📘 MATHS WALLAH • CLASS 9
MP Board Class 9 Maths Formulas & Definitions
Complete Chapter-Wise Formula Revision Guide
Definitions • Identities • Theorems • Properties • Rules
Welcome to the Maths Wallah Class 9 Mathematics Formula Book. This reference guide collects important definitions, mathematical formulas, algebraic identities, geometry properties and mensuration rules for quick revision.
Important syllabus note: This guide follows the traditional 15-chapter NCERT-based sequence commonly used in older Class 9 resources. The current MP Board textbook or syllabus may have a different chapter list. Check your prescribed textbook before treating this as the complete official syllabus.
📚 Table of Contents
01. Number Systems 02. Polynomials 03. Coordinate Geometry 04. Linear Equations in Two Variables 05. Introduction to Euclid's Geometry 06. Lines and Angles 07. Triangles 08. Quadrilaterals 09. Areas of Parallelograms and Triangles 10. Circles 11. Constructions 12. Heron's Formula 13. Surface Areas and Volumes 14. Statistics 15. Probability01. Number Systems
Important Definitions
Natural Numbers (N): Counting numbers 1, 2, 3, 4, ...
Whole Numbers (W): 0, 1, 2, 3, 4, ...
Integers (Z): Negative whole numbers, zero and positive whole numbers.
Rational Number: A number expressible as p/q, where p and q are integers and q ≠ 0.
Irrational Number: A real number that cannot be expressed as p/q for integers p and q, q ≠ 0.
Real Numbers (R): The set containing all rational and irrational numbers.
Terminating Decimal: A decimal representation with a finite number of digits after the decimal point.
Recurring Decimal: A decimal in which a digit or group of digits repeats indefinitely.
Laws of Exponents
am × an = am+n
am ÷ an = am−n, a ≠ 0
(am)n = amn
(ab)m = ambm
(a/b)m = am/bm, b ≠ 0
a0 = 1, a ≠ 0
a−m = 1/am, a ≠ 0
am/n = ⁿ√(am)
Surds and Rationalisation
√a × √b = √(ab), for a, b ≥ 0
√a / √b = √(a/b), for a ≥ 0, b > 0
(√a + √b)(√a − √b) = a − b
1/√a = √a/a, a > 0
Decimal Expansion Rule
A rational number in lowest terms has a terminating decimal expansion if its denominator has no prime factors other than 2 and 5. Otherwise, its decimal expansion is non-terminating recurring.
Every real number has a unique point on the number line.
02. Polynomials
Definitions
Polynomial: An algebraic expression in which the powers of variables are non-negative integers.
Degree: The highest power of the variable with a non-zero coefficient.
Zero of a Polynomial: A value a for which p(a) = 0.
Monomial: A polynomial with one term.
Binomial: A polynomial with two terms.
Trinomial: A polynomial with three terms.
Types by Degree
Constant: p(x) = c; Linear: p(x) = ax + b; Quadratic: p(x) = ax² + bx + c; Cubic: p(x) = ax³ + bx² + cx + d.
Important Algebraic Identities
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
a² − b² = (a − b)(a + b)
(x + a)(x + b) = x² + (a + b)x + ab
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca)
If a + b + c = 0, then a³ + b³ + c³ = 3abc.
Polynomial Theorems
Remainder Theorem: The remainder when p(x) is divided by (x − a) is p(a).
Factor Theorem: (x − a) is a factor of p(x) if and only if p(a) = 0.
For a linear polynomial ax + b, where a ≠ 0, its zero is x = −b/a.
03. Coordinate Geometry
Definitions
Coordinate Plane: A plane formed by two perpendicular number lines called the x-axis and y-axis.
Origin: The point where both axes meet, with coordinates (0, 0).
Abscissa: The x-coordinate of a point.
Ordinate: The y-coordinate of a point.
Ordered Pair: A pair (x, y) identifying the position of a point.
Quadrants and Axes
Quadrant I: (+, +)
Quadrant II: (−, +)
Quadrant III: (−, −)
Quadrant IV: (+, −)
Point on x-axis: y = 0
Point on y-axis: x = 0
Coordinate Rules
- A point is written as (x, y), in that order.
- The x-coordinate gives horizontal position; the y-coordinate gives vertical position.
- Points on the same vertical line have the same x-coordinate.
- Points on the same horizontal line have the same y-coordinate.
Curriculum note: Distance, midpoint and section formulas are often taught in later classes. Include them only if they appear in your prescribed Class 9 syllabus.
04. Linear Equations in Two Variables
Definitions
Linear Equation in Two Variables: An equation that can be written as ax + by + c = 0, where a, b and c are real numbers and a and b are not both zero.
Solution: An ordered pair (x, y) satisfying the equation.
Graph: The straight line representing all solutions of a linear equation.
Standard Forms
ax + by + c = 0
ax + by = c
y = mx + c, when b ≠ 0
If y = 0 and a ≠ 0, then x = −c/a for ax + by + c = 0.
If x = 0 and b ≠ 0, then y = −c/b for ax + by + c = 0.
Graph Properties
- A linear equation in two variables generally has infinitely many solutions.
- Every solution is represented by a point on its graph.
- Two distinct points determine a straight line.
- The graph of x = k is a vertical line; the graph of y = k is a horizontal line.
05. Introduction to Euclid's Geometry
Definitions
Axiom: A general statement accepted without proof.
Postulate: A basic geometrical assumption accepted without proof.
Theorem: A mathematical statement established by logical proof.
Point: An exact position with no dimensions.
Line: A straight path extending indefinitely in both directions.
Euclid's Five Postulates
- A straight line can be drawn from any point to any other point.
- A finite straight line can be extended continuously in a straight line.
- A circle can be drawn with any centre and radius.
- All right angles are equal to one another.
- If a transversal makes the interior angles on one side sum to less than two right angles, the two lines meet on that side when extended.
Important Axioms
- Things equal to the same thing are equal to one another.
- If equals are added to equals, the wholes are equal.
- If equals are subtracted from equals, the remainders are equal.
- Things that coincide with one another are equal.
- The whole is greater than the part.
- Things that are double of the same things are equal.
- Things that are halves of the same things are equal.
06. Lines and Angles
Definitions
Acute Angle: An angle less than 90°.
Right Angle: An angle equal to 90°.
Obtuse Angle: An angle greater than 90° but less than 180°.
Complementary Angles: Two angles whose sum is 90°.
Supplementary Angles: Two angles whose sum is 180°.
Transversal: A line intersecting two or more lines at distinct points.
Angle Formulas
Complementary angles: A + B = 90°
Supplementary angles: A + B = 180°
Linear pair: A + B = 180°
Angles around a point = 360°
Vertically opposite angles are equal.
Parallel Lines Cut by a Transversal
- Corresponding angles are equal.
- Alternate interior angles are equal.
- Alternate exterior angles are equal.
- Interior angles on the same side of a transversal sum to 180°.
Triangle Angle Properties
A + B + C = 180°
Exterior angle = sum of the two opposite interior angles.
An exterior angle and its adjacent interior angle sum to 180°.
07. Triangles
Definitions
Congruent Triangles: Triangles identical in shape and size; their corresponding sides and angles are equal.
Isosceles Triangle: A triangle with at least two equal sides.
Equilateral Triangle: A triangle with three equal sides and three 60° angles.
Right Triangle: A triangle with one 90° angle.
Congruence Criteria
SSS: Side-Side-Side
SAS: Side-Angle-Side
ASA: Angle-Side-Angle
AAS: Angle-Angle-Side, a consequence of angle sum and ASA.
RHS: Right angle-Hypotenuse-Side, for right triangles.
CPCT: Corresponding Parts of Congruent Triangles are equal.
Triangle Properties
A + B + C = 180°
Exterior angle = sum of the two opposite interior angles.
Sum of any two sides > the third side.
- Equal sides have equal opposite angles.
- Equal angles have equal opposite sides.
- The greater side lies opposite the greater angle.
- The greater angle lies opposite the greater side.
Pythagoras' Theorem
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
h² = p² + b²
h = √(p² + b²)
08. Quadrilaterals
Definitions
Quadrilateral: A polygon with four sides.
Parallelogram: A quadrilateral with both pairs of opposite sides parallel.
Rectangle: A parallelogram with four right angles.
Rhombus: A parallelogram with all sides equal.
Square: A quadrilateral with four equal sides and four right angles.
Trapezium: A quadrilateral with a pair of parallel opposite sides in the usual Indian school-textbook convention.
Angle Formula
A + B + C + D = 360°
Parallelogram Properties
- Opposite sides are equal and parallel.
- Opposite angles are equal.
- Adjacent angles are supplementary.
- Diagonals bisect each other.
- Each diagonal divides a parallelogram into two congruent triangles.
Special Quadrilaterals
Rectangle: Opposite sides equal; all angles 90°; diagonals equal and bisect each other.
Rhombus: All sides equal; diagonals bisect each other at right angles.
Square: All sides equal; all angles 90°; diagonals equal, perpendicular and bisect each other.
Midpoint Theorem
The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
DE ∥ BC; DE = ½ BC
Converse: A line through the midpoint of one side of a triangle and parallel to another side bisects the third side.
09. Areas of Parallelograms and Triangles
Definitions
Area: The measure of the surface enclosed by a plane figure.
Altitude: A perpendicular segment from a vertex to the line containing the opposite side.
Same Base and Same Parallels: Figures sharing a base and lying between the same parallel lines.
Area Formulas
Rectangle: A = l × b
Square: A = a²
Triangle: A = ½ × base × height
Parallelogram: A = base × height
Rhombus: A = ½ × d₁ × d₂
Important Theorems
- Parallelograms on the same base and between the same parallels have equal areas.
- Triangles on the same base and between the same parallels have equal areas.
- A diagonal of a parallelogram divides it into two triangles of equal area.
- A triangle and a parallelogram on the same base and between the same parallels have areas in the ratio 1:2.
10. Circles
Definitions
Circle: The set of all points in a plane at a fixed distance from a fixed point called the centre.
Radius (r): The distance from the centre to the circle.
Diameter (d): A chord passing through the centre.
Chord: A line segment joining two points on a circle.
Arc: A portion of the circumference.
Sector: A region enclosed by two radii and the included arc.
Segment: A region enclosed by a chord and its corresponding arc.
Circle Formulas
Diameter: d = 2r
Circumference: C = 2πr = πd
Area: A = πr²
π ≈ 22/7 or 3.14159...
Important Circle Theorems
- Equal chords of a circle subtend equal angles at the centre.
- Equal chords are equidistant from the centre.
- The perpendicular from the centre to a chord bisects the chord.
- There is one and only one circle passing through three non-collinear points.
- Equal chords subtend equal angles at the circumference when they lie in the same circle and subtend corresponding minor arcs.
- The angle subtended by an arc at the centre is twice the angle subtended at the remaining circumference.
- Angles in the same segment of a circle are equal.
- Opposite angles of a cyclic quadrilateral are supplementary.
11. Constructions
Definitions
Geometrical Construction: The accurate drawing of geometric figures using permitted instruments, usually a ruler and compass.
Perpendicular Bisector: A line that cuts a segment into two equal parts at 90°.
Angle Bisector: A ray that divides an angle into two equal angles.
Important Construction Properties
- Every point on the perpendicular bisector of a segment is equidistant from its endpoints.
- The converse also holds: a point equidistant from the endpoints lies on the perpendicular bisector.
- Every point on an angle bisector is equidistant from the two arms of the angle, measured perpendicularly.
- The three perpendicular bisectors of a triangle meet at the circumcentre.
- The three angle bisectors of a triangle meet at the incentre.
Triangle Construction Conditions
SSS: Three sides are given.
SAS: Two sides and the included angle are given.
ASA: Two angles and the included side are given.
RHS: A right angle, hypotenuse and one side are given.
A triangle cannot be constructed from arbitrary measurements. The given measurements must satisfy the triangle inequality and the required construction conditions.
12. Heron's Formula
Definitions
Semi-perimeter: Half the sum of the three sides of a triangle.
Heron's Formula: A formula for finding a triangle's area when its three side lengths are known.
Important Formulas
Semi-perimeter: s = (a + b + c)/2
Area = √[s(s − a)(s − b)(s − c)]
Equilateral triangle: A = (√3/4)a²
Triangle area: A = ½bh
Common Applications
- Finding the area of a triangle from its three sides.
- Finding the area of a triangular field.
- Finding the area of a quadrilateral by dividing it into two triangles and applying Heron's formula separately, when the necessary side lengths are known.
Here a, b and c are the three side lengths. The triangle inequality must be satisfied.
13. Surface Areas and Volumes
Definitions
Surface Area: The total area covering the outside of a three-dimensional solid.
Curved Surface Area (CSA): The area of the curved part of a solid.
Lateral Surface Area (LSA): The area of the side faces, excluding the top and bottom bases.
Volume: The space occupied by a solid.
Slant Height: The length measured along the sloping side of a cone.
Cube
LSA = 4a²
TSA = 6a²
Volume = a³
Space diagonal = a√3
Cuboid
LSA = 2h(l + b)
TSA = 2(lb + bh + hl)
Volume = lbh
Space diagonal = √(l² + b² + h²)
Cylinder
CSA = 2πrh
TSA = 2πr(r + h)
Volume = πr²h
Cone
Slant height: l = √(r² + h²)
CSA = πrl
TSA = πr(l + r)
Volume = ⅓πr²h
Sphere
Surface area = 4πr²
Volume = ⁴⁄₃πr³
Hemisphere
CSA = 2πr²
TSA = 3πr²
Volume = ⅔πr³
Units and Conversion
1 m = 100 cm
1 m² = 10,000 cm²
1 m³ = 1,000,000 cm³
1 litre = 1000 cm³
1 m³ = 1000 litres
Use square units for surface area and cubic units for volume. The formula for a solid must match its shape.
14. Statistics
Definitions
Data: A collection of observations or information.
Frequency: The number of times a value occurs.
Grouped Data: Data organised into class intervals.
Class Mark: The midpoint of a class interval.
Mean: The arithmetic average of the observations.
Median: The middle value when observations are arranged in order.
Mode: The value occurring most frequently.
Range: The difference between the greatest and smallest observations.
Statistical Formulas
Mean = Sum of observations / Number of observations
x̄ = Σx / n
Mean for frequency data: x̄ = Σfx / Σf
Class mark = (Upper class limit + Lower class limit) / 2
Range = Maximum value − Minimum value
Median for Ungrouped Data
- Arrange all observations in ascending or descending order.
- If n is odd, median is the value at position (n + 1)/2.
- If n is even, median is the average of the values at positions n/2 and (n/2 + 1).
Bar Graphs and Histograms
- A bar graph displays categories using separate bars.
- A histogram displays grouped continuous data using adjacent rectangles.
- For equal class widths, the histogram's rectangle height can represent frequency.
- For unequal class widths, frequency density is used: frequency density = frequency / class width.
Pie-chart sector angle = (Category frequency / Total frequency) × 360°
Some formulas, such as the grouped-data median and mode formulas, may belong to later classes or extensions; use them only if included in the prescribed syllabus.
15. Probability
Definitions
Random Experiment: An experiment whose exact outcome cannot be predicted with certainty.
Outcome: A possible result of a random experiment.
Sample Space: The set of all possible outcomes.
Event: A set of outcomes of a random experiment.
Experimental Probability: The ratio of the number of times an event occurs to the total number of trials.
Equally Likely Outcomes: Outcomes having the same chance of occurring.
Probability Formulas
Experimental probability: P(E) = Number of times E occurs / Total number of trials
For equally likely outcomes: P(E) = n(E) / n(S)
0 ≤ P(E) ≤ 1
P(impossible event) = 0
P(certain event) = 1
P(not E) = 1 − P(E)
P(E) + P(not E) = 1
Important Rules
- The probability of an event cannot be negative.
- The probability of an event cannot exceed 1.
- The probabilities of all outcomes in a finite sample space add up to 1 when the outcomes are considered individually.
- Experimental probability is based on observations and may differ from theoretical probability.
The formula n(E)/n(S) applies when the outcomes are equally likely. Do not apply it automatically to experiments with unequal outcome chances.
