📐 Mathematics & Formulas

Comprehensive mathematical calculators and formulas for geometry, algebra, trigonometry, calculus and statistics.

Calculator
Formulas
Examples
Reference
Guide

Interactive Math Calculator

Calculator Features: Select a mathematical category and formula type to perform calculations. All calculations show step-by-step solutions and explanations.

Mathematical Formulas

Geometry - Area Formulas
Rectangle: A = l × w
Circle: A = π × r²
Triangle: A = ½ × b × h
Square: A = s²

Essential area calculations for 2D shapes.

Geometry - Volume Formulas
Cube: V = s³
Sphere: V = (4/3) × π × r³
Cylinder: V = π × r² × h
Cone: V = (1/3) × π × r² × h

Volume calculations for 3D shapes.

Algebra - Quadratic Formula
x = (-b ± √(b²-4ac)) / 2a

For equation: ax² + bx + c = 0

Solves quadratic equations where a ≠ 0.

Trigonometry - Basic Functions
sin²θ + cos²θ = 1
tan θ = sin θ / cos θ
Law of Cosines: c² = a² + b² - 2ab cos C

Fundamental trigonometric relationships.

Calculus - Derivatives
d/dx[xⁿ] = n × xⁿ⁻¹
d/dx[eˣ] = eˣ
d/dx[ln x] = 1/x
d/dx[sin x] = cos x

Basic derivative formulas for common functions.

Statistics - Measures
Mean: x̄ = Σx / n
Variance: σ² = Σ(x-μ)² / n
Standard Deviation: σ = √σ²

Central tendency and dispersion measures.

Worked Examples

🔺 Triangle Area Calculation

Problem: Find the area of a triangle with base = 10 cm and height = 6 cm

Formula: A = ½ × base × height

Solution: A = ½ × 10 × 6 = 30 cm²

Answer: The area is 30 square centimeters

⭕ Circle Area & Circumference

Problem: Find area and circumference for radius = 5 cm

Area: A = π × r² = π × 5² = 25π ≈ 78.54 cm²

Circumference: C = 2πr = 2π × 5 = 10π ≈ 31.42 cm

📊 Quadratic Equation

Problem: Solve x² - 5x + 6 = 0

Values: a = 1, b = -5, c = 6

Discriminant: b² - 4ac = 25 - 24 = 1

Solutions: x = (5 ± 1)/2 = 3 or 2

📐 Pythagorean Theorem

Problem: Find hypotenuse when legs are 3 and 4

Formula: c² = a² + b²

Solution: c² = 3² + 4² = 9 + 16 = 25

Answer: c = √25 = 5

📈 Statistics Example

Data: [2, 4, 6, 8, 10]

Mean: (2+4+6+8+10)/5 = 6

Variance: [(2-6)²+(4-6)²+...]/5 = 8

Std Dev: √8 ≈ 2.83

🔄 Trigonometry

Problem: Find sin, cos, tan for 30°

sin 30°: 1/2 = 0.5

cos 30°: √3/2 ≈ 0.866

tan 30°: 1/√3 ≈ 0.577

Mathematical Reference

🔢 Common Constants
  • π (Pi): 3.14159265359... (circumference/diameter)
  • e (Euler's number): 2.71828182846...
  • φ (Golden ratio): 1.61803398875...
  • √2: 1.41421356237...
  • √3: 1.73205080757...
📏 Unit Conversions
  • Length: 1 m = 100 cm = 1000 mm
  • Area: 1 m² = 10,000 cm² = 10.764 ft²
  • Volume: 1 m³ = 1000 L = 264.17 gal
  • Angles: 180° = π radians
  • Temperature: °C = (°F - 32) × 5/9
📐 Angle Values
  • 0°: sin=0, cos=1, tan=0
  • 30°: sin=1/2, cos=√3/2, tan=1/√3
  • 45°: sin=√2/2, cos=√2/2, tan=1
  • 60°: sin=√3/2, cos=1/2, tan=√3
  • 90°: sin=1, cos=0, tan=undefined
🧮 Order of Operations
  • PEMDAS/BODMAS:
  • 1. Parentheses/Brackets
  • 2. Exponents/Orders
  • 3. Multiplication and Division (left to right)
  • 4. Addition and Subtraction (left to right)
Logarithm Rules
log(ab) = log(a) + log(b)
log(a/b) = log(a) - log(b)
log(aⁿ) = n × log(a)
Exponent Rules
aᵐ × aⁿ = aᵐ⁺ⁿ
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
(aᵐ)ⁿ = aᵐⁿ

Mathematics Guide

🎯 Getting Started
  • Select the mathematical category (Geometry, Algebra, etc.)
  • Choose the specific formula or calculation type
  • Enter the required values in the input fields
  • Click "Calculate" to see the result and steps
  • Use "Show Steps" for detailed explanations
📚 Study Tips
  • Practice regularly with different problem types
  • Understand the reasoning behind formulas
  • Check your work using different methods
  • Memorize key formulas and constants
  • Draw diagrams to visualize problems
🔍 Problem-Solving Strategy
  • Read: Understand what is being asked
  • Plan: Identify the appropriate formula
  • Solve: Apply the formula step by step
  • Check: Verify your answer makes sense
  • Review: Learn from any mistakes
⚠️ Common Mistakes
  • Forgetting to follow order of operations
  • Using wrong units or forgetting unit conversions
  • Confusing radius and diameter in circle formulas
  • Making sign errors in algebra
  • Rounding too early in multi-step calculations
🏆 Best Practices
  • Always label your answers with appropriate units
  • Show your work step by step
  • Double-check calculations
  • Use approximations to verify reasonableness
  • Keep track of significant figures
🔧 Calculator Features
  • Interactive visual previews for geometry
  • Step-by-step solution explanations
  • Multiple calculation modes per category
  • Copy results for use in other applications
  • Clear explanations of formulas used