All Maths Formulas for Class 9

# All Maths Formulas for Class 9

## 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.   ap.aq = ap + q                         ii.   (ap)q = apq

iii.  ap/aq = ap – q                      iv.  apbp = (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) = anxn + an – 1xn – 1 + . . . + a2x2 + a1x + a0.

Where a0, a1, a2, . . ., an are constants and an ≠ 0.

a0, a1, a2, . . ., an are respectively the coefficient of x0, x, x2, . . ., xn, and n is called the degree of the polynomial. Each of anxn + an – 1xn – 1 + . . . + a2x2 + a1x + a0 , with an ≠ 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 = x2 + y2 + z2 + 2xy + 2yz + 2zx

8.     (x + y)3 = x3 + y3 + 3xy(x + y)

9.     (x – y)3 = x3 – y3 – 3xy(x – y)

10. x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – 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

2.     You can calculate the area of a triangle if you know the lengths of all three sides, using a formula that has been known for nearly 2000 years. It is called "Heron's Formula" after Hero of Alexandria.

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 = 4a2, a is the side of the cube.

5.     Surface area of a cube = 6a2, a is the side of the cube.

6.     Volume of a cube = a3, 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€r2h, 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 2h, r is the radius of the circular base, l is the slant height of the cone.

13. Surface area of a sphere = 4Ï€r2, r is the radius of the sphere.

14. Surface area of a hemisphere = 2Ï€r2, r is the radius of the sphere.

15. Total surface area of a hemisphere = 3Ï€r2, r is the radius of the sphere.

16. Volume of a sphere = 4/3 Ï€r3, r is the radius of the sphere.

17. Volume of a hemisphere = 2/3 Ï€r3, r is the radius of the sphere.

## Mean

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

1.      If the numbers of observations are less, i.e., less than 25, then we simply add all the observations and divide the sum by the total number of observations.

For example, if five families have 0, 2, 2, 3 and 5 children respectively, then the mean number of children is (0 + 2 + 2 + 3 + 5)/5 = 12/5 = 2.4. This means that the five families have an average of 2.4 children. We can use this formula to calculate the mean.
Here, xi = Sum of all the individual values and n= Total number of items.

2.      If the numbers of observations are more, i.e., more than 25, then we find the frequency (f) of distinct observations (x) and multiply them to find fx. Now, add all the values of fx to find fixiAgain, add all the frequencies to get fand use this formula to calculate the mean.

## Median

In individual series, where data is given in the raw form, the first step towards median calculation is to arrange the data in ascending or descending order. Now count the number of observations denoted by nThe next step is decided by whether the value of n is even or odd.
1.      If the value of n is odd, then

2.      If the value of n is even, then

## Mode

The mode is the measure of central tendency that identifies the category or data value that occurs the most frequently within the distribution of data. In other words, it is the most common value or the score that appears the highest number of times in a distribution.
For example, suppose 100 families own the following pets:
• 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

2.      There are three different situations that we come across in our daily life. They may be
·        sure to happen
·        impossible to happen
·        having some chances of happening.

3.      A random experiment is a process whose outcome (result) cannot be predicted in advance.

4.      Outcomes of an experiment are equally likely if each has the same chance of occurring.

5.      The collection of outcomes is called an event.

6.      The sample space of an event is the list of all possible outcomes of an event.

7.      The probability of an event E is denoted by P(E) and defined as follows:

8.      Symbolically, for finite number of possible outcomes, Probability of an event E, P(E) = n(E)/n(S), where, n(E) is the number of favorable outcomes and n(S) is the number of possible outcomes.

9.      Probability of an event lies between 0 and 1, inclusive of both.

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