Maths formulas for class 8 include the formulas from all the chapters in NCERT class 8 Maths book. Here you will find all the formulas in a PDF file, including Rational Number, Algebraic Expressions, Exponents and Powers, Mensuration, and other chapters.

These formulas for class 8 are beneficial while preparing for your exam. You can get the printout or read the formulas online. Please note that you need to practice more and more to memorize all the maths formulas. You can also review the essential maths formulas for upcoming classes (i.e., Class 9 and Class 10). We have prepared this formula PDF sheet based on the NCERT & CBSE syllabus. We will soon update Hindi medium study materials.

Page Contents

## Rational Numbers Maths Formulas for Class 8

In class 8, you will learn about another type of number. i.e., Rational Numbers. In this section, we have provided maths formulas for class 8 to study rational numbers. You will also find essential properties of rational numbers and other numbers.

If r = p/q, then r is a rational number.

Where p and q are integers and q≠0.

### Properties of Numbers

#### Closure

Number Type | Operation | Condition | Conclusion |

Addition | for two whole numbers a and b, a+b is also a whole number. | Whole numbers are closed under addition | |

Whole Numbers | Subtraction | a-b is not always a whole number (e.g., 4-6 = -2) | Whole number are not closed under subtraction |

Multiplication | 3×5 = 15, a whole number | Whole numbers are closed under multiplication | |

Division | 4÷6 = 2÷3, not a whole number | Whole numbers are not closed under division | |

Addition | for two integers a and b, a+b is also an integer | Closed under addition | |

Integers | Subtraction | 4-6 = -2, an integer | Closed under subtraction |

Multiplication | -3×5 = -15, an integer | Closed under multiplication | |

Division | 4÷6 = 2÷3, not a whole number | Not closed under division | |

Addition | (1/2) + (2/3) = 7/6, a rational number | Rational numbers are closed under addition | |

Rational Numbers | Subtraction | (5/2) – (1/2) = 2, a rational number | Closed under subtraction |

Multiplication | (3/2) x (1/5) = 3/10, a rational number | Closed under multiplication | |

Division | (2/5) ÷ 0, not defined | Rational numbers are not closed under division |

#### Commutativity

Number Type | Operation | Condition | Conclusion |

Addition | a + b = b + a | Addition is commutative | |

Whole Numbers | Subtraction | a – b ≠ b – a | Subtraction is not commutative |

Multiplication | a x b = b x a | Multiplication is commutative | |

Division | a ÷ b ≠ b ÷ a | Division is not commutative | |

Addition | a + b = b + a | Addition is commutative | |

Integers | Subtraction | a – b ≠ b – a | Subtraction is not commutative |

Multiplication | a x b = b x a | Multiplication is commutative | |

Division | a ÷ b ≠ b ÷ a | Division is not commutative | |

Adition | (-3/8) + (1/9) = 1/9 + (-3/8) | Addition is commutative for rational numbers | |

Rational Numbers | Subtraction | (1/2) – (2/5) ≠ (2/5) – (1/2) | Subtraction is not commutative for rational numbers |

Multiplication | (2/5) x (3/8) = (3/8) x (2/5) | Multiplication is commutative for rational numbers | |

Division | (2/5) ÷ (4/7) ≠ (4/7) ÷ (2/5) | Division is not commutative for rational numbers. |

#### Associativity for Whole Numbers & Integers

Operation | Condition | Conclusion |

Addition | (a + b) + c = a + (b + c) | Associative |

Subtraction | (a – b) – c ≠ a – (b – c) | Not Associative |

Multiplication | (a x b) x c = a x (b x c) | Associative |

Division | (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) | Not Associative |

#### Associativity of Rational Numbers

- Addition

Hence, Addition is associative for Rational Numbers.

- Subtraction

Subtraction is not associative for Rational Numbers.

- Multiplication

Multiplication is associative for rational numbers.

- Division

Division is not associative for rational numbers.

### Other Rational Number Properties

- 1 is the multiplicative identity of any rational number. i.e., for any rational number a,
- ax1 = 1xa = a

- For a rational number a/b, – a/b is the negative inverse or additive inverse. i.e., (a/b) + (-a/b) = 0
- For two rational number a/b and (c/d), if (a/b) x (c/d) = 1, then c/d is called reciprocal or multiplicative inverse of a/b.
- Rational number can be represented on the number line and there can be infinite number of rational numbers between two numbers.
- Two find the rational numbers between any two given number, convert the given numbers to rational number with same denominator and then find the numbers between them.

## Linear Equations in one Variable

To solve a linear equation having the variable only at one side of the equation such as,

7x + 3 = 6

Add the negative invers of the constant to both sides,

i.e., 7x + 3 – 3 = 6 – 3

or 7x = 3, or x = 3/7

If a linear equation is given as

6x + 9 = 2x – 6

Then, to solve the above equation follow the below steps:

Transpose the variable to one side and constant to another side,

i.e., 6x – 2x = –6–9 (sign will change)

or 4x = – 15 or x = –15/4

## Quadrilaterals Formulas for Class 8

Learn about different types of quadrilaterals and solve angle sum-related problems quickly with these maths formulas of class 8. Please make sure you practice more to memorize all the properties of Parallelogram, Rhombus, Square, and other quadrilaterals.

### Polygons

Number of sides | Classification | Sum of interior angles |

3 | Triangle | 180° |

4 | Quadrilateral | 360° |

5 | Pentagon | 540° |

6 | Hexagon | 720° |

7 | Heptagon | 900° |

8 | Octagon | 1080° |

9 | Nonagon | 1260° |

10 | Decagon | 1440° |

n | n-gon | (n-2) x 180° |

**Convex Polygon**: No portion of diagonal is outside the polygon**Concave Polygon**: Some part of the diagonal is outside the polygon

### Angle Sum Property

- Sum of interior angles of a polygon = (n – 2) x 180°, where n is the number of sides.

e.g., Sum of all interior angles of a triangle = (3 – 2) x 180 = 180°

Sum of all interior angles of a quadrilateral = (4 – 2) x 180 = 360°

- Sum of all exterior angles of a polygon = 360°
- Sum of angles of external and internal angles at any vertex = 180°

### Quadrilaterals Properties

Type of Quadrilateral | Properties |

Parallelogram (each pair of opposite sides is parallel) | 1. Opposite sides are equal 2. Opposite angles are equal 3. Diagonals bisect each other 4. Adjacent angles are supplementary (i.e., their sum is 180°) |

Rhombus (parallelogram with all sides of equal length) | 1. All the properties of parallelogram 2. Diagonals are perpendicular to each other |

Rectangle (Parallelogram with right angle) | 1. All properties of a parallelogram 2. Each angle is a right angle 3. Diagonals are equal |

Square (Rectangle with sides of equal length) | All the properties of Rhombus, Parallelogram, and Rectangle |

Kite (exactly two distinct consecutive pairs of sides of equal length) | 1. Diagonals are perpendicular to one another 2. One of the diagonals bisects the other 3. One pair of opposite angles are equal |

Trapezium | 1. One pair of opposite sides is parallel 2. If non-parallel sides are of equal length then it’s called Isosceles Trapezium |

## Data Handling & Graph

There are two chapters to understand the use of Graphs and data in class 8. Although there are not many formulas in this section, still you can learn the important terms to solve problems conveniently.

**Data:**Collected information with some value.**Pictograph:**Pictorial representation of data using pictures.**Raw Data:**Data in unorganized form**Bar Graph:**Display of information using bars of uniform width, and height proportional to the respective values.**Double Bar Graph**: A bar graph with two sets of data used for comparison.**Frequency:**Number of times a particular entry occurs in a data.**Grouped Frequency Distribution:**Representation and distribution of data on the basis of frequency.**Histogram:**Representation of data in a bar graph using class-intervals.**Pie Chart**: Data representation in circular form. The whole circle is divided in sectors and the angles for each sector can be found using the following method.- Find the fraction of specific type.

- Multiply the fraction with total angle of circle i.e., 360°.

- Draw a circle and mark the first angle (usually the biggest)

A line graph displays data that changes continuously over periods of time. A line graph which is a whole unbroken line is called a linear graph.

**Probability**

## Square and Square Roots

### Square number table

Number | Square | Number | Square | Number | Square |

1 | 1 | 8 | 64 | 15 | 225 |

2 | 4 | 9 | 81 | 16 | 256 |

3 | 3 | 10 | 100 | 17 | 289 |

4 | 16 | 11 | 121 | 18 | 324 |

5 | 25 | 12 | 144 | 19 | 361 |

6 | 36 | 13 | 169 | 20 | 400 |

7 | 49 | 14 | 196 | 21 | 441 |

- Every square number has 1,3,4,5,6,9,0 in its unit place.
- Square numbers can only have even numbers of zero at the end.
- Sum of first n odd numbers is n
^{2}. - The square number must be the sum of successive odd numbers starting from 1. i.e., after subtracting successive odd numbers from a square number, the result must be 0, otherwise the number is not perfect square.
- Product of two consecutive even or odd numbers = (a + 1)(a – 1) = (a
^{2}– 1)

### Non-Square numbers between two squares

- Between n
^{2}and (n+1)^{2}, there are 2n non-square numbers. e.g., between 5^{2}and 6^{2}, there are 2×5 = 10 non-square numbers. - Difference of two squares – 1 = total number of non-square numbers between those two square numbers.

### Pythagorean Triplet

If for any natural number m>1, (2m)^{2} + (m^{2} – 1)^{2} = (m^{2} + 1)^{2} then, 2m, m^{2} – 1, and m^{2} + 1 forms a Pythagorean triplet.

In Pythagorean triplets, sum of two square numbers = third square number

e.g., 3^{2} + 4^{2} = 5^{2} and 6^{2} + 8^{2} = 10^{2}

- Smallest 3-digit perfect square = 100
- Greatest 3-digit perfect square = 961
- Smallest 4-digit perfect square = 1024
- Greatest 4-digit perfect square = 9801

### Square Root Table

Square Number | Square Root | Square Number | Square Root |

1 | √1 = 1 | 64 | √64 = 8 |

4 | √4 = 2 | 81 | √81 = 9 |

9 | √9 = 3 | 100 | √100 = 10 |

16 | √16 = 4 | 121 | √121 = 11 |

25 | √25 = 5 | 144 | √144 = 12 |

36 | √36 = 6 | 169 | √169 = 13 |

49 | √49 = 7 | 196 | √196 = 14 |

Finding square roots using Repeated Subtraction

- Subtract the successive odd numbers starting from 1
- Repeat the process until you get 0

Count the total number of steps taken to reach 0 or total number of odd numbers subtracted. That is the square root of the number.

You can also find the square root of a number by prime factorization. Find the prime factors and then pair them with the same factors to find the square root of a number.

e.g., prime factorisation of 324 = 2x2x3x3x3x3 = (2×2) x (3×3) x (3×3) = 2^{2} x 3^{2} x 3^{2} = (2x3x3)^{2}

or √324 = 18

## Cube and Cube Root

### Cube Numbers

Number | Cube | Number | Cube | Number | Cube |

1 | 1 | 8 | 512 | 15 | 3375 |

2 | 8 | 9 | 729 | 16 | 4096 |

3 | 27 | 10 | 1000 | 17 | 4913 |

4 | 64 | 11 | 1331 | 18 | 5832 |

5 | 125 | 12 | 1728 | 19 | 6859 |

6 | 216 | 13 | 2197 | 20 | 8000 |

7 | 343 | 14 | 2744 | 21 | 9261 |

- Cube of an even number is always even
- Cube of an odd number is always odd
- There are only 10 perfect cubes from 1 to 1000
- Each prime factor appears three times in a cube

### Cube Roots

The symbol ^{3}√ denotes ‘cube-root.’

Cube Number | Cube Root | Cube Number | Cube Root |

1 | ^{3}√1 = 1 | 216 | ^{3}√216 = 6 |

8 | ^{3}√8 = 2 | 343 | ^{3}√343 = 7 |

27 | ^{3}√27 = 3 | 512 | ^{3}√512 = 8 |

64 | ^{3}√64 = 4 | 729 | ^{3}√729 = 9 |

125 | ^{3}√125 = 5 | 1000 | ^{3}√1000 = 10 |

## Comparing Quantities

### Ratio

Comparing two quantities by using fractions, such as

= p:q or ‘p is to q’, where p and q are different quantities. e.g., ratio of 25 mangoes to 40 apples will be, 25/40 = 5/8 = 5:8

### Percentage

P% = p/100

Percentage | Value | Percentage | Value |

10% | 10/100 = 1/10 | 60% | 60/100 = 3/5 |

20% | 20/100 = 1/5 | 50% | 50/100 = 1/2 |

### Discounts

Discount = Marked price – Sale price

e.g., Marked Price = **₹2,000, Sale price = 1,500, Discount = 2000 – 1500 = ₹500.**

**Cost and Profit**

**Overhead charge:** expanses added to the buying or selling price. E.g., repair cost, labor cost, etc.

#### Profit

When Cost price < Selling price, CP<SP

Profit (P) = SP – CP

- Cost price may include additional expanses

#### Loss

When Cost Price > Selling Price, CP > SP

Loss (L) = SP – CP

#### Sales Tax (Value Added Tax)

Collected by the shopkeeper and transferred to the government. Tax is always applicable on the selling price. E.g., if the selling price of a chair is **₹**800 and 10% tax is added to it,

## Simple Interest and Compound Interest

Simple Interest (SI) = P x r x t, where p = principal amount, r = rate of interest, t = time

#### Compound Interest

Interest calculated on the previous year’s amount, A = P + I

Amount when interest is compounded half-yearly,

## Algebraic Expressions & Identities

Expressions: Formed using variables and constants

Terms: Added to form expressions. Terms are the product of factors

e.g., for an algebraic expression, x^{2} + 2xy + 9, x = variable, numbers = constants, X^{2}, 2xy, 9 = terms

**Coefficient:** Numerical factor of a term is called a coefficient.

**Monomial:**Expression with only one term**Binomial:**Expression with two terms**Trinomials:**Expression with three terms

Monomial, binomial, and trinomial are also called polynomials.

Like Terms: Terms with the same variable are called like terms. E.g., 2xy and 8yx are like terms, 5x^{2} and 6x^{2} are like terms.

5x^{2} and 2xy are not like terms, 9x and 9x^{2} are not like terms.

Addition and subtraction is performed on the coefficient of like terms. E.g., 2xy – xy = xy

Similarly, 9x^{2} + 4x^{2} = 13x^{2}

### Multiplication

- Coefficients of product = coefficient of first term x coefficient of second term
- Algebraic facto or product = algebraic factor of first term x algebraic factor of second term.

e.g., 9x^{2} . 4x = 9.4.x.x.x = 36x^{3} where . ‘dot’ represent the product.

- (a + b).c = ac + bc
- (a + b)(c + d) = ac + ad + bc + bd or a(c + d) + b(c + d), using the distributive law
- (a + b) (c
^{2}+ cd + d) = a(c^{2}+ cd + d) + b(c^{2}+ cd + d), using the distributive law

### Algebraic Identities

- (a + b)
^{2}= a^{2}+ b^{2}+ 2ab - (a – b)
^{2}= a^{2}+ b^{2}– 2ab - (a + b)(a – b) = a
^{2}– b^{2} - (x + a)(x + b) = x
^{2}+ (a + b)x + ab

## Mensuration Maths Formulas For Class 8

Here you will find area and volume formulas for class 8 and learn about solving various 2D and 3D (solid) shapes. These formulas are essential and you need to put more efforts so that you can solve questions without making any calculation mistakes. Download all mensuration formulas for class 8 and take a printout for the easy access to all formulas.

### 2D Shapes

Shape | Area | Perimeter |

Rectangle | a x b | 2(a + b) |

Square | a x a = a^{2} | 4a |

Triangle | (1/2) b x h (b = base, h = height) | a + b + c (a,b,c are sides) |

Parallelogram | b x h | 2 (s + b) |

Circle | πr2 | 2πr |

where d_{1}, d_{2} are diagonals of rhombus.

### Solid Shapes (3D)

#### Cuboid

l, b, h are the edges of a cuboid

- Surface area of a cuboid = 2(lb + bh + hl)
- Perimeter of cuboid = 4(l + b + h)
- Longest diagonal length in a cuboid = √(l
^{2}+ b^{2}+ h^{2}) - Volume of a cuboid = l x b x h

#### Cube

- Surface area of a cube = 6a
^{2} - Perimeter of a cube = 12a
- Volume of cube = a
^{3} - Diagonal of cube = √(3a)

#### Cylinder

- Curved surface area of a cylinder = 2πrh
- Total surface area of cylinder = 2πr(r + h)
- Volume of cylinder = πr
^{2}h - Area of two circular bases in a cylinder = πr
^{2}+ πr^{2}= 2πr^{2}

## Exponents and Powers

- a
^{m}x a^{n}= a^{m+n} - a
^{m}÷ a^{n}= a^{m-n} - (a
^{m})^{n}= a^{mn} - a
^{m}x b^{m}= (ab)^{m} - a
^{0}= 1 - a
^{m}/ b^{m}= (a/b)^{m} - a
^{-m}= 1/a^{m} - a
^{m}= 1/a^{-m}

## Direct and Inverse Proportions

x and y are in direct proportion if (x/y) = k, or x = ky, where k is any constant.

x and y are inversely proportional if xy = k,

## Test for Divisibility of Numbers

You have already learned about various divisibility rules in the previous class. Now is the time to revise those again. How many of the divisibility rules do you remember?

### Divisibility by 10

A number having 0 at the end is divisible by 10. i.e., 10,20,900,5640, are divisible by 10.

### Divisibility by 2

A number having 0,2,4,6, or 8 in its one’s place is divisible by 2. E.g., 24,2596, 4590, 5788 are divisible by 2.

### Divisibility by 3

If the sum of digits in a number is multiple of 3 then the number is divisible by 3.

E.g., 729 has digits sum as 7+2+9 = 18 which is a multiple of 3. Hence, 729 is divisible by 3.

### Divisibility by 4

A number having 3 or more than 3 digits is divisible by 4 if the number formed by the last two digits (i.e., ones and tens) is divisible by 4.

E.g., 9928 is divisible by 4 because 28 is divisible by 4. Similarly, 316 is divisible by 4 because 16 is divisible by 4.

### Divisibility by 5

If a number has 0 or 5 at one’s place, then the number is divisible by 5.

e.g., 255, 1250, 300, 995 are divisible by 5.

### Divisibility by 6

If a number is divisible by both 2 and 3, then it is also divisible by 6.

e.g., 216 is divisible by 2 and 3. Hence, it is also divisible by 6.

### Divisibility by 7

A number is divisible by 7 if the difference between twice of unit digit and the remaining part of the number is divisible by 7.

e.g., suppose we have 343. Now, twice of the unit digit, 3×2 = 6.

Difference between remaining part and twice of the unit digit, 34 – 6 = 28, which is divisible by 7. Hence, 343 is also divisible by 7.

Another example with, 819. Twice of the unit digit, 9×2 = 18.

The difference, 81-18 = 63, which is divisible by 7. Hence, 819 is also divisible by 7.

### Divisibility by 8

A number with 4 or more digits is divisible by 8 if the number formed by the last 3 digits is divisible by 8.

e.g., 8216 is divisible by 8 because 216 is also divisible by 8.

### Divisibility by 9

It is similar to the divisibility by 3 rule. If the sum of digits in a number is multiple of 9, then the number is divisible by 9.

e.g., 4608 is divisibly by 9 because the sum (4+6+0+8 = 18) is multiple of 9.

### Divisibility by 11

Follow the steps below to test the divisibility by 11 for any number:

- From the right, find sum of digits at odd place (ones, hundred, ten thousand, etc.)
- From the right, find sum of digits at even place (ten, thousand, lakh, etc.)
- Find the difference between both sums.
- If the difference is 0 or divisible by 11, then the number is also divisible by 11.

e.g., suppose we have 61809. Now,

the sum of odd place digits, 9+8+6 = 23, the sum of even place digits, 0+1=1.

The difference between both sums, 23-1 = 22, which is a multiple of 11, Hence 61809 is divisible by 11.

## Numbers in General Form

- Two digit number, ab = 10a + b
- Three digit number, abc = 100a + 10b + c
- Four digit number, abcd = 1000a + 100b + 10c + d
- Five digit number, abcde = 10000a + 1000b + 100c + 10d + e

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