Formula
A formula is a short mathematical rule that shows the relationship between quantities using symbols. In CBSE and NCERT Maths, formulas help you solve problems faster—whether it is finding area, perimeter, percentage, speed, or algebraic values. Instead of repeating long steps every time, you apply a trusted rule and substitute values correctly.
\[\text{Result} = \text{Rule (in symbols)}\;\Rightarrow\; \text{Substitute values}\]
What is a Formula?
A formula is a mathematical statement written using numbers, variables, and operations to represent a rule. It tells how one quantity depends on others. For example, the perimeter of a rectangle depends on its length and breadth, so we write \(P = 2(l+b)\). Here, \(l\) and \(b\) are variables, and the formula gives a direct method to calculate \(P\).
Formulas are used daily in real life. We use formulas to calculate total cost (price × quantity), speed (distance ÷ time), electricity units, discounts, and interest. In school, formulas help you solve problems quickly and avoid repeating the same reasoning again and again. However, using a formula correctly requires identifying the right values, choosing correct units, and understanding what each symbol means.
NCERT introduces formulas from early classes through geometry (perimeter, area, volume), arithmetic (fractions, decimals, percentage), and algebra (identities, linear equations). By Class 9–10, you use formulas in mensuration and statistics; by Class 11–12, formulas become more advanced (trigonometry, calculus). NCERT emphasizes understanding the meaning of a formula, not just memorizing it.
General Formula Structure
\[y = f(x)\]
- \(y\): the result/output quantity (what you want to find)
- \(x\): input quantity/variable(s) (values you know)
- \(f\): rule that connects input to output (operations like \(+,-,\times,\div\), powers, etc.)
Conditions / Notes:
- Use correct units (cm, m, s, ₹, etc.) before substituting values.
- Check domain restrictions (e.g., denominator \(\neq 0\), square root needs non-negative value for real numbers).
- Substitute values carefully and simplify step-by-step to avoid calculation mistakes.
Complete Formula List (Class 6–12)
Below is a comprehensive list of important formulas across Maths, Physics, Chemistry, Algebra, Geometry, and Trigonometry for CBSE Class 6–12 students.
Maths Formulas
| S.No | Formula Name | Formula | When to Use |
|---|---|---|---|
| 1 | Perimeter of Rectangle | \(P=2(l+b)\) | When length \(l\) and breadth \(b\) are given |
| 2 | Area of Rectangle | \(A=l\times b\) | To find region covered by a rectangle |
| 3 | Area of Triangle | \(A=\frac{1}{2}bh\) | When base \(b\) and height \(h\) are known |
| 4 | Area of Circle | \(A=\pi r^2\) | When radius \(r\) is given |
| 5 | Circumference of Circle | \(C=2\pi r\) | To find boundary length of a circle |
| 6 | Percentage Formula | \(\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times 100\%\) | To compare part with whole “per 100” |
| 7 | Speed Formula | \(\text{Speed}=\frac{\text{Distance}}{\text{Time}}\) | Motion problems |
| 8 | Simple Interest | \(\text{SI}=\frac{PRT}{100}\) | Bank/interest problems with simple interest |
| 9 | Compound Interest | \(A=P\left(1+\frac{R}{100}\right)^n\) | Interest compounded annually |
| 10 | Volume of Cuboid | \(V=l\times b\times h\) | 3D mensuration for cuboid |
| 11 | Volume of Cube | \(V=a^3\) | When side \(a\) of cube is given |
| 12 | Surface Area of Cube | \(SA=6a^2\) | Total surface area of cube |
| 13 | Volume of Cylinder | \(V=\pi r^2 h\) | Cylindrical containers, pipes |
| 14 | Volume of Cone | \(V=\frac{1}{3}\pi r^2 h\) | Cone-shaped objects |
| 15 | Volume of Sphere | \(V=\frac{4}{3}\pi r^3\) | Spherical objects like balls |
| 16 | Heron’s Formula | \(A=\sqrt{s(s-a)(s-b)(s-c)}\) | Area when all 3 sides are known |
| 17 | Section Formula | \(\left(\frac{m_1x_2+m_2x_1}{m_1+m_2}, \frac{m_1y_2+m_2y_1}{m_1+m_2}\right)\) | Point dividing line in ratio \(m_1:m_2\) |
| 18 | Midpoint Formula | \(\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)\) | Finding midpoint of a line segment |
| 19 | Centroid Formula | \(\left(\frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right)\) | Center of mass of triangle |
| 20 | Distance Formula | \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\) | Distance between two points |
| 21 | Slope of a Line | \(m=\frac{y_2-y_1}{x_2-x_1}\) | Coordinate geometry (Class 9–10+) |
| 22 | Quadratic Formula | \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) | To solve \(ax^2+bx+c=0\) |
| 23 | Pythagoras Theorem | \(c^2=a^2+b^2\) | Right-angled triangle with hypotenuse \(c\) |
| 24 | Ellipse Formula | \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) | Equation of ellipse (Class 11) |
| 25 | Ratio Formula | \(a:b = \frac{a}{b}\) | Comparing two quantities |
Algebra Formulas
| S.No | Identity/Formula Name | Formula | Application |
|---|---|---|---|
| 1 | Square of Sum | \((a+b)^2 = a^2 + 2ab + b^2\) | Expansion and simplification |
| 2 | Square of Difference | \((a-b)^2 = a^2 – 2ab + b^2\) | Expansion and simplification |
| 3 | Difference of Squares | \(a^2 – b^2 = (a+b)(a-b)\) | Factorisation |
| 4 | Cube of Sum | \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) | Cubic expansions |
| 5 | Cube of Difference | \((a-b)^3 = a^3 – 3a^2b + 3ab^2 – b^3\) | Cubic expansions |
| 6 | Sum of Cubes | \(a^3 + b^3 = (a+b)(a^2 – ab + b^2)\) | Factorisation of cubic expressions |
| 7 | Difference of Cubes | \(a^3 – b^3 = (a-b)(a^2 + ab + b^2)\) | Factorisation of cubic expressions |
| 8 | Arithmetic Progression (AP) | \(a_n = a + (n-1)d\) | Finding nth term of AP |
| 9 | Sum of AP | \(S_n = \frac{n}{2}[2a + (n-1)d]\) | Sum of first n terms of AP |
| 10 | Geometric Progression (GP) | \(a_n = ar^{n-1}\) | Finding nth term of GP |
| 11 | Sum of GP | \(S_n = a\cdot\frac{r^n-1}{r-1}\) (for \(r>1\)) | Sum of first n terms of GP |
| 12 | Logarithm Formula | \(\log_a(xy) = \log_a x + \log_a y\) | Simplifying logarithmic expressions |
Geometry Formulas
| S.No | Shape | Perimeter/Circumference | Area |
|---|---|---|---|
| 1 | Square | \(4a\) | \(a^2\) |
| 2 | Rectangle | \(2(l+b)\) | \(l \times b\) |
| 3 | Triangle | \(a+b+c\) | \(\frac{1}{2} \times b \times h\) |
| 4 | Equilateral Triangle | \(3a\) | \(\frac{\sqrt{3}}{4}a^2\) |
| 5 | Circle | \(2\pi r\) | \(\pi r^2\) |
| 6 | Parallelogram | \(2(a+b)\) | \(b \times h\) |
| 7 | Rhombus | \(4a\) | \(\frac{1}{2} \times d_1 \times d_2\) |
| 8 | Trapezium | \(a+b+c+d\) | \(\frac{1}{2}(a+b) \times h\) |
| 9 | Semicircle | \(\pi r + 2r\) | \(\frac{1}{2}\pi r^2\) |
| 10 | Sector of Circle | \(\frac{\theta}{360}\times 2\pi r + 2r\) | \(\frac{\theta}{360}\times \pi r^2\) |
Trigonometry Formulas
| S.No | Formula Name | Formula | Application |
|---|---|---|---|
| 1 | Sine Ratio | \(\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\) | Right triangle calculations |
| 2 | Cosine Ratio | \(\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\) | Right triangle calculations |
| 3 | Tangent Ratio | \(\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}\) | Heights and distances |
| 4 | Pythagorean Identity | \(\sin^2\theta + \cos^2\theta = 1\) | Simplifying trig expressions |
| 5 | Secant Identity | \(1 + \tan^2\theta = \sec^2\theta\) | Proving trig identities |
| 6 | Cosecant Identity | \(1 + \cot^2\theta = \csc^2\theta\) | Proving trig identities |
| 7 | Sine Rule | \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\) | Non-right triangles |
| 8 | Cosine Rule | \(c^2 = a^2 + b^2 – 2ab\cos C\) | Non-right triangles |
| 9 | Double Angle (Sin) | \(\sin 2\theta = 2\sin\theta\cos\theta\) | Compound angles (Class 11) |
| 10 | Double Angle (Cos) | \(\cos 2\theta = \cos^2\theta – \sin^2\theta\) | Compound angles (Class 11) |
Physics Formulas
| S.No | Formula Name | Formula | Application |
|---|---|---|---|
| 1 | Velocity Formula | \(v = \frac{d}{t}\) | Speed with direction |
| 2 | Acceleration Formula | \(a = \frac{v-u}{t}\) | Rate of change of velocity |
| 3 | First Equation of Motion | \(v = u + at\) | Kinematics problems |
| 4 | Second Equation of Motion | \(s = ut + \frac{1}{2}at^2\) | Distance in uniformly accelerated motion |
| 5 | Third Equation of Motion | \(v^2 = u^2 + 2as\) | When time is not given |
| 6 | Force Formula | \(F = ma\) | Newton’s Second Law |
| 7 | Momentum Formula | \(p = mv\) | Linear momentum |
| 8 | Kinetic Energy | \(KE = \frac{1}{2}mv^2\) | Energy of moving objects |
| 9 | Potential Energy | \(PE = mgh\) | Energy due to height |
| 10 | Power Formula | \(P = \frac{W}{t}\) | Rate of doing work |
| 11 | Ohm’s Law | \(V = IR\) | Electric circuits |
| 12 | Mirror Formula | \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\) | Spherical mirrors |
| 13 | Lens Formula | \(\frac{1}{f} = \frac{1}{v} – \frac{1}{u}\) | Thin lenses |
| 14 | Orbital Velocity | \(v_o = \sqrt{\frac{GM}{r}}\) | Satellite motion |
| 15 | Gravitational Force | \(F = \frac{Gm_1m_2}{r^2}\) | Universal gravitation |
Chemistry Formulas
| S.No | Formula/Compound Name | Chemical Formula | Use/Application |
|---|---|---|---|
| 1 | Water | \(\text{H}_2\text{O}\) | Universal solvent |
| 2 | Carbon Dioxide | \(\text{CO}_2\) | Photosynthesis, carbonated drinks |
| 3 | Methane | \(\text{CH}_4\) | Natural gas, fuel |
| 4 | Ethanol | \(\text{C}_2\text{H}_5\text{OH}\) | Alcoholic beverages, fuel |
| 5 | Ethyl Acetate | \(\text{CH}_3\text{COOC}_2\text{H}_5\) | Solvent, flavoring |
| 6 | Citric Acid | \(\text{C}_6\text{H}_8\text{O}_7\) | Food preservative, cleaning |
| 7 | Ammonium Carbonate | \((\text{NH}_4)_2\text{CO}_3\) | Baking powder, smelling salts |
| 8 | Bleaching Powder | \(\text{CaOCl}_2\) | Disinfectant, bleaching agent |
| 9 | Sulphuric Acid | \(\text{H}_2\text{SO}_4\) | King of chemicals, batteries |
| 10 | Sodium Hydroxide | \(\text{NaOH}\) | Soap making, drain cleaner |
| 11 | Glucose | \(\text{C}_6\text{H}_{12}\text{O}_6\) | Energy source, photosynthesis product |
| 12 | Mole Concept | \(n = \frac{\text{Mass}}{\text{Molar Mass}}\) | Stoichiometry calculations |
Calculus Formulas (Class 11-12)
| S.No | Formula Name | Formula | Application |
|---|---|---|---|
| 1 | Power Rule (Derivative) | \(\frac{d}{dx}(x^n) = nx^{n-1}\) | Differentiation of polynomials |
| 2 | Derivative of sin x | \(\frac{d}{dx}(\sin x) = \cos x\) | Trigonometric differentiation |
| 3 | Derivative of cos x | \(\frac{d}{dx}(\cos x) = -\sin x\) | Trigonometric differentiation |
| 4 | Derivative of \(e^x\) | \(\frac{d}{dx}(e^x) = e^x\) | Exponential functions |
| 5 | Derivative of ln x | \(\frac{d}{dx}(\ln x) = \frac{1}{x}\) | Logarithmic functions |
| 6 | Product Rule | \(\frac{d}{dx}(uv) = u’v + uv’\) | Derivative of product of functions |
| 7 | Quotient Rule | \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{u’v – uv’}{v^2}\) | Derivative of division of functions |
| 8 | Chain Rule | \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\) | Composite functions |
| 9 | Integration Power Rule | \(\int x^n dx = \frac{x^{n+1}}{n+1} + C\) | Basic integration |
| 10 | Integration of sin x | \(\int \sin x \, dx = -\cos x + C\) | Trigonometric integration |
Types (Variants) of Formulas with Examples
1) Geometry (Mensuration) Formulas
These formulas calculate perimeter, area, surface area, and volume of shapes.
\[A_{\text{circle}}=\pi r^2\]
Example: If \(r=7\) cm, then \(A=\pi\times 7^2=49\pi\) cm\(^2\).
2) Arithmetic Formulas (Percentage/Speed/Interest)
These formulas connect real-life quantities like money and time.
\[\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times 100\%\]
Example: If 18 out of 50 students are girls, percentage = \(\frac{18}{50}\times 100=36\%\).
3) Algebraic Formulas (Identities)
Algebraic identities help in expansion, factorisation, and simplification.
\[(a+b)^2=a^2+2ab+b^2\]
Example: \((x+3)^2=x^2+6x+9\).
4) Coordinate Geometry Formulas
These formulas are used for lines, distance, and section problems.
\[\text{Distance} = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\]
Example: Distance between \((0,0)\) and \((3,4)\) is \(\sqrt{9+16}=5\).
5) Advanced Formulas (Trigonometry/Calculus)
In Class 11–12, formulas help in trigonometric solutions and derivatives.
\[\frac{d}{dx}(x^n)=nx^{n-1}\]
Example: \(\frac{d}{dx}(x^4)=4x^3\).
Derivation (How a Formula is Formed)
Start with a definition or a known property from NCERT (for example, perimeter means total boundary length).
Express the situation using symbols/variables (like \(l\) and \(b\) for a rectangle).
Use basic operations to combine repeated terms (e.g., rectangle has two lengths and two breadths).
Simplify the expression into a compact symbolic rule.
Check the rule with a small numerical example to verify it works.
Therefore, a formula is a simplified symbolic rule derived from definitions, properties, and logical steps, so that we can substitute values and compute results quickly.
How to Use Any Formula (Step-by-Step)
- Read the question: identify what is asked (output/result).
- Choose the correct formula: match the topic (area, percentage, speed, etc.).
- List given values: write all known quantities with units.
- Substitute carefully: put values into the formula in the correct places.
- Simplify and write answer: calculate step-by-step and mention unit.
Quick Tip: Before calculating, quickly check units (cm vs m, minutes vs hours). Many CBSE errors happen due to unit mismatch, not the formula.
CBSE Class-wise Formula Focus (NCERT Reference)
NCERT builds formula understanding step-by-step. Here is a practical class-wise view of where formulas matter most.
Class 6 (NCERT Maths: Basic Geometry & Arithmetic)
- Focus: perimeter, area basics, fractions-decimals conversions.
- Examples: \(P=2(l+b)\), \(A=l\times b\).
Class 7–8 (NCERT Maths: Comparing Quantities & Mensuration)
- Focus: percentage, profit-loss basics, simple interest, area/volume expansion.
- Examples: \(\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times 100\%\), \(\text{SI}=\frac{PRT}{100}\).
Class 9–10 (NCERT Maths: Mensuration, Algebra, Coordinate Geometry)
- Focus: areas and surface areas, quadratic equations, coordinate geometry formulas.
- Examples: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\), \(m=\frac{y_2-y_1}{x_2-x_1}\).
Class 11–12 (NCERT Maths: Trigonometry & Calculus)
- Focus: trigonometric identities, derivatives, integrals, applications.
- Example: \(\frac{d}{dx}(x^n)=nx^{n-1}\).
Download Free PDF: Maths Formulas PDF (Class 6–12)
Solved Examples (Easy to Board Style)
Example 1 (Easy): Area of Rectangle
Given: Length \(l=12\) cm, breadth \(b=5\) cm. Find area.
Formula: \(A=l\times b\)
Steps: \(A=12\times 5=60\)
Answer: \(60\) cm\(^2\)
Example 2 (Medium): Percentage Score
Given: Marks obtained = 72, total marks = 80. Find percentage.
Formula: \(\text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times 100\%\)
Steps: \(\frac{72}{80}\times 100=90\%\)
Answer: \(90\%\)
Example 3 (Hard): Simple Interest
Given: \(P=₹5000\), \(R=8\%\) per year, \(T=3\) years. Find SI.
Formula: \(\text{SI}=\frac{PRT}{100}\)
Steps: \(\text{SI}=\frac{5000\times 8\times 3}{100}=1200\)
Answer: ₹1200
Example 4 (Board Style): Quadratic Formula
Given: Solve \(2x^2-3x-2=0\).
Formula: \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
Steps:
- \(a=2,\; b=-3,\; c=-2\)
- Discriminant \(b^2-4ac = (-3)^2-4(2)(-2)=9+16=25\)
- \(x=\frac{-(-3)\pm\sqrt{25}}{2\cdot 2}=\frac{3\pm 5}{4}\)
- \(x=\frac{8}{4}=2\) or \(x=\frac{-2}{4}=-\frac{1}{2}\)
Answer: \(x=2\) and \(x=-\frac{1}{2}\)
Practice Problems (Mixed Formulas)
Try these problems to practice choosing the correct formula.
- Find the perimeter of a rectangle with \(l=14\) cm and \(b=9\) cm.
- Find the area of a triangle with base \(b=12\) cm and height \(h=7\) cm.
- Find \(15\%\) of 260.
- A bike covers 120 km in 3 hours. Find its speed.
- Find SI on ₹8000 at \(7.5\%\) per year for 2 years.
- Find the circumference of a circle of radius 3.5 cm (use \(\pi=\frac{22}{7}\)).
- Solve \(x^2-5x+6=0\).
Answers (Open after solving)
1) \(P=2(l+b)=2(14+9)=46\) cm
2) \(A=\frac{1}{2}bh=\frac{1}{2}\cdot 12\cdot 7=42\) cm\(^2\)
3) \(\frac{15}{100}\times 260=39\)
4) Speed = \(\frac{120}{3}=40\) km/h
5) SI = \(\frac{8000\times 7.5\times 2}{100}=1200\)
6) \(C=2\pi r=2\cdot \frac{22}{7}\cdot 3.5=22\) cm
7) \((x-2)(x-3)=0\Rightarrow x=2,3\)
Tips, Speed Tricks & Common Mistakes
Speed Trick
- Write “Given → Required → Formula” first: This 10-second habit helps you select the correct formula and prevents wrong substitution.
3 Common Mistakes
- ❌ Using a formula without unit conversion. ✓ Convert first (cm to m, minutes to hours) then substitute.
- ❌ Mixing up symbols (like taking \(b\) as breadth in one step and base in another). ✓ Clearly write what each symbol means for that question.
- ❌ Incorrect bracket handling. ✓ Use brackets carefully: \(2(l+b)\neq 2l+b\).
Classroom Tip
Create a one-page formula sheet by chapter (mensuration, algebra, percentage). Revise it daily for 5 minutes—this improves speed and accuracy in CBSE exams.
Frequently Asked Questions on Formula
A formula is a rule written using symbols and variables that shows how to calculate a quantity. Example: \(A=l\times b\) for the area of a rectangle.
Formulas save time and make problem-solving faster. Instead of repeating long reasoning, you substitute values and compute the answer.
Learn formulas chapter-wise, practice 5–10 problems per formula, and revise a short formula sheet regularly.
An equation states that two expressions are equal and may need solving (like \(2x+3=7\)). A formula is a general rule to calculate a specific quantity (like \(A=\pi r^2\)).
Variables are symbols like \(x, y, l, b\) that can take different values. They represent quantities that change.
Speed formula: \(\text{Speed}=\frac{\text{Distance}}{\text{Time}}\). It is used to find how fast a vehicle is moving.
The most common mistake is wrong substitution or wrong units. Always convert units first and use brackets carefully.
Mensuration (areas/volumes), Comparing Quantities (percentage/interest), Algebra (identities/quadratic), and Coordinate Geometry contain many important formulas.
Both are important. Understanding helps you remember longer and apply correctly, while memorization improves speed in exams.
Do a quick estimate (approximation) and check units. If the unit or size looks wrong, recheck substitution and calculation.