# Maths Formulas for Class 9 PDF

Here you will find all the maths formulas for class 9. You can also download these formulas of class 9 in a PDF file and take the printout for fast practice. There are fifteen chapters in NCERT class 9 Maths book. We have included all the formulas for these chapters, including Numbers Systems, Polynomials, Linear Equations, Triangles & Quadrilaterals, Area & Volume, and other vital formulas.

The syllabus of maths for class 9 and class 10 is the same in most topics. But, you will learn the basics in class 9, and things will get a bit complex in class 10. If you are preparing for NTSE or STSE examination, you should clarify the basic concepts in mathematics. So, get all the maths formulas for the class 9 PDF from the download link below and start your preparation today.

Page Contents

## Number Systems Formulas for Class 9

You have already studied about number system in previous classes. Number systems Maths formulas for class 9 include important properties and formulas for Rational Numbers and Real Numbers. You have studied the rational numbers in class 8.

### Rational Number (r)

r = $$\frac{p}{q}$$

where p and q are integers and q≠0.

### Equivalent Rational Number (Fractions)

$$\frac{p}{q}$$ = $$\frac{1}{2}$$ = $$\frac{4}{8}$$ = $$\frac{5}{10}$$ = $$\frac{10}{20}$$ = $$\frac{26}{52}$$ , every number is equivalent fraction of $$\frac{1}{2}$$.
• Every integer is a rational number
• Not all rational numbers are integers
• There can be unlimited (infinite) numbers of rational number between any two given rational numbers.

### Irrational Number (s)

s ≠ $$\frac{p}{q}$$ where, p and q are integers and q≠0. e.g., $$\sqrt{2}$$ is not rational number as it cannot be expressed into p/q form.

### Real Numbers (R)

• Collection of rational numbers and irrational numbers
• Every point on the number line represents a unique real number

#### Decimal Expansion of Real Numbers

• A number is rational number if its decimal expansion is terminating or non-terminating recurring.
• A number is irrational number if its decimal expansion is non-terminating non-recurring.

### Real Number Properties

If r is rational and s is irrational, then r+s and r-s are irrational, and rs and r/s are irrational r≠0.

For the positive real numbers a and b:

1. $$\sqrt{\text{ab}}$$ = $$\sqrt{a}\sqrt{b}$$
2. $$\sqrt{\frac{a}{b}}$$ = $$\frac{\sqrt{a}}{\sqrt{b}}$$
3.$$\left( \sqrt{a} – \sqrt{b} \right)\left( \sqrt{a} + \sqrt{b} \right)\$$= $$a – b$$
4.$$\left( a + \sqrt{b} \right)\left( a – \sqrt{b} \right)$$ = $$a^{2} – b$$
5.$$\left( \sqrt{a} + \sqrt{b} \right)$$^{2} = $$a + 2\sqrt{\text{ab}} + b$$

Let a > 0 be a real number and p and q be rational numbers. Then,

• ap.aq = ap+q
• (ap)q = apq
• apbp = (ab)p
• ap / aq = ap-q
To rationalize the denominators of $$\frac{1}{\sqrt{a} + b}$$ we multiply this by $$\frac{\sqrt{a} – b}{\sqrt{a} – b}$$ , where a and b are integers.

## Polynomials Formulas For Class 9

Here you will find Polynomials maths formulas for class 9 and learn about different types of polynomials. You should know how to solve the polynomials and find the zeroes and degree of these polynomials. You will also find algebraic identities in this section.

Degree of polynomials = highest power of the variable. E.g., x2+5x3 + 4x5 – 2 has 5 degrees of polynomials as it is the highest power of the variable x.

• Linear polynomials: Polynomials of degree 1, i.e., 2x-3
• Quadric Polynomials: Polynomials of degree 2, i.e., 3x2-4x+7
• Cubic Polynomials: Polynomials of degree 3, i.e., 4x3+3x2+2x+9
• Monomial: Polynomial with one term
• Binomial: Polynomial with two terms
• Trinomial: Polynomial with three terms
• Degree of non-zero constant polynomial is zero, e.g., degree of 2 is 0.
• Degree of zero polynomial is not defined. i.e., degree of 0 is no

### Zeroes of polynomials

a zero of a polynomial p(x) is a number c such that p(c) = 0.

e.g., for a polynomial p(x) = x2 – 1,  p(1) = 12 – 1 = 0. Hence, 1 is the zero of given polynomial.

We can find the zeros of a polynomial by equating them to zero. Hence, 1 in the above example is also the root of the polynomial equation, x2 – 1 = 0

• A non-zero constant polynomial has no zero
• Every real number is a zero of zero polynomial

### Remainder Theorem

Let p(x) be any polynomial of degree ≥ 1, and let a be any real number. If p(x) is divided by the linear polynomial x – a, then the remainder is p(a).

If p(x) and g(x) are two polynomials such that degree of p(x) ≥ degree of g(x) and g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that:

p(x) = g(x)q(x) + r(x),

where r(x) = 0 or degree of r(x) < degree of g(x). Here we say that p(x) divided by g(x), gives q(x) as quotient and r(x) as remainder.

### 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.

### Algebraic Identities

These are some algebraic maths formulas for class 9. You will study more of these formulas in next classes. However, if you want you can get more algebraic formulas by clicking here.

## Coordinate Geometry

To locate an object or a point in a plane (Cartesian, Coordinate Plane), we require two perpendicular lines (Horizontal and Vertical). These lines are called coordinate axes.

• x-axis: The horizontal line of coordinate plane
• y-axis: The vertical line in the coordinate plane
• Quadrants: Coordinate axes divides the plane in four parts. These are called quadrants.
• Origin: Point of intersection of axes.
• x-coordinate: distance of a point from y-axis
• y-coordinate: distance of a point from x-axis
• Coordinates of a point: distance from y-axis and x-axis. It is denoted as (x,y). where x = x-coordinate and y = y-coordinate
• Coordinates of point on the x-axis = (x,0)
• Coordinates of point on the y-axis = (0,y)
• Coordinates of origin = (0,0)

If x ≠ y, then (x, y) ≠ (y, x)
If x = y, then (x, y) = (y, x)

## Lines, Angles, Triangles, and Quadrilaterals

Here are some other maths formulas for class 9 that include Lines, Angles, Triangles, and Quadrilaterals. You need to learn the properties and theorems alongside the formulas while practicing the geometrical problems. For example, you should know the difference between various quadrilaterals.

• Collinear Points: Three or more points lie on the same line.
• Non-Collinear Points: If the points do not lie on the same line.
• If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
• If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
• Transversal Line: Line which intersects two or more lines at distinct points. If a transversal intersects two parallel lines, then
• each pair of corresponding angles is equal,
• each pair of alternate interior angles is equal,
• each pair of interior angles on the same side of the transversal is supplementary.

If two lines are parallel to a given line, then they are parallel to each other.

### Triangle Properties

• The sum of the three angles of a triangle is 180°.
• If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles.
• Angles opposite to equal sides are equal and vice versa
• Angle opposite to longer side is larger/greater and vice versa
• The Sum of the two sides is greater than the third side.
• Each angle of an equilateral triangle is 60°.
• Line-segment joining the mid-points of any two sides is parallel to the third side and is half of it.
• A line through the mid-point of a side parallel to another side bisects the third side.

### Congruency

Two figures are congruent if they are of the same shape and the same size. Circles are congruent of the same radii, and squares are congruent if they have the same sides.

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

• SAS Congruence Rule: When two sides and the included angle of one triangle are equal to another.
• ASA Congruence Rule: When two angles and the included side of one triangle are equal to another.
• AAS Congruence Rule: When two angles and the side are equal to the corresponding side and angles of another triangle.
• SSS Congruence Rule: If three sides of one triangle are equal to another triangle.
• RHS Congruence Rule: In two right triangles, the hypotenuse and one side of the triangle are equal to the hypotenuse and one side of another triangle.

• The Sum of angles of quadrilaterals is 360°
• Opposite sides are equal or,
• Opposite angles are equal or,
• Diagonals bisect each other or,
• A pair of opposite sides is equal and parallel
• Quadrilateral formed by joining mid-points of the sides of a quadrilateral (in order) is parallelogram
• Rectangle: Diagonals bisect each other and are equal and vice-versa
• Rhombus: Diagonals bisect each other at right angle and vice-versa
• Square: Diagonals bisect each other at right angles, and are equal and vice-versa

## Area of Parallelograms and Triangles

Area of a Parallelogram = base x corresponding altitude (height)

• Parallelograms on the same base (or equal bases) and between the same parallels are equal in area and vice versa.
• 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.

Area of a Triangle = ½ (base x height)

• Triangles on the same and between the same parallels are equal in area and vice versa.
• A median of a triangle divides it into two triangles of equal areas.

## Circles

• Equal chords of a circle subtend equal angles at the centre and vice versa.
• Perpendicular from the centre of a circle to the chord bisects the chord and vice versa.
• Only one circle passes through three non-collinear points.
• Equal chords of a circle (or of congruent circle) are equidistant from the circle and vice versa.
• Congruent arcs of a circle subtend equal angles at the centre.

The angle subtended by an arc at the centre = 2 x (angle subtended by that arc at any point on the remaining part of the circle)

• Angles in the same segment of a circle are equal.
• Angle in a semicircle is a right angle.

Sum of either pair of opposite angles = 180o

If the sum of either pair of a quadrilateral is 180o, then the quadrilateral is cyclic.

## Heron’s Formula

Area of a triangle = $$\sqrt{s\left( s – a \right)\left( s – b \right)\left( s – c \right)}$$

Where a,b, and c are sides of triangle, s = semi-perimeter = (a+b+c)/2

• If the sides and one diagonal of a quadrilateral are given, then its area can be calculated by dividing the quadrilateral into two triangles and then using the Heron’s formula.

## Surface Area and Volume

This section contains all the mensuration formulas for class 9. You need to practice problems based on various shapes (cube, cylinder, sphere, etc.) to memorize these maths formulas.

### 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 = √(l2 + b2 + h2)
• Volume of a cuboid = l x b x h

### Cube

• Surface area of a cube = 6a2
• Perimeter of a cube = 12a
• Volume of cube = a3
• Diagonal of cube = √(3a)

### Cylinder

• Curved surface area of a cylinder = 2πrh
• Total surface area of cylinder = 2πr(r + h)
• Volume of cylinder = πr2h
• Area of two circular bases in a cylinder = πr2 + πr2 = 2πr2

### Cone

• Curved surface area = πrl
• Total surface area of right-circular cone = πr(r + l)
• Volume of cone = 1/3 (πr2h)
• Slant height of cone = √(r+ h2)

### Sphere

• Surface area of sphere with radius r = 4πr2
• Curved surface area of hemisphere = 2πr2
• Total surface area of a hemisphere = 3πr2
• Volume of a sphere = (4/3)πr3
• Volume of hemisphere = (2/3)πr3

## Statistics Formulas

Class – Mark = (Upper Limit + Lower Limit) / 2

### Mean

$$\overline{x} = \frac{\text{Sum of all the observations}}{T\text{otal number of observations}}$$
or, $$\overline{x} = \ \frac{\sum_{i = 1}^{n}x_{i}}{n}$$
For an ungrouped frequency distribution, $\overline{x} = \ \frac{\sum_{i = 1}^{n}{f_{i}x_{i}}}{\sum_{i = 1}^{n}f_{i}}$

### Median

It is the value of middle-most observation

If n is an odd number, then Median = value of the $$\left( \frac{n + 1}{2} \right)^{\text{th}}$$ observation
If n is an even number, then Median = Mean of the values of $$\left( \frac{n}{2} \right)^{\text{th}}$$ and $$\left( \frac{n}{2} + 1 \right)^{\text{th}}$$ observation.

### Mode

It is the most frequently occurring observation.

## Probability Maths Formulas for Class 9

There is only one exercise for Probability problems in class 9. This exercise will use only one basic experimental formula to find out the probability of a given event.

P(E) = $$\frac{\text{Number of trials in which the event happened}}{\text{Total Number of trials}}$$

For example, if you throw a dice, then the probability of getting the six will be; 1/6. Here, the total trials will be 6 (there are 6 numbers in a dice), and the event can happen only once.

P(E1) + P(E2) + P(E3) + P(E4) + …………+ P(En) = 1

• The probability of an event lies between 0 and 1 (0 and 1 inclusive)

### Playing Cards – total 52 in sets of four

You will study Probability in detail in upcoming classes.

Those were all of the maths formulas for class 9. You can download the maths formulas of the class 9 PDF by clicking on the download link shared below. We will upload more study materials, including notes and solutions. Till then, practice the maths problems and memorize the formulas quickly. Subscribe to get the latest updates about more study materials and news on Vector Tutorials.