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	(𝑎+𝑏)²=𝑎²+2𝑎𝑏+𝑏²
Geometric Formulas	𝐴=𝜋𝑟²
Trigonometric Formulas	sin⁡²𝜃+cos⁡²𝜃=1
Calculus Formulas	𝑑/𝑑𝑥(∫𝑓(𝑥) 𝑑𝑥)=𝑓(𝑥)
Statistical Formulas	𝑥ˉ=(∑𝑖₌₁𝑛𝑥ᵢ)/𝑛​​
Probability Formulas	𝑃(𝐴∪𝐵)=𝑃(𝐴)+𝑃(𝐵)−𝑃(𝐴∩𝐵)
Number Theory Formulas	𝑎²+𝑏²=𝑐²
Differential Equations	𝑑𝑦/𝑑𝑥=𝑘𝑦
Matrix Formulas	𝐴⋅𝐵≠𝐵⋅𝐴
Geometry Formulas	𝑉=4/3𝜋𝑟³

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.

• a² – b² = (a – b)(a + b)
• (a + b)² = a² + 2ab + b²
• a²+ b² = (a + b)² – 2ab
• (a – b)² = a² – 2ab + b²
• (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
• (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)
• (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
• (a – b)^⁴ = a⁴ – 4a³b + 6a²b² – 4ab³ + b⁴
• a⁴– b⁴ = (a – b)(a + b)(a² + b²)
• (aᵐ)(aⁿ) = aᵐ⁺ⁿ
• (ab)ᵐ = a^ᵐb^ᵐ
• (aᵐ)ⁿ = aᵐⁿ

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: 𝐴=𝑠² Perimeter: 𝑃=4𝑠P=4s
Rectangle	Area: 𝐴=𝑙×𝑤 Perimeter: 𝑃=2𝑙+2w
Triangle	Area: 𝐴=1/2×𝑏×ℎ Perimeter: 𝑃=𝑎+𝑏+𝑐
Circle	Area: 𝐴=𝜋𝑟² Circumference: 𝐶=2𝜋𝑟
Parallelogram	Area: 𝐴=𝑏×ℎ Perimeter: 𝑃=2(𝑎+𝑏)
Trapezoid	Area: 𝐴=1/2(𝑏₁+𝑏₂)×ℎ Perimeter: 𝑃=𝑎+𝑏₁+𝑐+𝑏₂

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=s³ Surface Area: 𝑆𝐴=6𝑠²
Rectangular Prism	Volume: 𝑉=𝑙×𝑤×ℎ Surface Area: 𝑆𝐴=2𝑙𝑤+2𝑙ℎ+2𝑤ℎ
Sphere	Volume: 𝑉=4/3𝜋𝑟³ Surface Area: 𝑆𝐴=4𝜋𝑟²
Cylinder	Volume: 𝑉=𝜋𝑟²ℎ Lateral Surface Area: 𝐿𝑆𝐴=2𝜋𝑟ℎ Total Surface Area: 𝑇𝑆𝐴=2𝜋𝑟(𝑟+ℎ)
Cone	Volume: 𝑉=1/3𝜋𝑟²ℎ Lateral Surface Area: 𝐿𝑆𝐴=𝜋𝑟𝑙 Total Surface Area: 𝑇𝑆𝐴=𝜋𝑟(𝑟+𝑙)
Prism	Volume: 𝑉=𝐵ℎ Lateral Surface Area: 𝐿𝑆𝐴=𝑃ℎ Total Surface Area: 𝑇𝑆𝐴=𝑃ℎ+2𝐵
Pyramid	Volume: 𝑉=1/3𝐵ℎ Lateral Surface Area: 𝐿𝑆𝐴=1/2𝑃𝑙 Total Surface Area: 𝑇𝑆𝐴=1/2𝑃𝑙+𝐵

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	𝑃(𝐸)=Number of favorable outcomes / Total number of outcomes​
Probability of the Complement of an Event	𝑃(𝐸′)=1−𝑃(𝐸)
Conditional Probability	𝑃(𝐴∣𝐵)=𝑃(𝐴∩𝐵)/𝑃(𝐵)
Addition Rule	𝑃(𝐴∪𝐵)=𝑃(𝐴)+𝑃(𝐵)−𝑃(𝐴∩𝐵)
Multiplication Rule (Independent Events)	𝑃(𝐴∩𝐵)=𝑃(𝐴)×𝑃(𝐵)
Multiplication Rule (Dependent Events)	𝑃(𝐴∩𝐵)=𝑃(𝐴)×𝑃(𝐵∣𝐴)
Bayes’ Theorem	𝑃(𝐴∣𝐵)={𝑃(𝐵∣𝐴)×𝑃(𝐴)}/𝑃(𝐵)

Trigonometric Formulas

Basic Trigonometric Formulas of Addition

Formula Type	Formula
Sine of Sum	sin⁡(𝑎+𝑏)=sin⁡𝑎 cos⁡𝑏+cos⁡𝑎 sin⁡𝑏
Cosine of Sum	cos⁡(𝑎+𝑏)=cos⁡𝑎 cos⁡𝑏−sin⁡𝑎 sin⁡𝑏
Tangent of Sum	tan⁡(𝑎+𝑏)=(tan⁡𝑎+tan⁡𝑏)/1−tan⁡𝑎tan⁡𝑏​

Formula Type	Formula
Sine of Difference	sin⁡(𝑎−𝑏)=sin⁡𝑎 cos⁡𝑏−cos⁡𝑎 sin⁡𝑏sin
Cosine of Difference	cos⁡(𝑎−𝑏)=cos⁡𝑎 cos⁡𝑏+sin⁡𝑎 sin⁡𝑏
Tangent of Difference	tan⁡(𝑎−𝑏)=tan⁡𝑎−tan⁡𝑏1+tan⁡𝑎 tan⁡𝑏​

Formula Type	Formula
Double Angle for Sine	sin⁡2𝜃=2sin⁡𝜃cos⁡𝜃sin2θ=2sinθcosθ
Double Angle for Cosine	cos⁡2𝜃=cos⁡2𝜃−sin⁡2𝜃cos2θ=cos2θ−sin2θ
Double Angle for Tangent	tan⁡2𝜃=2tan⁡𝜃1−tan⁡2𝜃tan2θ=1−tan2θ2tanθ​

Trigonometric Formulas for Division

Formula Type	Formula
Sine Division	sin⁡𝑎/sin⁡𝑏​
Cosine Division	cos⁡𝑎/cos⁡𝑏​
Tangent Division	tan⁡𝑎/tan⁡𝑏=(sin⁡𝑎/cos⁡𝑎)/(sin⁡𝑏/cos⁡𝑏)=(sin⁡𝑎 cos⁡𝑏) / (cos⁡𝑎 sin⁡𝑏) ​
Cosecant Division	csc⁡𝑎/csc⁡𝑏=sin⁡𝑏/sin⁡𝑎​
Secant Division	sec⁡𝑎/sec⁡𝑏=cos⁡𝑏/cos⁡𝑎​
Cotangent Division	cot⁡𝑎/cot⁡𝑏=(sin⁡𝑏 cos⁡𝑎) / (sin⁡𝑎 cos⁡𝑏 ​)

Other Trigonometric Formulas

Formula Type	Formula
Sine (sin)	sin⁡𝜃=opposite / hypotenuse
Cosine (cos)	cos⁡𝜃=adjacent / hypotenuset​
Tangent (tan)	tan⁡𝜃=opposite / adjacent
Cosecant (csc)	csc⁡𝜃=1/sin⁡𝜃
Secant (sec)	sec⁡𝜃=1/cos⁡𝜃
Cotangent (cot)	cot⁡𝜃=1/tan⁡𝜃
Pythagorean Identity	sin⁡²𝜃+²𝜃=1
Sine of Sum	sin⁡(𝑎+𝑏)=sin⁡𝑎 cos⁡𝑏+cos⁡𝑎 sin⁡𝑏
Cosine of Sum	cos⁡(𝑎+𝑏)=cos⁡𝑎 cos⁡𝑏−sin⁡𝑎 sin⁡𝑏
Tangent of Sum	tan⁡(𝑎+𝑏)=(tan⁡𝑎+tan⁡𝑏)/ (1−tan⁡𝑎 tan⁡𝑏)
Sine of Difference	sin⁡(𝑎−𝑏)=sin⁡𝑎 cos⁡𝑏−cos⁡𝑎 sin⁡𝑏
Cosine of Difference	cos⁡(𝑎−𝑏)=cos⁡𝑎 cos⁡𝑏+sin⁡𝑎 sin⁡𝑏
Tangent of Difference	tan⁡(𝑎−𝑏)= (tan⁡𝑎−tan⁡𝑏) / (1+tan⁡𝑎 tan⁡𝑏)​
Double Angle for Sine	sin⁡2𝜃=2sin⁡𝜃 cos⁡𝜃
Double Angle for Cosine	cos⁡2𝜃=cos⁡²𝜃−sin⁡²𝜃c
Double Angle for Tangent	tan⁡2𝜃=2tan⁡𝜃1−tan⁡2𝜃t​
Half-Angle for Sine	sin⁡𝜃/2=±√ 1−cos⁡𝜃/2​​
Half-Angle for Cosine	cos⁡𝜃/2=±√ 1+cos⁡𝜃/2​​
Half-Angle for Tangent	tan⁡𝜃/2=±√ 1−cos⁡𝜃 / 1+cos⁡𝜃​​

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 × c) + b]/c
• (a/b + d/b) = (a + d)/b
• (a/b + c/d) = (a × d + b × c)/(b × d)
• a/b × c/d = ac/bd
• (a/b)/(c/d) = a/b × 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) × 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:

1. 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.
2. Practice: Practice applying the formula to different problems. This builds familiarity and confidence, making it easier to recall and use the formula when needed.
3. Memorization: Some formulas are fundamental and are worth committing to memory. Flashcards or repetitive practice can help with this.
4. Visualization: Visualize what the formula represents geometrically or conceptually. This can aid in understanding and recalling the formula.
5. Analogies: Relate new formulas to ones you already know. Drawing parallels can help you grasp new concepts faster.
6. 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.
7. Application: Look for opportunities to apply the formula in real-world scenarios or in solving problems. Practical application reinforces learning and understanding.
8. 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.
9. Simplify: Break down complex formulas into smaller parts or terms. Understanding each component separately can make the overall formula more manageable.
10. Review: Regularly review formulas to keep them fresh in your memory. This prevents forgetting and ensures you can recall them when needed.
11. Resources: Use textbooks, online resources, or educational videos to supplement your learning. Different explanations or perspectives can sometimes clarify concepts.
12. 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 × 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: 𝑒𝑖𝜋+1=0. It elegantly combines five fundamental mathematical constants (e, π, 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: 𝑥=−𝑏±𝑏²−4𝑎𝑐²𝑎 . 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.