MATHEMATICS IMPORTANT FORMULAE CLASS - 9

Important Formulae for class 9 mathematics
Updated for new syllabus - 2026

10. Reflection in Coordinate Axes

Reflection in x-axis

$$ (x,y)\rightarrow(x,-y) $$

Reflection in y-axis

$$ (x,y)\rightarrow(-x,y) $$

Reflection in Origin

$$ (x,y)\rightarrow(-x,-y) $$

Important Formulae: Mathematics Class - 9
Chapter -2

Important Formulae

1. General Form of a Linear Polynomial

$$ P(x)=ax+b $$

where \(a\neq0\).

2. General Form of a Linear Equation

$$ ax+b=c $$

3. Value of a Linear Polynomial

If

$$ P(x)=ax+b $$

then for \(x=k\),

$$ P(k)=ak+b $$

4. General Form of a Linear Relationship

$$ y=ax+b $$

5. General Form of Linear Growth

$$ y=a+bn $$

6. General Form of Linear Decay

$$ y=a-bn $$

7. Linear Pattern Formula

$$ T_n=an+b $$

For the square tile pattern discussed in this chapter:

$$ T_n=2n-1 $$

Role of a and b in Linear Equations

1. In the Equation

$$ y=ax $$

• \(a\) is called the coefficient or slope. It determines how quickly the value of \(y\) changes when \(x\) changes.
• If \(a>0\), the graph rises from left to right.
• If \(a<0\), the graph falls from left to right.
• If \(|a|\) is larger, the line becomes steeper.
• Since there is no constant term, the graph always passes through the origin \((0,0)\).

2. In the Equation

$$ y=ax+b $$

• \(a\) is the coefficient or slope. It determines the rate of increase or decrease of the line.
• \(b\) is called the constant term or y-intercept.
• The value of \(b\) gives the value of \(y\) when \(x=0\).
• If \(b>0\), the graph cuts the y-axis above the origin.
• If \(b<0\), the graph cuts the y-axis below the origin.
• If \(b=0\), the equation becomes $$ y=ax $$ and the graph passes through the origin.

Summary

Important Formulae: Mathematics Class - 9
Chapter -3

Important Formulae

1. Set of Natural Numbers

$$ \mathbb{N}=\{1,2,3,4,\ldots\} $$

2. Set of Integers

$$ \mathbb{Z}=\{\ldots,-3,-2,-1,0,1,2,3,\ldots\} $$

3. Set of Rational Numbers

$$ \mathbb{Q}=\left\{\frac{p}{q}:p,q\in\mathbb{Z},\;q\neq0\right\} $$

4. Definition of Zero

$$ a-a=0 $$

5. Addition with Zero

$$ a+0=a $$

6. Subtraction with Zero

$$ a-0=a $$

7. Multiplication with Zero

$$ a\times0=0 $$

8. Equality of Two Rational Numbers

If $$ \frac{a}{b} \quad\text{and}\quad \frac{c}{d} $$ then $$ ad=bc $$

9. Addition of Rational Numbers

For the same denominator, $$ \frac{a}{b}+\frac{c}{b} = \frac{a+c}{b} $$

10. Subtraction of Rational Numbers

$$ \frac{a}{b}-\frac{c}{b} = \frac{a-c}{b} $$

11. Multiplication of Rational Numbers

$$ \frac{a}{b}\times\frac{c}{d} = \frac{ac}{bd} $$

12. Division of Rational Numbers

$$ \frac{a}{b}\div\frac{c}{d} = \frac{a}{b}\times\frac{d}{c} = \frac{ad}{bc} $$

13. Commutative Property of Addition

$$ \frac{a}{b}+\frac{c}{d} = \frac{c}{d}+\frac{a}{b} $$

14. Commutative Property of Multiplication

$$ \frac{a}{b}\times\frac{c}{d} = \frac{c}{d}\times\frac{a}{b} $$

15. Distributive Property

$$ p(q+r)=pq+pr $$

16. Absolute Value

$$ |x|\ge0 $$

17. Distance Between Two Rational Numbers

$$ \text{Distance}=|a-b| $$

18. Average of Two Rational Numbers

$$ \frac{a+b}{2} $$

19. Equivalent Rational Numbers

$$ \frac{a}{b} = \frac{ka}{kb}, \qquad k\neq0 $$

20. Absolute Value Examples

$$ |a|=a,\qquad a\ge0 $$

$$ |-a|=a $$

$$ |0|=0 $$

1. Pythagoras Theorem

In a right-angled triangle,

$$ (\text{Hypotenuse})^2=(\text{Perpendicular})^2+(\text{Base})^2 $$

or

$$ c^2=a^2+b^2 $$

2. Converse of Pythagoras Theorem

If the sides of a triangle satisfy

$$ c^2=a^2+b^2, $$

then the triangle is a right-angled triangle.

3. Terminating Decimal

A rational number

$$ \frac{p}{q} $$

is a terminating decimal if, after simplifying, the denominator has only the prime factors 2 and/or 5.

$$ q=2^m\times5^n $$

where \(m,n\ge0\).

Examples:

$$ \frac{1}{2}=0.5 $$

$$ \frac{3}{8}=0.375 $$

$$ \frac{7}{25}=0.28 $$

4. Non-Terminating Recurring Decimal

A rational number

$$ \frac{p}{q} $$

is a non-terminating recurring decimal if, after simplification, the denominator contains any prime factor other than 2 or 5.

Examples of such prime factors are:

$$ 3,\;7,\;11,\;13,\ldots $$

Examples:

$$ \frac{1}{3}=0.3333\ldots $$

$$ \frac{2}{7}=0.285714285714\ldots $$

$$ \frac{5}{6}=0.83333\ldots $$

5. Condition for Decimal Expansion

Important Formulae: Mathematics Class - 9
Chapter -4

Important Algebraic Identities

• $$ (x+y)^2=x^2+2xy+y^2 $$
• $$ (x-y)^2=x^2-2xy+y^2 $$
• $$ (x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx $$
• $$ (x+y)(x-y)=x^2-y^2 $$
• $$ (x+a)(x+b)=x^2+(a+b)x+ab $$
• $$ (ax+b)(cx+d)=acx^2+(ad+bc)x+bd $$
• $$ x^3-y^3=(x-y)(x^2+xy+y^2) $$
• $$ x^3+y^3=(x+y)(x^2-xy+y^2) $$
• $$ (x+y)^3=x^3+3x^2y+3xy^2+y^3 $$
• $$ (x-y)^3=x^3-3x^2y+3xy^2-y^3 $$
• $$ x^3+y^3+z^3-3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx) $$

Important Formulae: Mathematics Class - 9
Chapter -5

Definitions Related to Circles

Important Theorems of Circles

Theorem 1: Equal Chords Subtend Equal Angles at the Centre

In the same circle or in congruent circles, equal chords subtend equal angles at the centre.

Theorem 2: Equal Angles at the Centre Subtend Equal Chords

In the same circle or in congruent circles, if two angles subtended at the centre are equal, then the corresponding chords are equal.

Theorem 3: Perpendicular from the Centre to a Chord Bisects the Chord

The perpendicular drawn from the centre of a circle to a chord bisects the chord.

Theorem 4: Line Joining the Centre to the Midpoint of a Chord is Perpendicular to the Chord

The line joining the centre of a circle to the midpoint of a chord is perpendicular to the chord.

Theorem 5: Equal Chords are Equidistant from the Centre

In the same circle or in congruent circles, equal chords are equidistant from the centre.

Theorem 6: Chords Equidistant from the Centre are Equal

In the same circle or in congruent circles, chords that are equidistant from the centre are equal.

Theorem 7: Among Unequal Chords, the Longer Chord is Nearer to the Centre

In the same circle or in congruent circles, the longer of two unequal chords is nearer to the centre.

Theorem 8: Among Chords Unequally Distant from the Centre, the Nearer Chord is Longer

In the same circle or in congruent circles, the chord which is nearer to the centre is longer.

Theorem 9: Angle Subtended by an Arc at the Centre is Twice the Angle Subtended at the Remaining Part of the Circle

The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the remaining part of the circle.

Theorem 10: Angles in the Same Segment are Equal

Angles subtended by the same chord (or the same arc) at points on the same segment of a circle are equal.

Theorem 11: Angle in a Semicircle is a Right Angle

The angle subtended by a diameter at any point on the remaining part of the circle is always 90°.

Theorem 12: Opposite Angles of a Cyclic Quadrilateral are Supplementary

If the vertices of a quadrilateral lie on the same circle, then each pair of opposite angles is supplementary.

$$ \angle A+\angle C=180^\circ $$ $$ \angle B+\angle D=180^\circ $$

Theorem 13: Converse of the Cyclic Quadrilateral Theorem

If a pair of opposite angles of a quadrilateral are supplementary, then the four vertices of the quadrilateral lie on the same circle.

Theorem 14: Diameter is the Longest Chord of a Circle

The diameter is the longest chord of a circle.

Important Formulae: Mathematics Class - 9
Chapter -6

Important Formulae

1. Value of π

$$ \pi \approx \frac{22}{7} \approx 3.14 $$

For every circle,

$$ \pi=\frac{\text{Circumference}}{\text{Diameter}} $$

2. Circumference of a Circle

$$ C=2\pi r $$

3. Length of an Arc

$$ l=\frac{\theta}{360^\circ}\times2\pi r $$

where \( \theta \) is the central angle.

4. Area of a Triangle

$$ A=\frac{1}{2}\times\text{Base}\times\text{Height} $$

5. Heron's Formula

$$ A=\sqrt{s(s-a)(s-b)(s-c)} $$

where

$$ s=\frac{a+b+c}{2} $$

6. Area of a Circle

$$ A=\pi r^2 $$

7. Approximate Values of π

Archimedes:

$$ 3\frac{10}{71}<\pi<3\frac{1}{7} $$

Chongzhi:

$$ \pi\approx\frac{355}{113} $$

Aryabhata:

$$ \pi\approx3.1416 $$

Mādhava's Infinite Series:

$$ \pi=4\left(1-\frac13+\frac15-\frac17+\cdots\right) $$

8. π is an Irrational Number

The value of \( \pi \) is an irrational number. It cannot be expressed in the form

$$ \frac{p}{q} $$

where \(p\) and \(q\) are integers and \(q\neq0\).

9. Area of a Sector

$$ A=\frac{\theta}{360^\circ}\times\pi r^2 $$

where \( \theta \) is the central angle.

10. Brahmagupta's Formula

For a cyclic quadrilateral,

$$ A=\sqrt{(s-a)(s-b)(s-c)(s-d)} $$

where

$$ s=\frac{a+b+c+d}{2} $$

Important Formulae: Mathematics Class - 9
Chapter -7

Important Formulae

1. Probability of an Event

For any event \(E\),

$$ 0 \leq P(E) \leq 1 $$

2. Experimental Probability

$$ \text{Experimental Probability} = \frac{\text{Number of times the event occurred}} {\text{Total number of trials}} $$

or

$$ P(E)= \frac{\text{Number of times the event occurred}} {\text{Total number of trials}} $$

3. Theoretical Probability

For an event \(A\),

$$ P(A)= \frac{\text{Number of favourable outcomes}} {\text{Total number of possible outcomes}} $$

4. Sample Space

The sample space is represented by

$$ S=\{\text{Outcome}_1,\;\text{Outcome}_2,\;\text{Outcome}_3,\;\ldots,\;\text{Outcome}_n\} $$

5. Complementary Event

If \(A\) is an event, then its complement is denoted by

$$ A' $$

and

$$ P(A')=1-P(A) $$

6. Sum of Probabilities of an Event and its Complement

$$ P(A)+P(A')=1 $$

Important Concepts of Probability

1. Random Experiment

A random experiment is an experiment whose outcome cannot be predicted with certainty before it is performed, although all possible outcomes are known.

2. Outcome

An outcome is a possible result of a random experiment.

Example: When a die is rolled, getting 4 is an outcome.

3. Sample Space

The sample space is the set of all possible outcomes of a random experiment.

$$ S=\{\text{Outcome}_1,\text{Outcome}_2,\ldots,\text{Outcome}_n\} $$

Example:

$$ S=\{1,2,3,4,5,6\} $$

4. Event

An event is one or more outcomes selected from the sample space.

Example:

Getting an even number when a die is rolled:

$$ E=\{2,4,6\} $$

5. Equally Likely Outcomes

Outcomes having the same chance of occurring are called equally likely outcomes.

Example: Each face of a fair die has an equal chance of appearing.

6. Favourable Outcomes

The outcomes that satisfy a given event are called favourable outcomes.

7. Experimental Probability

Experimental probability is obtained by performing an experiment several times.

$$ P(E)= \frac{\text{Number of times the event occurred}} {\text{Total number of trials}} $$

8. Theoretical Probability

Theoretical probability is calculated using all possible outcomes without performing the experiment.

$$ P(E)= \frac{\text{Number of favourable outcomes}} {\text{Total number of possible outcomes}} $$

9. Impossible Event

An event that can never occur is called an impossible event.

$$ P(E)=0 $$

Example: Getting 7 on a standard die.

10. Certain Event

An event that always occurs is called a certain event.

$$ P(E)=1 $$

Example: Getting a number less than 7 on a standard die.

11. Complementary Event

If an event is denoted by \(A\), then the event "not A" is called its complement.

$$ A' $$

Its probability is

$$ P(A')=1-P(A) $$

12. Probability Range

The probability of any event always lies between 0 and 1.

$$ 0\le P(E)\le1 $$

13. Sum of Complementary Probabilities

$$ P(A)+P(A')=1 $$

14. Fair Experiment

A fair experiment is one in which every outcome has an equal chance of occurring.

Examples: Tossing a fair coin and rolling a fair die.

15. Coin Toss

Sample space:

$$ S=\{H,T\} $$

16. Rolling a Die

Sample space:

$$ S=\{1,2,3,4,5,6\} $$

17. Tossing Two Coins

Sample space:

$$ S=\{HH,HT,TH,TT\} $$

18. Types of Events

• Simple Event: An event having only one outcome.
• Compound Event: An event having more than one outcome.
• Impossible Event: Probability is 0.
• Certain Event: Probability is 1.

19. Tree Diagram

A tree diagram is a branching diagram used to list all possible outcomes of a random experiment. It helps in solving probability problems involving two or more stages.

20. Important Facts

• The probability of an event can never be negative.
• The probability of an event can never be greater than 1.
• The total probability of all possible outcomes of a random experiment is always equal to 1.
• Probability is used in games, weather forecasting, insurance, statistics, sports, and scientific research.

Important Formulae: Mathematics Class - 9
Chapter -8

Important Formulae

1. Sequence

A sequence is an ordered list of numbers. Each number in the sequence is called a term.

2. General (Explicit) Formula of a Sequence

An explicit formula gives the value of the nth term directly.

$$ t_n=f(n) $$

3. Recursive Formula

A recursive formula gives each term using one or more previous terms.

General form:

$$ t_n=f(t_{n-1}) $$

4. Triangular Numbers

Sequence:

$$ 1,\;3,\;6,\;10,\;15,\;\ldots $$

The nth triangular number is

$$ t_n=\frac{n(n+1)}{2} $$

5. Sum of First n Natural Numbers

$$ S_n=\frac{n(n+1)}{2} $$

6. Arithmetic Progression (AP)

An Arithmetic Progression (AP) is a sequence in which the difference between two consecutive terms is constant.

7. General Form of an AP

$$ a,\;a+d,\;a+2d,\;a+3d,\;\ldots $$

8. Common Difference

The common difference is

$$ d=a_2-a_1 $$

or

$$ d=a_n-a_{n-1} $$

9. nth Term of an AP

$$ t_n=a+(n-1)d $$

where

• \(a\) = First term
• \(d\) = Common difference
• \(n\) = Term number

10. Last Term of an AP

$$ l=a+(n-1)d $$

11. Geometric Progression (GP)

A Geometric Progression (GP) is a sequence in which each term is obtained by multiplying the previous term by a fixed number called the common ratio.

12. General Form of a GP

$$ a,\;ar,\;ar^2,\;ar^3,\;\ldots $$

13. Common Ratio

$$ r=\frac{a_2}{a_1} $$

or

$$ r=\frac{a_n}{a_{n-1}} $$

14. nth Term of a GP

$$ t_n=ar^{\,n-1} $$

where

• \(a\) = First term
• \(r\) = Common ratio
• \(n\) = Term number

15. Increasing AP

If

$$ d>0 $$

then the AP is increasing.

16. Decreasing AP

If

$$ d<0 $$

then the AP is decreasing.

17. Constant AP

If

$$ d=0 $$

then all the terms are equal.

18. Increasing GP

If

$$ r>1 $$

then the GP is generally increasing.

19. Decreasing GP

20. Constant GP

If r = 1 then all the terms are equal.

21. Important Symbols

• \(a\) = First term
• \(d\) = Common difference (AP)
• \(r\) = Common ratio (GP)
• \(t_n\) = nth term
• \(n\) = Position of the term
• \(l\) = Last term
• \(S_n\) = Sum of first n terms