Maths Formulas are the foundation of every calculation, proof, and problem-solving strategy in school and competitive mathematics. These formulas cover algebra, geometry, trigonometry, calculus, and statistics — from CBSE Class 6 all the way to Class 12. Students preparing for board exams, JEE Main, JEE Advanced, or NEET will find this complete formula reference essential. This article covers all key mathematical formulas, their derivations, solved examples, and exam tips for 2025-26.
Key Maths Formulas at a Glance
Quick reference for the most important formulas across all classes.
- Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)
- Pythagoras Theorem: \( a^2 + b^2 = c^2 \)
- Area of Circle: \( A = \pi r^2 \)
- Compound Interest: \( A = P\left(1 + \frac{r}{100}\right)^n \)
- Binomial Theorem: \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)
- Trigonometric Identity: \( \sin^2\theta + \cos^2\theta = 1 \)
- Derivative of \( x^n \): \( \frac{d}{dx}(x^n) = nx^{n-1} \)
What are Maths Formulas?
Maths Formulas are precise mathematical expressions that define relationships between quantities. They allow students to solve problems efficiently without re-deriving every result from scratch. In NCERT textbooks, formulas are introduced progressively — starting with basic arithmetic in Class 6 and advancing to differential calculus in Class 12.
Every formula encodes a mathematical law or identity. For example, the area of a rectangle is always length multiplied by breadth. This relationship holds universally, making the formula a reliable tool. CBSE examiners expect students to recall, apply, and sometimes derive these formulas under timed conditions.
Formulas span multiple branches of mathematics. Algebra provides identities and equations. Geometry provides area, perimeter, and volume formulas. Trigonometry provides ratio and identity formulas. Calculus provides differentiation and integration formulas. Statistics provides mean, median, and standard deviation formulas. Mastering all these categories is essential for scoring full marks in CBSE board exams and cracking JEE or NEET.
The NCERT curriculum introduces formulas in a structured way. Class 6 to 8 covers basic algebra and mensuration. Class 9 to 10 covers quadratic equations, coordinate geometry, and trigonometry. Class 11 to 12 covers advanced algebra, calculus, vectors, and probability. This article organises all formulas class-wise and topic-wise for easy revision.
Core Maths Formula Expressions and Variables
Below are the most fundamental mathematical formulas used across CBSE classes. Each formula is presented with its expression, variable definitions, and units where applicable.
Algebraic Identities
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
\[ (a – b)^2 = a^2 – 2ab + b^2 \]
\[ (a + b)(a – b) = a^2 – b^2 \]
\[ (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \]
Quadratic Formula
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
| Symbol | Quantity | Notes |
|---|---|---|
| x | Roots of the quadratic equation | Two values: x₁ and x₂ |
| a | Coefficient of x² | a ≠ 0 |
| b | Coefficient of x | Can be positive, negative, or zero |
| c | Constant term | Independent of x |
| b²−4ac | Discriminant (D) | Determines nature of roots |
Derivation of the Quadratic Formula
Start with the standard quadratic equation: \( ax^2 + bx + c = 0 \). Divide every term by \( a \) to get \( x^2 + \frac{b}{a}x + \frac{c}{a} = 0 \). Move the constant to the right side: \( x^2 + \frac{b}{a}x = -\frac{c}{a} \). Complete the square by adding \( \left(\frac{b}{2a}\right)^2 \) to both sides. This gives \( \left(x + \frac{b}{2a}\right)^2 = \frac{b^2 – 4ac}{4a^2} \). Taking the square root of both sides and solving for \( x \) yields the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \). This derivation appears in NCERT Class 10, Chapter 4.
Trigonometric Ratios and Identities
\[ \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad \cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}, \quad \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \]
\[ \sin^2\theta + \cos^2\theta = 1 \]
\[ 1 + \tan^2\theta = \sec^2\theta \]
\[ 1 + \cot^2\theta = \csc^2\theta \]
Mensuration Formulas
| Shape | Area Formula | Perimeter/Volume |
|---|---|---|
| Rectangle | \( A = l \times b \) | \( P = 2(l + b) \) |
| Circle | \( A = \pi r^2 \) | \( C = 2\pi r \) |
| Triangle | \( A = \frac{1}{2} \times b \times h \) | Sum of all sides |
| Sphere | \( SA = 4\pi r^2 \) | \( V = \frac{4}{3}\pi r^3 \) |
| Cylinder | \( CSA = 2\pi rh \) | \( V = \pi r^2 h \) |
| Cone | \( CSA = \pi rl \) | \( V = \frac{1}{3}\pi r^2 h \) |
Complete Maths Formula Sheet — Class 6 to 12
This comprehensive table covers all major formulas from NCERT textbooks, Class 6 through Class 12. Use this as your master revision reference.
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Simple Interest | \( SI = \frac{P \times R \times T}{100} \) | P=Principal, R=Rate, T=Time | Rupees (₹) | Class 7, Ch 8 |
| Compound Interest | \( A = P\left(1 + \frac{r}{100}\right)^n \) | P=Principal, r=Rate, n=Years | Rupees (₹) | Class 8, Ch 8 |
| Pythagoras Theorem | \( a^2 + b^2 = c^2 \) | a,b=legs, c=hypotenuse | m or cm | Class 10, Ch 6 |
| Quadratic Formula | \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \) | a,b,c=coefficients | Dimensionless | Class 10, Ch 4 |
| Distance Formula | \( d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2} \) | Coordinates (x₁,y₁) and (x₂,y₂) | Units of length | Class 10, Ch 7 |
| Section Formula | \( \left(\frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n}\right) \) | m,n=ratio, x,y=coordinates | Units of length | Class 10, Ch 7 |
| Arithmetic Progression (nth term) | \( a_n = a + (n-1)d \) | a=first term, d=common difference | Dimensionless | Class 10, Ch 5 |
| AP Sum Formula | \( S_n = \frac{n}{2}[2a + (n-1)d] \) | n=number of terms | Dimensionless | Class 10, Ch 5 |
| Geometric Progression (nth term) | \( a_n = ar^{n-1} \) | a=first term, r=common ratio | Dimensionless | Class 11, Ch 9 |
| GP Sum (finite) | \( S_n = \frac{a(r^n – 1)}{r – 1} \) | r ≠ 1 | Dimensionless | Class 11, Ch 9 |
| Binomial Theorem | \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \) | n=positive integer | Dimensionless | Class 11, Ch 8 |
| Permutation | \( P(n,r) = \frac{n!}{(n-r)!} \) | n=total, r=selected | Dimensionless | Class 11, Ch 7 |
| Combination | \( C(n,r) = \frac{n!}{r!(n-r)!} \) | n=total, r=selected | Dimensionless | Class 11, Ch 7 |
| Derivative (Power Rule) | \( \frac{d}{dx}(x^n) = nx^{n-1} \) | n=any real number | Depends on context | Class 11, Ch 13 |
| Integration (Power Rule) | \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) | n ≠ −1, C=constant | Depends on context | Class 12, Ch 7 |
| Mean (Statistics) | \( \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \) | f=frequency, x=value | Same as data units | Class 10, Ch 14 |
| Standard Deviation | \( \sigma = \sqrt{\frac{\sum (x_i – \bar{x})^2}{n}} \) | xᵢ=data points, n=count | Same as data units | Class 11, Ch 15 |
| Probability | \( P(E) = \frac{\text{Favourable outcomes}}{\text{Total outcomes}} \) | 0 ≤ P(E) ≤ 1 | Dimensionless | Class 10, Ch 15 |
Maths Formulas — Solved Examples
Example 1 (Class 9-10 Level) — Quadratic Equation
Problem: Find the roots of the equation \( 2x^2 – 5x + 3 = 0 \) using the quadratic formula.
Given: \( a = 2,\ b = -5,\ c = 3 \)
Step 1: Write the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \)
Step 2: Calculate the discriminant: \( D = (-5)^2 – 4(2)(3) = 25 – 24 = 1 \)
Step 3: Find the roots: \( x = \frac{5 \pm \sqrt{1}}{4} = \frac{5 \pm 1}{4} \)
Step 4: Compute both values: \( x_1 = \frac{6}{4} = 1.5 \) and \( x_2 = \frac{4}{4} = 1 \)
Answer
The roots of the equation are \( x = 1.5 \) and \( x = 1 \).
Example 2 (Class 11-12 Level) — Arithmetic Progression and Sum Formula
Problem: The first term of an AP is 7 and the common difference is 3. Find the sum of the first 20 terms.
Given: \( a = 7,\ d = 3,\ n = 20 \)
Step 1: Write the AP sum formula: \( S_n = \frac{n}{2}[2a + (n-1)d] \)
Step 2: Substitute the values: \( S_{20} = \frac{20}{2}[2(7) + (20-1)(3)] \)
Step 3: Simplify inside the bracket: \( 2(7) = 14 \) and \( 19 \times 3 = 57 \). So \( 14 + 57 = 71 \).
Step 4: Multiply: \( S_{20} = 10 \times 71 = 710 \)
Answer
The sum of the first 20 terms is \( S_{20} = 710 \).
Example 3 (JEE/NEET Level) — Binomial Theorem Application
Problem: Find the middle term in the expansion of \( \left(x + \frac{1}{x}\right)^8 \).
Given: \( n = 8,\ a = x,\ b = \frac{1}{x} \)
Step 1: For even \( n \), the middle term is the \( \left(\frac{n}{2} + 1\right)\text{th} \) term. So it is the 5th term, \( T_5 \).
Step 2: Use the general term formula: \( T_{r+1} = \binom{n}{r} a^{n-r} b^r \). For \( T_5 \), set \( r = 4 \).
Step 3: Substitute: \( T_5 = \binom{8}{4} x^{8-4} \left(\frac{1}{x}\right)^4 = 70 \cdot x^4 \cdot \frac{1}{x^4} \)
Step 4: Simplify: \( T_5 = 70 \cdot x^{4-4} = 70 \cdot x^0 = 70 \)
Answer
The middle term in the expansion is \( 70 \) (a constant, independent of \( x \)).
CBSE Exam Tips 2025-26
- Create a formula diary: Write every new formula in a dedicated notebook. Review it daily for 10 minutes. We recommend colour-coding by topic — algebra in blue, geometry in green, trigonometry in red.
- Memorise discriminant conditions: CBSE frequently asks about the nature of roots. Remember: D > 0 means two distinct real roots; D = 0 means equal roots; D < 0 means no real roots.
- Learn both forms of AP sum: The formula \( S_n = \frac{n}{2}(a + l) \) is faster when the last term \( l \) is given. Use \( S_n = \frac{n}{2}[2a + (n-1)d] \) otherwise.
- Practice formula derivations: CBSE Class 10 and 12 boards sometimes ask for derivations worth 3-5 marks. Knowing the proof helps you re-derive if you forget the formula.
- Use formula sheets in mock tests: In our experience, students who revise formulas 48 hours before the exam score 8-12% higher. Do a timed formula-recall test the evening before your exam.
- Verify units and signs: Many students lose marks by forgetting negative signs in algebraic identities or using wrong units in mensuration. Always double-check before writing the final answer.
Common Mistakes to Avoid with Maths Formulas
- Confusing (a+b)² with a²+b²: Many students forget the middle term \( 2ab \). The correct expansion is \( (a+b)^2 = a^2 + 2ab + b^2 \). Omitting \( 2ab \) is one of the most common algebraic errors in CBSE exams.
- Wrong discriminant sign: Students often compute \( b^2 + 4ac \) instead of \( b^2 – 4ac \). Always subtract \( 4ac \) from \( b^2 \). A positive discriminant means real roots; a negative one means complex roots.
- Mixing up AP and GP formulas: The nth term of an AP is \( a + (n-1)d \), while for a GP it is \( ar^{n-1} \). These look similar but behave very differently. Confusing addition with multiplication here costs marks.
- Forgetting the constant of integration: In Class 12 integration, every indefinite integral must include \( + C \). Omitting it is penalised in CBSE marking schemes.
- Applying Pythagoras to non-right triangles: The formula \( a^2 + b^2 = c^2 \) applies only to right-angled triangles. For other triangles, use the cosine rule: \( c^2 = a^2 + b^2 – 2ab\cos C \).
JEE/NEET Application of Maths Formulas
In our experience, JEE aspirants who master the complete set of maths formulas from Classes 9 to 12 gain a significant advantage. JEE Main and JEE Advanced test formula recall and application under time pressure. NEET also requires strong command of statistics and basic algebra for the Biology and Chemistry sections.
Pattern 1 — Algebra and Quadratic Equations in JEE
JEE Main regularly features questions on the sum and product of roots. For \( ax^2 + bx + c = 0 \), remember: sum of roots \( = -\frac{b}{a} \) and product of roots \( = \frac{c}{a} \). These Vieta’s formulas let you solve problems without finding individual roots. JEE Advanced combines this with inequalities and complex numbers.
Pattern 2 — Sequences and Series in JEE
Questions on AP, GP, and their sums appear in almost every JEE Main paper. The formula for the sum of an infinite GP, \( S_\infty = \frac{a}{1-r} \) (where \( |r| < 1 \)), is especially important. JEE problems often ask you to identify whether a sequence is AP or GP before applying the correct formula.
Pattern 3 — Calculus Formulas in JEE Advanced
Differentiation and integration formulas are the backbone of JEE Advanced Paper 2. Our experts suggest memorising the derivatives of all six trigonometric functions, exponential functions (\( \frac{d}{dx}e^x = e^x \)), and logarithmic functions (\( \frac{d}{dx}\ln x = \frac{1}{x} \)). Integration by parts, \( \int u\, dv = uv – \int v\, du \), appears in 2-3 questions every year.
NEET-Specific Formula Tips
NEET Physics requires formulas from kinematics, thermodynamics, and optics — all of which rely on algebraic and trigonometric maths formulas. Students who are comfortable with \( \sin \), \( \cos \), and \( \tan \) values for standard angles (0°, 30°, 45°, 60°, 90°) solve Physics MCQs 40% faster. Statistical formulas also appear in NEET Biology data interpretation questions.
FAQs on Maths Formulas
Explore More Formula Articles
Strengthen your understanding of individual formulas with our dedicated topic pages. Visit the Algebra Formulas hub for a complete collection of algebraic identities and equations. For physics-related maths, explore the Velocity Formula page and the Acceleration Formula guide. You can also check the Force Formula article for Newton’s laws in mathematical form. For official NCERT resources, visit the NCERT official website to download textbooks and exemplar problems.