 × 7

Every integer greater than 1 has a unique prime factorisation, apart from the order of its factors.

Arithmetic Sequences

When a pattern increases or decreases by a fixed amount, it is an arithmetic sequence.

Next term = Previous term + Common difference aₙ = a₁ + (n − 1)d

The nth-term rule is a useful extension for a sequence with first term a₁ and common difference d.

Example: 4, 7, 10, 13, ... has common difference 3.

4. Quadrilaterals

A quadrilateral is a polygon with four sides. Its properties help us classify shapes and calculate missing angles.

Sum of Interior Angles

Interior angle sum of a quadrilateral = 360°

Example: Three angles are 80°, 90° and 100°. The fourth angle is 360° − 270° = 90°.

Parallelogram

Opposite sides are parallel and equal.
Opposite angles are equal.
Adjacent angles add up to 180°.
Diagonals bisect each other.

Rectangle

All angles are 90°.
Opposite sides are equal and parallel.
Diagonals are equal and bisect each other.

Rhombus

All four sides are equal.
Opposite angles are equal.
Diagonals bisect each other at right angles.

Square

All sides are equal.
All angles are 90°.
Diagonals are equal, perpendicular and bisect each other.

Interior Angle Sum of a Polygon

Interior angle sum = (n − 2) × 180°

This is a useful general geometry formula for a simple n-sided polygon.

Example: For a pentagon, the sum is (5 − 2) × 180° = 540°.

5. Number Play

Number play develops logical reasoning with digits, factors, multiples, patterns and numerical relationships.

Even and Odd Numbers

Even + Even = Even Odd + Odd = Even Even + Odd = Odd Odd × Odd = Odd Even × Any Integer = Even

Factors, Multiples and Common Factors

HCF = Highest common factor LCM = Least common positive multiple

Example: HCF(18, 24) = 6 and LCM(18, 24) = 72.

For two positive integers: HCF × LCM = Product of the integers

Square and Triangular Patterns

Square number = n² Triangular number = n(n + 1)/2

Triangular numbers: 1, 3, 6, 10, 15, 21, ...

Number puzzle strategy: Test whether a pattern depends on digit sums, divisibility, parity, multiplication, differences between terms or alternating rules. Explain why your answer works.

6. We Distribute, Yet Things Multiply

This chapter connects multiplication, distribution and algebraic expressions. The distributive property is one of the most useful tools for simplifying expressions.

Distributive Property

a(b + c) = ab + ac a(b − c) = ab − ac

Example: 7 × 23 = 7 × (20 + 3) = 140 + 21 = 161.

Expanding Brackets

3(x + 4) = 3x + 12 5(2x − 3) = 10x − 15

Multiply every term inside the bracket by the term outside it.

Taking Out a Common Factor

ab + ac = a(b + c) 6x + 12 = 6(x + 2)

Factoring reverses the distributive property.

Useful Algebraic Identities

(a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b² (a + b)(a − b) = a² − b²

Use these identities when appropriate, and verify expansions by multiplying each term.

7. Proportional Reasoning–1

Proportional reasoning compares quantities and helps solve problems involving equivalent ratios, scaling and rates.

Ratio

Ratio of a to b = a : b = a/b

Here b must be non-zero. Quantities compared in a ratio should be expressed in compatible units.

Example: 2 red pens and 3 blue pens give a red-to-blue ratio of 2 : 3.

Equivalent Ratios

a : b = ka : kb, for k ≠ 0

Example: 2 : 3 = 4 : 6 = 6 : 9.

Direct Proportion

When one quantity increases or decreases by a factor, the other changes by the same factor in direct proportion.

y = kx y/x = k, when x ≠ 0

Example: If 3 notebooks cost ₹90, then 5 notebooks at the same price cost 5 × ₹30 = ₹150.

Unit Rate

Unit rate = Total quantity ÷ Number of units

Example: 240 km in 4 hours gives an average rate of 60 km/h.

Direct proportion is not suitable for every real-world situation. Confirm that the quantities change at a constant rate before applying the rule.

8. Fractions in Disguise

Fractions can represent division, ratios and quantities. Equivalent fractions allow different-looking fractions to express the same value.

Equivalent Fractions

a/b = (a × k)/(b × k), for k ≠ 0

Example: 3/4 = 6/8 = 9/12.

Addition and Subtraction

For equal denominators:

a/c + b/c = (a + b)/c a/c − b/c = (a − b)/c

For different denominators, first find a common denominator.

Example: 2/3 + 1/4 = 8/12 + 3/12 = 11/12.

Multiplication

a/b × c/d = ac/bd

Example: 2/5 × 3/4 = 6/20 = 3/10.

Division

a/b ÷ c/d = a/b × d/c

The divisor must be non-zero. Example: 3/4 ÷ 1/2 = 3/4 × 2 = 3/2.

Fraction of a Quantity

a/b of N = (a × N)/b

Example: 3/5 of 40 = (3 × 40)/5 = 24.

Converting Fractions to Decimals

Divide the numerator by the denominator.

3/8 = 3 ÷ 8 = 0.375

Some fractions have terminating decimals; others have repeating decimal expansions.

9. The Baudhāyana–Pythagoras Theorem

The Pythagoras theorem applies to right-angled triangles. It relates the lengths of the two perpendicular sides to the hypotenuse.

Pythagoras Theorem

c² = a² + b²

Here a and b are the perpendicular sides and c is the hypotenuse, the side opposite the right angle.

Example: If a = 3 cm and b = 4 cm, then c² = 9 + 16 = 25, so c = 5 cm.

Finding a Missing Perpendicular Side

a² = c² − b² b² = c² − a²

Example: If the hypotenuse is 13 cm and one side is 5 cm, the other side is √(169 − 25) = √144 = 12 cm.

Recognising Pythagorean Triples

A set of positive integers satisfying a² + b² = c² is called a Pythagorean triple.

3² + 4² = 5² 5² + 12² = 13² 8² + 15² = 17²

Important: The theorem works for right-angled triangles. Do not apply it to an arbitrary triangle without first establishing that one angle is 90°.

10. Proportional Reasoning–2

This section extends proportional thinking to practical situations involving rates, scaling and percentages as useful mathematical tools.

Direct Proportion

y = kx

If x doubles, y doubles, provided the relationship is directly proportional.

Example: At ₹40 per kilogram, 3 kg costs ₹120 and 5 kg costs ₹200.

Inverse Proportion

In inverse proportion, the product of two non-zero quantities remains constant.

xy = k x₁y₁ = x₂y₂

Example: If 4 workers take 6 days for a job, 8 equally productive workers would take 3 days under the same ideal conditions.

Percentage

Percentage = (Part ÷ Whole) × 100% p% of N = (p/100) × N

Example: 15% of 200 = 15/100 × 200 = 30.

Percentage Increase and Decrease

Percentage change = (Change ÷ Original value) × 100%

Example: A price rises from ₹80 to ₹100. Increase = ₹20, so percentage increase = 20/80 × 100% = 25%.

Direct and inverse proportion describe different relationships. For workers and time, inverse proportion is valid only when the work rate per worker and the total work remain comparable.

11. Exploring Some Geometric Themes

Geometry uses properties of points, lines, angles and shapes to describe relationships and prove results.

Angle Facts

Angles on a straight line = 180° Angles around a point = 360° Vertically opposite angles are equal.

Example: If one angle at an intersection is 65°, its vertically opposite angle is also 65°.

Parallel Lines

When a transversal crosses parallel lines:

Corresponding angles are equal.
Alternate interior angles are equal.
Interior angles on the same side add up to 180°.

Triangle Facts

Interior angle sum of a triangle = 180° Exterior angle = Sum of the two opposite interior angles

Example: If the two opposite interior angles are 40° and 70°, the exterior angle is 110°.

Polygon Angle Sum

Interior angle sum = (n − 2) × 180°

For a hexagon: (6 − 2) × 180° = 720°.

For a regular polygon, each interior angle is (n − 2) × 180°/n.

12. Tales by Dots and Lines

Diagrams, dots, lines and geometric representations help us recognise patterns, relationships and mathematical structures.

Coordinate Pairs

A point on a coordinate grid can be represented as (x, y). Read the horizontal coordinate first, then the vertical coordinate.

(3, 2) means 3 units horizontally and 2 units vertically from the origin in the positive directions.

Patterns from Dots

Count the dots in successive figures and compare the differences to identify a possible pattern.

1, 3, 6, 10, 15, ...

These are triangular numbers, with nth term n(n + 1)/2.

Lines and Intersections

When lines intersect, vertically opposite angles are equal. When several lines meet, the angles around the common point total 360°.

Sum of angles around one point = 360°

The key goal in diagram-based problems is to identify what is given, what is unknown and which geometric relationship connects them. Draw clear diagrams and label the values.

13. Algebra Play

Algebra represents numbers using letters. It allows us to express general rules, simplify expressions and solve equations.

Combining Like Terms

3x + 5x = 8x 9a − 4a = 5a

Only like terms can be combined by adding or subtracting their coefficients.

Distributive Property

a(b + c) = ab + ac a(b − c) = ab − ac

Example: 4(x + 3) = 4x + 12.

Linear Equations

x + a = b → x = b − a x − a = b → x = b + a ax = b → x = b/a, for a ≠ 0 x/a = b → x = ab, for a ≠ 0

Example: 3x + 5 = 20. Subtract 5: 3x = 15. Divide by 3: x = 5.

Identities

(a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b² a² − b² = (a − b)(a + b)

These identities can help simplify and factorise algebraic expressions.

Check your solution: Substitute the value of the unknown back into the original equation. If both sides are equal, the solution satisfies the equation.

14. Area

Area measures the surface enclosed by a two-dimensional shape. Always use consistent units and write area in square units.

Square

Area = side² Perimeter = 4 × side

Example: Side 6 cm gives area 36 cm².

Rectangle

Area = length × breadth Perimeter = 2(length + breadth)

Example: 8 cm × 5 cm gives area 40 cm².

Triangle

Area = ½ × base × perpendicular height

Example: Base 10 cm and height 6 cm give area 30 cm².

Parallelogram

Area = base × perpendicular height

The height is the perpendicular distance between the parallel sides, not necessarily the slant side.

Trapezium

Area = ½ × (sum of parallel sides) × height

Example: Parallel sides 8 cm and 12 cm, height 5 cm: area = 50 cm².

Circle — Useful Revision

Circumference = 2πr Area = πr²

These are useful general geometry formulas; check whether circle area is included in your current chapter requirements.

Area of a Composite Shape

Total area = Sum of non-overlapping component areas

For a shape with a cut-out:

Remaining area = Outer area − Cut-out area

Example: A 10 cm × 8 cm rectangle has area 80 cm². If a 2 cm × 3 cm rectangle is removed, remaining area = 80 − 6 = 74 cm².

Units to remember: 1 m² = 10,000 cm². Area is measured in square units, whereas perimeter is measured in linear units such as cm or m.

📌 Quick Revision Table

Topic	Formula or Rule
Square number	n² = n × n
Cube number	n³ = n × n × n
Power product	aᵐ × aⁿ = aᵐ⁺ⁿ
Power quotient	aᵐ ÷ aⁿ = aᵐ⁻ⁿ, for a ≠ 0 and m ≥ n
Distributive property	a(b + c) = ab + ac
Triangle angle sum	180°
Quadrilateral angle sum	360°
Polygon interior angle sum	(n − 2) × 180°
HCF and LCM	HCF × LCM = product of two positive integers
Direct proportion	y = kx
Inverse proportion	xy = k
Fraction multiplication	a/b × c/d = ac/bd
Fraction division	a/b ÷ c/d = a/b × d/c
Pythagoras theorem	c² = a² + b²
Percentage	(Part ÷ Whole) × 100%
Linear equation	Perform the same valid operation on both sides
Square area	side²
Rectangle area	length × breadth
Triangle area	½ × base × perpendicular height
Parallelogram area	base × perpendicular height
Trapezium area	½ × sum of parallel sides × height

🧠 Interactive Practice Questions

Try each question before opening the answer.

Q1. Find 15².

Answer: 15 × 15 = 225.

Q2. Simplify 2³ × 2⁴.

Answer: 2⁷ = 128.

Q3. Calculate 3(x + 4) when x = 2.

Answer: 3(2 + 4) = 18.

Q4. Three angles of a quadrilateral are 70°, 80° and 110°. Find the fourth.

Answer: 360° − 260° = 100°.

Q5. Find the HCF of 24 and 36.

Answer: 12.

Q6. Calculate 2/3 + 1/6.

Answer: 4/6 + 1/6 = 5/6.

Q7. A right triangle has perpendicular sides 6 cm and 8 cm. Find the hypotenuse.

Answer: c² = 36 + 64 = 100, so c = 10 cm.

Q8. Find 25% of 240.

Answer: 25/100 × 240 = 60.

Q9. Solve 4x − 7 = 21.

Answer: 4x = 28, so x = 7.

Q10. Find the area of a parallelogram with base 12 cm and height 5 cm.

Answer: 12 × 5 = 60 cm².

Q11. Find the next term: 5, 9, 13, 17, ...

Answer: 21. Add 4 each time.

Q12. If 4 identical items cost ₹100, what do 7 items cost at the same rate?

Answer: One item costs ₹25; 7 items cost ₹175.

❓ Frequently Asked Questions

1. What is the name of the new NCERT Class 8 Maths book?

The new Class 8 Mathematics textbook series is called Ganita Prakash and is divided into Part I and Part II.

2. How many chapters are there in the new Class 8 Maths books?

The chapter sequence covered in this guide contains 7 chapters in Part I and 7 chapters in Part II, for a total of 14.

3. What is the most important formula in Class 8 Maths?

There is no single most important formula. Powers, distributive properties, fractions, proportional reasoning, the Pythagoras theorem and area formulas are important revision areas, depending on your prescribed chapters.

4. What is the Pythagoras theorem?

For a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the two perpendicular sides: c² = a² + b².

5. What is the distributive property?

a(b + c) = ab + ac. Multiply the term outside the brackets by each term inside the brackets.

6. What is the difference between direct and inverse proportion?

In direct proportion, y/x remains constant. In inverse proportion, xy remains constant. These rules apply only when the quantities follow the corresponding relationship.

7. Does every chapter contain formulas?

No. Some chapters mainly develop mathematical reasoning, diagrams, number patterns and geometric understanding. Learn the relevant methods and properties for these chapters.

8. Is this article enough for the NCERT examination?

Use this article as a revision companion. Read your complete prescribed textbook, solve its exercises and follow the topics specified by your teacher.

9. How can I remember formulas for longer?

Understand what each formula means, draw diagrams, practise examples without looking at the solution and revise the questions you get wrong.

10. Where can I access official NCERT books?

Visit the official NCERT textbook portal and select the relevant Class VIII Mathematics book and chapter.

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