Mathematics is a universe of numbers, shapes, and patterns, each governed by its own set of rules and formulas. From the basic equations that define the very fabric of reality to the complex algorithms that drive modern computing, mathematics is the language of logic and precision. Below, we present an extensive list of mathematical formulas, divided into key sections for ease of navigation and understanding.
Geometry Formulas
Geometry, the branch of mathematics concerned with the properties and relations of points, lines, angles, surfaces, and solids, offers a fundamental understanding of the physical world.
Plane Geometry
| Name | Formula | Basic Description |
|---|---|---|
| Area of a Square | Side x Side | Represents area enclosed by a square. |
| Perimeter of a Square | 4 x Side | Distance around a square. |
| Area of a Rectangle | Length x Width | Represents area enclosed by a rectangle. |
| Perimeter of a Rectangle | 2 x (Length + Width) | Distance around a rectangle. |
| Area of a Triangle | 0.5 x Base x Height | Represents area enclosed by a triangle. |
| Circumference of a Circle | 2 x Pi x Radius | Distance around a circle. |
| Area of a Circle | Pi x Radius^2 | Represents area enclosed by a circle. |
| Area of a Parallelogram | Base x Height | Represents area enclosed by a parallelogram. |
| Area of a Trapezoid | 0.5 x (Base1 + Base2) x Height | Represents area enclosed by a trapezoid. |
| Volume of a Cube | Side^3 | Represents space enclosed by a cube. |
| Volume of a Rectangular Prism | Length x Width x Height | Represents space enclosed by a rectangular prism. |
| Volume of a Cylinder | Pi x Radius^2 x Height | Represents space enclosed by a cylinder. |
| Surface Area of a Sphere | 4 x Pi x Radius^2 | Represents the area around a sphere. |
| Volume of a Sphere | 4/3 x Pi x Radius^3 | Represents space enclosed by a sphere. |
| Volume of a Cone | 1/3 x Pi x Radius^2 x Height | Represents space enclosed by a cone. |
| Surface Area of a Cylinder | 2 x Pi x Radius x (Radius + Height) | Represents the area around a cylinder. |
Algebra Formulas
Algebra, the branch of mathematics dealing with symbols and the rules for manipulating these symbols, is fundamental in exploring and understanding mathematical relationships.
| Name | Formula | Basic Description |
|---|---|---|
| Solving a Linear Equation | mx + b = y | Basic linear equation form. |
| Quadratic Equation | ax^2 + bx + c = 0 | Represents a quadratic equation. |
| Slope of a Line | (y2 - y1) / (x2 - x1) | Rate of change between two points. |
| Pythagorean Theorem | a^2 + b^2 = c^2 | Relationship in a right triangle. |
| Distance Formula | sqrt((x2 - x1)^2 + (y2 - y1)^2) | Distance between two points in a plane. |
| Permutation | nPr = n! / (n-r)! | Number of ways r items can be selected from n items and arranged. |
| Combination | nCr = n! / r!(n-r)! | Number of ways r items can be selected from n items. |
Trigonometry Formulas
Trigonometry, the branch of mathematics dealing with the relationships between the angles and sides of triangles, is essential in fields ranging from engineering to physics.
| Name | Formula | Basic Description |
|---|---|---|
| Sine (sin) | Opposite / Hypotenuse | Ratio of the opposite side to the hypotenuse of a right-angled triangle. |
| Cosine (cos) | Adjacent / Hypotenuse | Ratio of the adjacent side to the hypotenuse of a right-angled triangle. |
| Tangent (tan) | Opposite / Adjacent | Ratio of the opposite side to the adjacent side of a right-angled triangle. |
| Sine Rule | a/sin(A) = b/sin(B) = c/sin(C) | Relationship between sides and angles in any triangle. |
| Cosine Rule | c^2 = a^2 + b^2 - 2abcos(C) | Relates the lengths of the sides of a triangle to the cosine of one of its angles. |
Probability Formulas
Probability, the branch of mathematics concerned with analyzing random events, is all about determining the likelihood of various outcomes.
| Name | Formula | Basic Description |
|---|---|---|
| Simple Probability | P(A) = Number of favorable outcomes / Total outcomes | Calculates the likelihood of a single event occurring. |
| Compound Probability | P(A and B) = P(A) x P(B) | Probability of two independent events both occurring. |
| Conditional Probability | P(A | B) = P(A and B) / P(B) |
| Bayes' Theorem | ||
| Probability of Either Event Occurring | P(A or B) = P(A) + P(B) - P(A and B) | Probability of at least one of two events occurring. |
| Expected Value | E(X) = Σ [x * P(x)] | The average of all possible outcomes, weighted by their probabilities. |
| Variance | Var(X) = Σ [(x - μ)^2 * P(x)] | Measures the spread of a set of outcomes. |
| Standard Deviation | SD(X) = sqrt(Var(X)) | The square root of the variance, measuring dispersion. |
Combinatorial Formulas Used in Probability
Combinatorial analysis provides a way to quantify the probabilities of complex events by counting the number of possible outcomes.
| Name | Formula | Basic Description |
|---|---|---|
| Factorial (n!) | n! = n x (n-1) x ... x 2 x 1 | The product of all positive integers up to n. |
| Permutations (nPr) | nPr = n! / (n-r)! | The number of ways to arrange r objects from a set of n. |
| Combinations (nCr) | nCr = n! / [r! x (n-r)!] | The number of ways to choose r objects from a set of n without regard to order. |