Class 9 Maths All Formulas: The Complete CBSE Formula Sheet (2026)

Rational numbers: Numbers of the form $\frac{p}{q}$ where $p, q$ are integers and $q \ne 0$. Their decimal expansion is either terminating or repeating.

Irrational numbers: Numbers that cannot be expressed as $\frac{p}{q}$. Their decimal expansion is non-terminating and non-repeating. Examples: $\sqrt{2}, \sqrt{3}, \pi$.

Real numbers: The set of all rational and irrational numbers together.

Laws of Exponents for Real Numbers:

Rationalising the denominator:

Fractional exponents:

Memory tip: To rationalise, multiply numerator and denominator by the conjugate. The conjugate of $a + \sqrt{b}$ is $a - \sqrt{b}$.

Standard Algebraic Identities:

Special case: If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$.

Remainder Theorem: If a polynomial $p(x)$ is divided by $(x - a)$, the remainder is $p(a)$.

Factor Theorem: $(x - a)$ is a factor of $p(x)$ if and only if $p(a) = 0$.

Zeroes of a polynomial: The values of $x$ for which $p(x) = 0$. A polynomial of degree $n$ has at most $n$ zeroes.

Memory tip: For the identity $a^3 + b^3 + c^3 - 3abc$, remember the special case first. If the sum is zero, the cubes add up to $3abc$. This is asked in almost every exam.

Cartesian plane: Two perpendicular number lines (axes) meeting at the origin $O(0, 0)$.

Coordinates of a point: Written as $(x, y)$ where $x$ is the abscissa (horizontal distance) and $y$ is the ordinate (vertical distance).

Sign conventions by quadrant:
- Quadrant I: $(+, +)$
- Quadrant II: $(-, +)$
- Quadrant III: $(-, -)$
- Quadrant IV: $(+, -)$

Points on axes:
- On the x-axis: $y = 0$, so point is $(x, 0)$
- On the y-axis: $x = 0$, so point is $(0, y)$
- Origin: $(0, 0)$

Distance formula (not in the Class 9 syllabus but useful to know):

Memory tip: Remember "run before you rise" -- the x-coordinate (horizontal run) comes before the y-coordinate (vertical rise).

General form of a linear equation in two variables:

Solution of a linear equation: An ordered pair $(x, y)$ that satisfies the equation. A linear equation in two variables has infinitely many solutions.

Graph: The graph of a linear equation in two variables is always a straight line.

Special cases:
- $x = k$ (constant): Vertical line parallel to the y-axis
- $y = k$ (constant): Horizontal line parallel to the x-axis
- $y = mx$: Line passing through the origin with slope $m$
- $y = mx + c$: Line with slope $m$ and y-intercept $c$

To plot a linear equation:
1. Find at least two solutions (ordered pairs)
2. Plot the points on the Cartesian plane
3. Join them with a straight line

Memory tip: Every point on the line is a solution. Every solution lies on the line. The line extends infinitely, which is why there are infinitely many solutions.

Euclid's Axioms (universal truths):
1. Things equal to the same thing are equal to one another.
2. If equals are added to equals, the wholes are equal.
3. If equals are subtracted from equals, the remainders are equal.
4. Things which coincide with one another are equal.
5. The whole is greater than the part.

Euclid's Postulates (geometry-specific):
1. A straight line may be drawn from any point to any other point.
2. A terminated line can be extended indefinitely.
3. A circle can be drawn with any centre and any radius.
4. All right angles are equal to one another.
5. (Parallel postulate) If a straight line falling on two straight lines makes the interior angles on the same side less than $180°$, the two lines will meet on that side.

Key result: Two distinct lines cannot have more than one point in common.

Memory tip: Axioms are general truths. Postulates are specific to geometry. Both are accepted without proof.

Linear pair: If two angles form a linear pair, their sum is $180°$.

Vertically opposite angles: When two lines intersect, vertically opposite angles are equal.

Angles on a straight line: Angles on one side of a straight line add up to $180°$.

Angles at a point: Angles around a point add up to $360°$.

Parallel lines cut by a transversal:
- Corresponding angles are equal: $\angle 1 = \angle 5$
- Alternate interior angles are equal: $\angle 3 = \angle 5$
- Alternate exterior angles are equal: $\angle 1 = \angle 7$
- Co-interior (same-side interior) angles are supplementary: $\angle 3 + \angle 6 = 180°$

Converse: If any of the above angle relationships hold, the lines are parallel.

Angle sum property of a triangle:

Exterior angle theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

Memory tip: For parallel lines, corresponding angles make an "F" shape, alternate angles make a "Z" shape, and co-interior angles make a "U" shape.

Congruence criteria for triangles:

• SAS (Side-Angle-Side): Two sides and the included angle are equal.
  - ASA (Angle-Side-Angle): Two angles and the included side are equal.
  - AAS (Angle-Angle-Side): Two angles and a non-included side are equal.
  - SSS (Side-Side-Side): All three sides are equal.
  - RHS (Right angle-Hypotenuse-Side): Right angle, hypotenuse, and one side are equal.

Properties of isosceles triangles:
- Angles opposite to equal sides are equal.
- Sides opposite to equal angles are equal.

Triangle inequality theorem:
The sum of any two sides of a triangle is greater than the third side.

Inequality result: In a triangle, the side opposite to the larger angle is longer. Conversely, the angle opposite to the longer side is larger.

Memory tip: For congruence rules, remember "SAS, ASA, AAS, SSS, RHS." Note that AAA and SSA are NOT valid congruence criteria. AAA proves similarity, not congruence.

Angle sum property of a quadrilateral:

Properties of a parallelogram:
- Opposite sides are equal: $AB = CD$ and $BC = DA$
- Opposite angles are equal: $\angle A = \angle C$ and $\angle B = \angle D$
- Diagonals bisect each other
- Consecutive angles are supplementary: $\angle A + \angle B = 180°$

Converse results:
- If opposite sides of a quadrilateral are equal, it is a parallelogram.
- If opposite angles are equal, it is a parallelogram.
- If diagonals bisect each other, it is a parallelogram.
- If one pair of opposite sides is both equal and parallel, it is a parallelogram.

Mid-Point Theorem:
The line segment joining the mid-points of two sides of a triangle is parallel to the third side and half its length.

If $D$ is the midpoint of $AB$ and $E$ is the midpoint of $AC$, then:

Converse: A line through the mid-point of one side of a triangle, parallel to another side, bisects the third side.

Memory tip: Think of a parallelogram as having "opposite everything equal." Opposite sides equal, opposite angles equal, and diagonals bisect each other (split each other into equal halves).

Area of a parallelogram:

Area of a triangle:

Key theorems:

1. Parallelograms on the same base and between the same parallels have equal area.

2. Triangles on the same base and between the same parallels have equal area.

3. A triangle and a parallelogram on the same base and between the same parallels: the area of the triangle is half the area of the parallelogram.

4. A median of a triangle divides it into two triangles of equal area.

(where $AD$ is a median from $A$ to $BC$)

Memory tip: "Same base, same parallels, same area" for parallelograms. For triangles, same base and same parallels give equal area (not the same shape, but the same area).

Key definitions:
- Radius $r$, Diameter $d = 2r$
- Chord, arc (minor and major), sector, segment

Theorem 1: Equal chords of a circle subtend equal angles at the centre.

Converse: If angles subtended by two chords at the centre are equal, the chords are equal.

Theorem 2: The perpendicular from the centre of a circle to a chord bisects the chord.

Converse: A line from the centre that bisects a chord is perpendicular to it.

Theorem 3: Equal chords are equidistant from the centre.

Theorem 4: The angle subtended by an arc at the centre is double the angle subtended at any point on the remaining arc.

Theorem 5: Angles in the same segment of a circle are equal.

Theorem 6: Angle in a semicircle is $90°$.

Cyclic quadrilateral: A quadrilateral inscribed in a circle.

Opposite angles of a cyclic quadrilateral are supplementary.

Memory tip: The central angle is always twice the inscribed angle on the same arc. This single fact is behind most circle theorem problems.

Construction 1: Bisect a given angle using compass and straightedge.

Construction 2: Construct the perpendicular bisector of a given line segment.

Construction 3: Construct an angle of $60°$ (equilateral triangle method).

From $60°$, you can construct:
- $30°$ (bisect $60°$)
- $90°$ (construct $60°$ then add $30°$, or use perpendicular bisector)
- $45°$ (bisect $90°$)
- $120°$ (construct $60°$ twice, or use the supplement of $60°$)

Construction 4: Construct a triangle given its base, a base angle, and the sum of the other two sides.

Construction 5: Construct a triangle given its base, a base angle, and the difference of the other two sides.

Construction 6: Construct a triangle given its perimeter and two base angles.

Memory tip: All constructions in this chapter use only a compass and a straightedge (unmarked ruler). No measurements with a protractor.

Semi-perimeter:

where $a, b, c$ are the sides of the triangle.

Heron's Formula:

This formula works for any triangle: scalene, isosceles, or equilateral.

**Area of an equilateral triangle with side $a$:**

(This is a special case of Heron's formula with $a = b = c$.)

Application to quadrilaterals: To find the area of a quadrilateral, divide it into two triangles using a diagonal. Find the area of each triangle using Heron's formula and add them.

Example: Find the area of a triangle with sides $5, 6, 7$.

Memory tip: Remember the four terms under the square root: $s$, then $s$ minus each side. The semi-perimeter $s$ always appears first.

Mean (average) of ungrouped data:

where $x_i$ are the observations and $n$ is the total number of observations.

Mean using frequency distribution:

Median: The middle value when data is arranged in ascending order.
- If $n$ is odd: Median $= \left(\frac{n+1}{2}\right)\text{th observation}$
- If $n$ is even: Median $= \frac{\left(\frac{n}{2}\right)\text{th} + \left(\frac{n}{2} + 1\right)\text{th observation}}{2}$

Mode: The value that appears most frequently in the data. A data set can have no mode, one mode, or multiple modes.

Range:

Memory tip: Mean is the mathematical average. Median is the physical middle. Mode is the most popular. Think of "mode" as "most" since both start with "mo."

Experimental (empirical) probability:

This is based on actual experiments and observations.

Key facts:
- $0 \le P(E) \le 1$ for any event $E$
- $P(\text{sure event}) = 1$
- $P(\text{impossible event}) = 0$
- $P(E) + P(\overline{E}) = 1$ where $\overline{E}$ means "E does not happen"
- As the number of trials increases, experimental probability tends to stabilise and approach the theoretical probability.

Example: A coin is tossed 100 times. Heads appears 47 times.

Memory tip: Experimental probability is about what actually happened, not what should theoretically happen. More trials give a more reliable estimate.