## Math Formulas

## What are Math Formulas?

**Math formulas are concise mathematical expressions that represent relationships between quantities, properties, or operations**. They are used to describe and solve mathematical problems across various fields such as algebra, geometry, calculus, and statistics. Formulas often consist of variables, constants, and mathematical symbols, enabling calculations and predictions in a structured manner. These formulas serve as fundamental tools for understanding and solving mathematical problems efficiently.

## List of Basic Math formulas

Essential mathematical formulas cover arithmetic operation, algebra, geometry, and more. They include equations for basic operations like addition and multiplication, as well as formulas for calculating areas, volumes, and solving equations. These formulas form the foundation of mathematical understanding and problem-solving.

Algebraic Formulas | (๐+๐)ยฒ=๐ยฒ+2๐๐+๐ยฒ |

Geometric Formulas | ๐ด=๐๐ยฒ |

Trigonometric Formulas | sinโกยฒ๐+cosโกยฒ๐=1 |

Calculus Formulas | ๐/๐๐ฅ(โซ๐(๐ฅ)โ๐๐ฅ)=๐(๐ฅ) |

Statistical Formulas | ๐ฅห=(โ๐โโ๐๐ฅแตข)/๐โโ |

Probability Formulas | ๐(๐ดโช๐ต)=๐(๐ด)+๐(๐ต)โ๐(๐ดโฉ๐ต) |

Number Theory Formulas | ๐ยฒ+๐ยฒ=๐ยฒ |

Differential Equations | ๐๐ฆ/๐๐ฅ=๐๐ฆ |

Matrix Formulas | ๐ดโ ๐ตโ ๐ตโ ๐ด |

Geometry Formulas | ๐=4/3๐๐ยณ |

## Basic Algebra Formulas

Fundamental rules essential for solving algebraic problems efficiently and accurately. These formulas cover concepts like linear equations, quadratic equations, slopes, distances, and geometric shapes, forming the building blocks for more advanced algebraic manipulations.

- aยฒย โ bยฒย = (a โ b)(a + b)
- (a + b)ยฒ = aยฒ + 2ab + bยฒ
- aยฒ+ bยฒ = (a + b)ยฒ โ 2ab
- (a โ b)ยฒ = aยฒ โ 2ab + bยฒ
- (a + b + c)ยฒ = aยฒ + bยฒ + cยฒ + 2ab + 2bc + 2ca
- (a โ b โ c)ยฒ = aยฒ + bยฒ + cยฒ โ 2ab + 2bc โ 2ca
- (a + b)ยณ = aยณ + 3aยฒb + 3abยฒ + bยณ
- (a โ b)ยณ = aยณ โ 3aยฒb + 3abยฒ โ bยณ
- aยณ โ bยณ = (a โ b)(aยฒ + ab + bยฒ)
- aยณ + bยณ = (a + b)(aยฒ โ ab + bยฒ)
- (a + b)โด = aโด + 4aยณb + 6aยฒbยฒ + 4abยณ + bโด
- (a โ b)
^{โด}= aโด โ 4aยณb + 6aยฒbยฒ โ 4abยณ + bโด - aโดโ bโด = (a โ b)(a + b)(aยฒ + bยฒ)
- (aแต)(aโฟ) = aแตโบโฟ
- (ab)แต = a
^{แต}b^{แต} - (aแต)โฟ = aแตโฟ

## Geometry Formulas

Geometric formulas essential for calculating properties of shapes, areas, volumes, and angles. These formulas are crucial in geometry, helping solve problems related to triangles, circles, rectangles, and other geometric figures.

We study Geometry formulas under two headings that are,

- 2-D Formulas
- 3-D Formulas

### 2-D Formulas

Key formulas for 2D geometry, including those for calculating area, perimeter, and properties of shapes like triangles, squares, circles, and polygons

Shape | Formula |
---|---|

Square | Area: ๐ด=๐ ยฒ Perimeter: ๐=4๐ P=4s |

Rectangle | Area: ๐ด=๐ร๐คPerimeter: ๐=2๐+2w |

Triangle | Area: ๐ด=1/2ร๐รโ Perimeter: ๐=๐+๐+๐ |

Circle | Area: ๐ด=๐๐ยฒCircumference: ๐ถ=2๐๐ |

Parallelogram | Area: ๐ด=๐รโPerimeter: ๐=2(๐+๐) |

Trapezoid | Area: ๐ด=1/2(๐โ+๐โ)รโ Perimeter: ๐=๐+๐โ+๐+๐โ |

### 3-D Formulas

Essential formulas for 3D geometry, covering calculations for volume, surface area, and characteristics of three-dimensional shapes such as spheres, cubes, cylinders, cones, and prisms.

Shape | Formula |
---|---|

Cube | Volume: V=sยณ Surface Area: ๐๐ด=6๐ ยฒ |

Rectangular Prism | Volume: ๐=๐ร๐ครโSurface Area: ๐๐ด=2๐๐ค+2๐โ+2๐คโ |

Sphere | Volume: ๐=4/3๐๐ยณSurface Area: ๐๐ด=4๐๐ยฒ |

Cylinder | Volume: ๐=๐๐ยฒโLateral Surface Area: ๐ฟ๐๐ด=2๐๐โTotal Surface Area: ๐๐๐ด=2๐๐(๐+โ) |

Cone | Volume: ๐=1/3๐๐ยฒโLateral Surface Area: ๐ฟ๐๐ด=๐๐๐Total Surface Area: ๐๐๐ด=๐๐(๐+๐) |

Prism | Volume: ๐=๐ตโLateral Surface Area: ๐ฟ๐๐ด=๐โTotal Surface Area: ๐๐๐ด=๐โ+2๐ต |

Pyramid | Volume: ๐=1/3๐ตโLateral Surface Area: ๐ฟ๐๐ด=1/2๐๐Total Surface Area: ๐๐๐ด=1/2๐๐+๐ต |

### Probability Formulas

Probability can simply be defined as the possibility of the occurrence of an event. It is expressed on a linear scale from 0 to 1. There are three types of probability : theoretical probability , and subjective probability.

**P(A) = n(A)/n(S)**

Where,**P(A)** is the Probability of an Event.**n(A)** is the Number of Favourable Outcomes**n(S)** is the Total Number of Events

Formula Type | Formula |
---|---|

Probability of an Event | ๐(๐ธ)=Number of favorable outcomes / Total number of outcomesโ |

Probability of the Complement of an Event | ๐(๐ธโฒ)=1โ๐(๐ธ) |

Conditional Probability | ๐(๐ดโฃ๐ต)=๐(๐ดโฉ๐ต)/๐(๐ต) |

Addition Rule | ๐(๐ดโช๐ต)=๐(๐ด)+๐(๐ต)โ๐(๐ดโฉ๐ต) |

Multiplication Rule (Independent Events) | ๐(๐ดโฉ๐ต)=๐(๐ด)ร๐(๐ต) |

Multiplication Rule (Dependent Events) | ๐(๐ดโฉ๐ต)=๐(๐ด)ร๐(๐ตโฃ๐ด) |

Bayes’ Theorem | ๐(๐ดโฃ๐ต)={๐(๐ตโฃ๐ด)ร๐(๐ด)}/๐(๐ต) |

### Trigonometric Formulas

**Basic Trigonometric Formulas of Addition**

Formula Type | Formula |
---|---|

Sine of Sum | sinโก(๐+๐)=sinโก๐ cosโก๐+cosโก๐ sinโก๐ |

Cosine of Sum | cosโก(๐+๐)=cosโก๐ cosโก๐โsinโก๐ sinโก๐ |

Tangent of Sum | tanโก(๐+๐)=(tanโก๐+tanโก๐)/1โtanโก๐tanโก๐โ |

**Basic Trigonometric Formulas of Subtraction**

Formula Type | Formula |
---|---|

Sine of Difference | sinโก(๐โ๐)=sinโก๐ cosโก๐โcosโก๐ sinโก๐sin |

Cosine of Difference | cosโก(๐โ๐)=cosโก๐ cosโก๐+sinโก๐ sinโก๐ |

Tangent of Difference | tanโก(๐โ๐)=tanโก๐โtanโก๐1+tanโก๐ tanโก๐โ |

**Basic Trigonometric Formulas of Multiplication**

Formula Type | Formula |
---|---|

Double Angle for Sine | sinโก2๐=2sinโก๐cosโก๐sin2ฮธ=2sinฮธcosฮธ |

Double Angle for Cosine | cosโก2๐=cosโก2๐โsinโก2๐cos2ฮธ=cos2ฮธโsin2ฮธ |

Double Angle for Tangent | tanโก2๐=2tanโก๐1โtanโก2๐tan2ฮธ=1โtan2ฮธ2tanฮธโ |

**Trigonometric Formulas for Division**

Formula Type | Formula |
---|---|

Sine Division | sinโก๐/sinโก๐โ |

Cosine Division | cosโก๐/cosโก๐โ |

Tangent Division | tanโก๐/tanโก๐=(sinโก๐/cosโก๐)/(sinโก๐/cosโก๐)=(sinโก๐ cosโก๐) / (cosโก๐ sinโก๐) โ |

Cosecant Division | cscโก๐/cscโก๐=sinโก๐/sinโก๐โ |

Secant Division | secโก๐/secโก๐=cosโก๐/cosโก๐โ |

Cotangent Division | cotโก๐/cotโก๐=(sinโก๐ cosโก๐) / (sinโก๐ cosโก๐ โ) |

### Other Trigonometric Formulas

Formula Type | Formula |
---|---|

Sine (sin) | sinโก๐=opposite / hypotenuse |

Cosine (cos) | cosโก๐=adjacent / hypotenusetโ |

Tangent (tan) | tanโก๐=opposite / adjacent |

Cosecant (csc) | cscโก๐=1/sinโก๐ |

Secant (sec) | secโก๐=1/cosโก๐ |

Cotangent (cot) | cotโก๐=1/tanโก๐ |

Pythagorean Identity | sinโกยฒ๐+ยฒ๐=1 |

Sine of Sum | sinโก(๐+๐)=sinโก๐ cosโก๐+cosโก๐ sinโก๐ |

Cosine of Sum | cosโก(๐+๐)=cosโก๐ cosโก๐โsinโก๐ sinโก๐ |

Tangent of Sum | tanโก(๐+๐)=(tanโก๐+tanโก๐)/ (1โtanโก๐ tanโก๐) |

Sine of Difference | sinโก(๐โ๐)=sinโก๐ cosโก๐โcosโก๐ sinโก๐ |

Cosine of Difference | cosโก(๐โ๐)=cosโก๐ cosโก๐+sinโก๐ sinโก๐ |

Tangent of Difference | tanโก(๐โ๐)= (tanโก๐โtanโก๐) / (1+tanโก๐ tanโก๐)โ |

Double Angle for Sine | sinโก2๐=2sinโก๐ cosโก๐ |

Double Angle for Cosine | cosโก2๐=cosโกยฒ๐โsinโกยฒ๐c |

Double Angle for Tangent | tanโก2๐=2tanโก๐1โtanโก2๐tโ |

Half-Angle for Sine | sinโก๐/2=ยฑโ 1โcosโก๐/2โโ |

Half-Angle for Cosine | cosโก๐/2=ยฑโ 1+cosโก๐/2โโ |

Half-Angle for Tangent | tanโก๐/2=ยฑโ 1โcosโก๐ / 1+cosโก๐โโ |

### Fraction Formula

A fraction represents a numerical value expressed as the quotient of two integers. The top number, called the numerator, indicates how many parts of the whole are being considered, while the bottom number, the denominator, denotes the total number of equal parts that make up the whole.

- (a + b/c) = [(a ร c) + b]/c
- (a/b + d/b) = (a + d)/b
- (a/b + c/d) = (a ร d + b ร c)/(b ร d)
- a/b ร c/d = ac/bd
- (a/b)/(c/d) = a/b ร d/c

### Percentage Formula

A percentage represents a ratio or a numerical value expressed as a part of 100. It is commonly denoted by the “%” symbol.

Percentage = (Given Value/Total Value) ร 100

## Tips for using Math Formulas

Using math formulas efficiently requires understanding and practice. Here are some tips to help you make the most of them:

**Understanding**: Ensure you understand the concepts behind the formula. Knowing how and why a formula works will help you apply it correctly and recognize when it’s applicable.**Practice**: Practice applying the formula to different problems. This builds familiarity and confidence, making it easier to recall and use the formula when needed.**Memorization**: Some formulas are fundamental and are worth committing to memory. Flashcards or repetitive practice can help with this.**Visualization**: Visualize what the formula represents geometrically or conceptually. This can aid in understanding and recalling the formula.**Analogies**: Relate new formulas to ones you already know. Drawing parallels can help you grasp new concepts faster.**Context**: Understand the context in which the formula is used. This helps you identify when to apply it and when alternative approaches might be more appropriate.**Application**: Look for opportunities to apply the formula in real-world scenarios or in solving problems. Practical application reinforces learning and understanding.**Derivation**: If possible, try to understand how the formula is derived. This deepens your understanding and can help you generalize the formula to different situations.**Simplify**: Break down complex formulas into smaller parts or terms. Understanding each component separately can make the overall formula more manageable.**Review**: Regularly review formulas to keep them fresh in your memory. This prevents forgetting and ensures you can recall them when needed.**Resources**: Use textbooks, online resources, or educational videos to supplement your learning. Different explanations or perspectives can sometimes clarify concepts.**Seek Help**: Don’t hesitate to seek help if you’re struggling with a particular formula. Teachers, tutors, or online communities can provide guidance and support.

### Best Formula to Learn Math

The best formula to learn math is one that’s fundamental and widely applicable, such as the Pythagorean theorem, which relates to geometry and has real-world applications in areas like engineering and architecture.

### Math Formula for Kids

A simple and intuitive formula for kids is the area formula for squares and rectangles (Area = length ร width). It’s easy to understand and apply, making it a foundational concept in early math education.

### Most Famous Formula in Math

The most famous formula in math is arguably Euler’s identity: ๐๐๐+1=0. It elegantly combines five fundamental mathematical constants (e, ฯ, i, 1, and 0) in a single equation, demonstrating the beauty and interconnectedness of mathematics.

### Most Used Formula in Math

One of the most used formulas in math is the quadratic formula: ๐ฅ=โ๐ยฑ๐ยฒโ4๐๐ยฒ๐ . It’s vital for solving quadratic equations and has applications in various fields, including physics, engineering, and economics.

### Golden Rule of Algebra

The golden rule of algebra is to maintain equality by performing the same operation on both sides of an equation. This ensures that the equation remains balanced and valid, allowing for accurate mathematical manipulation and problem-solving.