**All Maths Formulas for Class 9**

If you are
searching for all Maths formulas of Class 9 at one place, then you are surely
come to the right place. Maths-formula.com brings you all the important
Maths formulas for class 9 to help you in your preparation for class 8
examination. These formulas are extremely important from the examinations’
point of view.

Maths is a subject
where reasoning and logic are very important. Students should have a clear
understanding of the underlying theories, concepts and formulas. They must
understand that what are the different formulas and what they mean actually.
Only then, they will be able to crack the mathematics questions asked in the
examination. Here, we are providing all the formulas chapter-wise for maths
class 9.

**Chapter-wise
Maths Formulas for Class 9**

**Maths Formulas
for Class 9 Chapter 1 Number System**

1. A number is called a rational number, if it
can be written in the form p/q ,
where p, q ∈ Z and q ≠ 0.

2. A number is called an irrational number, if it
cannot be written in the form p/q ,
where p, q ∈ Z and q ≠ 0.

3. The decimal expansion of a rational number is
either terminating or non-terminating (recurring). Moreover, a number whose
decimal expansion is terminating or non-terminating (recurring) is rational.

4. The decimal expansion of an irrational number
is non-terminating non-recurring. Moreover, a number whose decimal expansion is
non-terminating non-recurring is irrational.

5. The collection of rational and irrational
numbers is called the collection of real numbers.

6. If r is a rational number and s is an
irrational number, then r + s, r – s, r × s and r/s are irrational numbers, r ≠ 0.

7. For positive real
numbers a and b, the following identities hold true:

8. To rationalize the denominator of 1/(√a
+ b), we multiply this by (√a – b)/ (√a –
b), where a and b are integers.

9. Let a ˃ 0 be a real number and p and q
be rational numbers, then

i. *a ^{p}.a^{q} = a^{p + q}*

^{ }ii.

*(a*

^{p})^{q}= a^{pq}iii. *a ^{p}/a^{q} = a^{p – q} *iv.

*a*

^{p}b^{p}= (ab)^{p}**Maths Formulas
for Class 9 Chapter 2 Polynomials**

1. A polynomial p(x) in one variable x is an
algebraic expression in x of the form *p(x) = a _{n}x^{n} + a_{n
– 1}x^{n – 1} + . . . + a_{2}x^{2} + a_{1}x
+ a_{0}.*

Where
*a _{0}, a_{1}, a_{2}, . . ., a_{n}* are
constants and a

_{n}≠ 0.

*a _{0},
a_{1}, a_{2}, . . ., a_{n }*are respectively the coefficient of

*x*, and n is called the degree of the polynomial. Each of

^{0}, x, x^{2}, . . ., x^{n}*a*, with a

_{n}x^{n}+ a_{n – 1}x^{n – 1}+ . . . + a_{2}x^{2}+ a_{1}x + a_{0 }_{n}≠ 0, is called a term of the polynomial p(x).

2. A polynomial with one, two and three terms are
called monomial, binomial and trinomial, respectively.

3. A polynomial of degree one is called a linear
polynomial, a polynomial of degree two is called a quadratic polynomial and
polynomial of degree three is called a cubic polynomial.

4. A real number ‘a’ is a **zero of a polynomial
**p(x) if p(a) = 0. In this case, a is also called a **root of the equation**
p(x) = 0.

5. **Remainder Theorem: **If p(x) is any polynomial of degree greater
than or equal to 1 and p(x) is divided by the linear polynomial (x – a), then
the remainder is p(a).

6. **Factor Theorem: (**x – a) is a factor of the polynomial p(x), if
p(a) = 0. Also, if **(**x – a) is a factor of p(x), then p(a) = 0.

7. *(x + y + z) ^{2} = x^{2} + y^{2}
+ z^{2} + 2xy + 2yz + 2zx*

8. *(x + y) ^{3} = x^{3} + y^{3}
+ 3xy(x + y)*

9. *(x – y) ^{3} = x^{3} – y^{3}
– 3xy(x – y)*

10. *x ^{3} + y^{3} + z^{3}
– 3xyz = (x + y + z) (x^{2} + y^{2} + z^{2} – xy – yz –
zx)*

**Maths Formulas
for Class 9 Chapter 3 Coordinate Geometry**

1. The plane is called the Cartesian or
coordinate plane, and the lines are called the coordinate axes.

2. The horizontal line is called the x-axis and
the vertical line is called the y-axis.

3. The coordinate axes divide the plane into four
parts called quadrants.

4. The point of intersection of the axes is
called the origin. The coordinates of the origin are (0, 0).

5. The coordinates of a point are of the form (+,
+) in the first quadrant, (–, +) in the second quadrant, (–, –) in the third
quadrant and (+, –) in the fourth quadrant, where + denotes a positive real
number and – denotes a negative real number.

6. If x ≠ y, then (x, y) ≠ (y, x), and if x = y,
then (x, y) = (y, x).

**Maths Formulas
for Class 9 Chapter 4 Linear Equations in Two Variables**

1.
An
equation of the form *ax + by + c = 0*, where a, b and c are real numbers
such that a and b are not both zero, is called a linear equation in two
variables.

2.
A
linear equation in two variables has infinitely many solutions.

3.
The
graph of every linear equation in two variables is a straight line.

4.
The
equation of the x-axis is *y = 0* and the equation of the y-axis is *x =
0*.

5.
The
equation of a straight line parallel to y-axis is *x = a*.

6.
The
equation of a straight line parallel to x-axis is *y = a*.

7.
An
equation of the type *y = mx *represents a line passing through the
origin.

8. Every point on the graph of a linear equation
in two variables is a solution of the linear equation. Moreover, every solution
of the linear equation is a point on the graph of the linear equation.

**Maths Formulas
for Class 9 Chapter 5 Introduction to Euclid's Geometry**

1.
Axioms
or postulates are the assumptions which are obvious universal truths. They are
not proved.

2.
Theorems
are statements which are proved, using definitions, axioms, previously proved
statements and deductive reasoning.

3.
**Some
of Euclid’s axioms were:**

i.
Things
which are equal to the same thing are equal to one another.

ii.
If
equals are added to equals, the wholes are equal.

iii.
If
equals are subtracted from equals, the remainders are equal.

iv.
Things
which coincide with one another are equal to one another.

v.
The
whole is greater than the part.

vi.
Things
which are double of the same things are equal to one another.

vii.
Things
which are halves of the same things are equal to one another.

4. **Euclid’s postulates were:**

**Postulate
1: **A straight line
may be drawn from any one point to any other point.

**Postulate
2: **A terminated line
can be produced indefinitely.

**Postulate
3: **A circle can be
drawn with any center and any radius.

**Postulate
4: **All right angles
are equal to one another.

**Postulate
5: **If a straight
line falling on two straight lines makes the interior angles on the same side
of it taken together less than two right angles, then the two straight lines,
if produced indefinitely, meet on that side on which the sum of angles is less
than two right angles.

5. Two equivalent versions of Euclid’s fifth
postulate are:

i.
For
every line *l* and for every point P not lying on *l*, there exists a
unique line *m* passing through P and parallel to *l*.

ii.
Two
distinct intersecting lines cannot be parallel to the same line.

**Maths Formulas
for Class 9 Chapter 6 Lines and Angles**

1.
If
a ray stands on a line, then the sum of the two adjacent angles so formed is
180° and vice-versa. This property is called as the linear pair axiom.

2.
If
two lines intersect each other, then the vertically opposite angles are equal.

3.
If
a transversal intersects two parallel lines, then

i.
each
pair of corresponding angles is equal.

ii.
each
pair of alternate interior angles is equal.

iii.
each
pair of interior angles on the same side of the transversal is supplementary.

4. If a transversal intersects two lines such
that, either

i.
any
one pair of corresponding angles is equal, or

ii.
any
one pair of alternate interior angles is equal, or

iii.
any
one pair of interior angles on the same side of the transversal is
supplementary, then the lines are parallel.

5. Lines which are parallel to a given line are
parallel to each other.

6. The sum of the three angles of a triangle is
180°.

7. If a side of a triangle is produced, the
exterior angle so formed is equal to the sum of the two interior opposite
angles.

**Maths Formulas
for Class 9 Chapter 7 Triangles**

1. Two figures are said to be congruent, if they
are of the same shape and of the same size.

2. Two circles are said to be congruent, if they
have the same radii.

3. Two squares are said to be congruent, if they
have the equal sides.

4. If two triangles PQR and ABC are congruent
under the correspondence P ↔ A, Q ↔ B and R ↔ C, then symbolically, it is
expressed as ∆PQR ≅ ∆ABC.

5. **Theorem 1:** Two
triangles are congruent, if two sides and the included angled of one are
respectively equal to the two sides and the included angle of the other. This
is called SAS congruency condition.

6. **Theorem 2:** Two
triangles are congruent, if two angles and the included side of one are
respectively equal to the two angles and the included side of the other.

This is called ASA congruency condition.

7. **Theorem 3:** Two
triangles are congruent, if two angles and a side opposite to one angle of one
triangle are respectively equal to the two angles and a side opposite to one
angle of other triangle. This is called AAS congruency condition.

8. **Theorem 4:** Two
triangles are congruent, if the three sides of one triangle are respectively
equal to the three sides of the other triangle. This is called SSS congruency
condition.

9. **Theorem 5:** Two
right-angled triangles are congruent, if the hypotenuse and one side of one
triangle are respectively equal to the hypotenuse and one side of the other
triangle. This is called RHS congruency condition.

10. **Theorem 6: **Angles opposite
to equal sides of a triangle are equal.

11. **Theorem 7: **Sides opposite to
equal angles of a triangle are equal.

12. **Theorem 8: **In a triangle,
angle opposite to the longer side is greater.

13. **Theorem 9: **In a triangle,
side opposite to the greater angle is longer.

14. **Theorem 10: **Sum of any two
sides of a triangle is greater than the third side.

** **

**Maths Formulas
for Class 9 Chapter 8 Quadrilaterals**

1.
The
sum of the angles of a quadrilateral is 360°.

2.
**Properties
of a parallelogram:**

i.
A
diagonal of a parallelogram divides it into two congruent triangles.

ii.
The
opposite sides of a parallelogram are equal.

iii.
The
opposite angles of a parallelogram are equal.

iv.
The
diagonals of a parallelogram bisect each other.

3. A quadrilateral is a parallelogram, if its

i.
opposite
sides are equal, or

ii.
opposite
angles are equal, or

iii.
diagonals
bisect each other, or

iv.
a
pair of opposite sides is parallel and equal.

4. **Mid-point Theorem: **The line segment joining the mid-points of any
two sides of a triangle is parallel to the third side and is half of it.

5. **Converse of Mid-point Theorem: **The line drawn through the mid-point of one
side of a triangle, parallel to another side bisects the third side.

**Maths Formulas
for Class 9 Chapter 9 Area of Parallelograms and Triangles**

1.
Area
of a figure is the part of the plane enclosed by that figure.

2.
Two
congruent figures have equal areas but the converse need not be true.

3.
Parallelograms
on the same base (or equal base) and between the same parallels are equal in
area.

4.
The
area of a parallelogram = Base × Corresponding Height.

5.
Parallelograms
on the same base (or equal base) and having equal areas lie between the same
parallels.

6.
If
a parallelogram and a triangle are on the same base and between the same
parallels, then the area of the triangle is half the area of the parallelogram.

7.
Triangles
on the same base (or equal base) and between the same parallels are equal in
area.

8.
Area
of a triangle = ½ × Base × Perpendicular Height

9.
Triangles
on the same base (or equal base) and having equal areas lie between the same
parallels.

**Maths Formulas
for Class 9 Chapter 10 Circles**

1.
Equal
chords of a circle (or of congruent circles) subtend equal angles at the center
of the circle.

2.
If
the angles subtended by two chords of a circle (or of congruent circles) at the
center are equal, the chords are equal.

3.
The
perpendicular from the center of a circle to a chord bisects the chords.

4.
The
line drawn from the center of a circle to bisect a chord is perpendicular to
the chord.

5.
Equal
chords of a circle (or of congruent circles) are equidistant from the center of
the circle.

6.
Chords
equidistant from the center (or corresponding centers) of a circle (or of
congruent circles) are equal.

7.
If
two arcs of a circle are congruent, then their corresponding chords are equal and
conversely if the chords of a circle are equal, then their corresponding arcs
(minor and major) are congruent.

8.
Congruent
arcs of a circle subtend equal angles at the center.

9.
The
angle subtended by an arc at the center is double the angle subtended by it at
any point on the remaining part of the circle.

10. Angles in the same segment of a circle are
equal.

11. Angle in a semicircle is a right angle.

12. If a line segment joining two points subtends
equal angles at two other points lying on the same side of the line containing
the line segment, the four points lie on a circle.

13. The sum of a pair of opposite angles of a
cyclic quadrilateral is 180°.

14. If sum of a pair of opposite angles of a
quadrilateral is 180°, the quadrilateral is a cyclic quadrilateral.

**Maths Formulas
for Class 9 Chapter 11 Constructions**

1.
We
can construct a bisector of a given angle.

2.
We
can bisect the given line segment using ruler and compasses.

3.
We
can construct the angles of 30°, 45°, 60°, 90°, 120°, etc. using ruler and a
pair of compasses.

4.
We
can construct a triangle with different conditions.

**Maths Formulas
for Class 9 Chapter 12 Heron’s Formula**

1.
Area
of a triangle = ½ × base × height

3.
Just use this two steps process:

**Step 1:** Calculate **"s"** (half
of the triangles perimeter):

**Step
2:** Then calculate the **Area **using
the following formula:

**Maths Formulas
for Class 9 Chapter 13 Surface Area and Volume**

1.
Lateral
surface area of a cuboid = 2(l + b)h, where l, b, and h are the length,
breadth and height of a cuboid.

2.
Surface
area of a cuboid = 2(lb + bh + hl), where l, b, and h are the length,
breadth and height of a cuboid.

3.
Volume of
a cuboid = l × b × h, where l, b and h are the length, breadth and height of a
cuboid.

4.
Lateral
surface area of a cube = 4a^{2}, a is the side of the cube.

5.
Surface
area of a cube = 6a^{2}, a is the side of the cube.

6.
Volume of
a cube = a^{3}, a is the side of the cube.

7.
Lateral or
curved surface area of a cylinder = 2Ï€rh, r is the radius of circular base and
h is the height of the cylinder.

8.
Total surface
area of a cylinder = 2Ï€r(r + h), r is the radius of circular base and h is the
height of the cylinder.

9.
Volume of
a cylinder = Ï€r^{2}h, r is the radius of circular base and h is the
height of the cylinder.

10.
Lateral or
curved surface area of a cone = Ï€rL, r is the radius of the circular base,
L is the slant height of the cone.

11.
Surface
area of a cone = Ï€r(L + r), r is the radius of the circular base, l is the
slant height of the cone.

12.
Volume of
a cone = 1/3 Ï€r ^{2}h, r is the radius of the circular base, l is
the slant height of the cone.

13.
Surface
area of a sphere = 4Ï€r^{2}, r is the radius of the sphere.

14.
Surface
area of a hemisphere = 2Ï€r^{2}, r is the radius of the sphere.

15.
Total surface
area of a hemisphere = 3Ï€r^{2}, r is the radius of the sphere.

16.
Volume of
a sphere = 4/3 Ï€r^{3}, r is the radius of the sphere.

17.
Volume of
a hemisphere = 2/3 Ï€r^{3}, r is the radius of the sphere.

**Maths Formulas
for Class 9 Chapter 14 Statistics**

**Mean**

** **To calculate the mean of ungrouped data, we can use the following formulas:

*x*= Sum of all the individual values and n= Total number of items.

_{i}*f*) of distinct observations (

*x*) and multiply them to find

*fx*. Now, add all the values of

*fx*to find ∑

*f*. Again, add all the frequencies to get ∑

_{i}x_{i}*f*and use this formula to calculate the mean.

_{i }**Median**

*n***.**The next step is decided by whether the value of

*n*is even or odd.

**Mode**

- Dog: 80
- Cat: 25
- Duck: 11
- Parrot: 5
- Fish: 14

Here, the mode is "dog" since more families own a dog than any other animal. Note that the mode is always expressed as the category or score, not the frequency of that score. In the above example, the mode is "dog" not 80, which is the number of times dog appears.

**Maths Formulas
for Class 9 Chapter 15 Probability**

10. The probability of a sure event is always 1 and the probability of an impossible event is always 0.