# Quick Reference: Common Formulas & Constants

## Physical Constants

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Speed of Light (c) | 299,792,458 m/s ≈ 3×10⁸ m/s | In vacuum; approximately 300,000 km/s or 186,000 mi/s |
| Speed of Sound | ~343 m/s (1,235 km/h or 767 mph) at 20°C | Varies with temperature: v ≈ 331 + 0.6T (where T is °C) |
| Gravity (g) | 9.8 m/s² (Earth surface) | Often approximated as 10 m/s² for quick calculations |
| Pi (π) | 3.14159265... | Commonly used as 3.14 or 22/7 |
| Euler's Number (e) | 2.71828... | Base of natural logarithms |
| Golden Ratio (φ) | 1.618033... | (1 + √5) / 2 |

## Weather & Atmospheric

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Thunder Distance | Distance (km) = Seconds / 3 | Count seconds between lightning and thunder; divide by 5 for miles |
| Lightning Speed | ~220,000 km/h (1/3 speed of light) | Thunder travels at speed of sound |
| Temperature Conversion | °F = (°C × 9/5) + 32; °C = (°F - 32) × 5/9 | Kelvin: K = °C + 273.15 |
| Heat Index Estimate | Feels hotter when humidity >60% | Rule: Each 10% humidity increase feels like ~1°F warmer above 80°F |
| Wind Chill Rough | Feels colder with wind | At 0°C with 50 km/h wind feels like -10°C |

## Basic Trigonometry

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Sine (sin) | sin(θ) = opposite/hypotenuse | In right triangle |
| Cosine (cos) | cos(θ) = adjacent/hypotenuse | In right triangle |
| Tangent (tan) | tan(θ) = opposite/adjacent = sin/cos | In right triangle |
| Pythagorean Theorem | a² + b² = c² | c is hypotenuse of right triangle |
| SOH-CAH-TOA | Sine=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj | Memory aid for trig ratios |
| Common Angles | sin(30°)=0.5, sin(45°)≈0.707, sin(60°)≈0.866 | cos(30°)≈0.866, cos(45°)≈0.707, cos(60°)=0.5 |
| Radian Conversion | Radians = Degrees × π/180 | 180° = π radians; 360° = 2π radians |
| Circle Circumference | C = 2πr = πd | r=radius, d=diameter |
| Circle Area | A = πr² | r=radius |

## Motion & Physics

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Distance | d = v × t | Distance = velocity × time |
| Acceleration | a = (v - u) / t | a=acceleration, v=final velocity, u=initial velocity, t=time |
| Force | F = m × a | Newton's 2nd Law; mass × acceleration |
| Kinetic Energy | KE = ½mv² | m=mass, v=velocity |
| Potential Energy | PE = mgh | m=mass, g=gravity, h=height |
| Work | W = F × d | Force × distance (in direction of force) |
| Power | P = W/t = E/t | Work or energy per unit time |
| Momentum | p = m × v | Mass × velocity |
| Impulse | J = F × t = Δp | Force × time = change in momentum |
| Free Fall Distance | d = ½gt² | Distance fallen after t seconds (starting from rest) |
| Free Fall Velocity | v = gt | Velocity after t seconds of free fall |

## Electricity & Electronics

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Ohm's Law | V = I × R | Voltage = Current × Resistance |
| Power (Electrical) | P = V × I = I²R = V²/R | Watts = Volts × Amps |
| Series Resistance | R_total = R₁ + R₂ + R₃ + ... | Resistances add in series |
| Parallel Resistance | 1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ... | For two: R = (R₁×R₂)/(R₁+R₂) |
| Battery Life | Hours = mAh / mA | Battery capacity divided by current draw |

## Area & Volume

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Rectangle Area | A = length × width | Also for square: A = s² |
| Triangle Area | A = ½ × base × height | Or use Heron's formula for 3 sides |
| Trapezoid Area | A = ½(b₁ + b₂) × h | b₁, b₂ are parallel sides, h is height |
| Sphere Volume | V = 4/3 πr³ | r=radius |
| Sphere Surface Area | A = 4πr² | r=radius |
| Cylinder Volume | V = πr²h | r=radius, h=height |
| Cylinder Surface Area | A = 2πr² + 2πrh | Two circles plus side |
| Cone Volume | V = 1/3 πr²h | One-third of cylinder |
| Cube Volume | V = s³ | s=side length |
| Cube Surface Area | A = 6s² | Six square faces |
| Rectangular Prism Volume | V = l × w × h | Length × width × height |

## Probability & Statistics

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Mean (Average) | x̄ = Σx / n | Sum of values divided by count |
| Probability | P(event) = favorable outcomes / total outcomes | Value between 0 and 1 |
| Percentage | Percentage = (part/whole) × 100 | To find what % one number is of another |
| Percent Change | % change = [(new - old) / old] × 100 | Positive = increase, negative = decrease |
| Compound Interest | A = P(1 + r/n)^(nt) | P=principal, r=rate, n=compounds/year, t=years |
| Simple Interest | I = P × r × t | P=principal, r=rate, t=time |

## Distance & Speed Conversions

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| mph to km/h | km/h = mph × 1.609 | 60 mph ≈ 97 km/h |
| km/h to m/s | m/s = km/h ÷ 3.6 | 100 km/h ≈ 27.8 m/s |
| mph to m/s | m/s = mph × 0.447 | 60 mph ≈ 26.8 m/s |
| Nautical Mile | 1 nautical mile = 1.852 km | Used in maritime and aviation |
| Knot | 1 knot = 1 nautical mile/hour ≈ 1.852 km/h | Speed measurement at sea |

## Length Conversions

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Inch to cm | cm = inches × 2.54 | 1 inch = exactly 2.54 cm |
| Foot to meter | meters = feet × 0.3048 | 1 foot ≈ 30.48 cm |
| Mile to km | km = miles × 1.609 | 1 mile ≈ 1.6 km |
| Meter to yard | yards = meters × 1.094 | 1 meter ≈ 1.09 yards |

## Volume & Weight Conversions

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Liter to gallon (US) | gallons = liters × 0.264 | 1 liter ≈ 0.264 US gallons |
| Gallon to liter | liters = gallons × 3.785 | 1 US gallon ≈ 3.785 liters |
| Kilogram to pound | pounds = kg × 2.205 | 1 kg ≈ 2.2 lbs |
| Pound to kg | kg = pounds × 0.454 | 1 lb ≈ 0.45 kg |
| Ounce to gram | grams = ounces × 28.35 | 1 oz ≈ 28.35 g |
| Cup to mL | mL = cups × 237 | 1 US cup ≈ 237 mL |

## Cooking & Kitchen

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Tablespoon to teaspoon | 1 Tbsp = 3 tsp | Common cooking conversion |
| Cup measurements | 1 cup = 16 Tbsp = 48 tsp | Also 1 cup = 8 fluid oz |
| Boiling point water | 100°C = 212°F (at sea level) | Decreases with altitude |
| Freezing point water | 0°C = 32°F | Standard pressure |
| Oven temp conversion | Gas Mark 4 ≈ 180°C ≈ 350°F | Common baking temperature |

## Time Calculations

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Seconds in a day | 86,400 seconds | 24 × 60 × 60 |
| Days in a year | 365.25 days (average) | Accounts for leap years |
| Hours in a year | 8,760 hours (regular year) | 8,784 in leap year |
| Rule of 72 | Years to double = 72 / interest rate | Estimate investment doubling time |

## Geometry & Angles

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Triangle angles | Sum = 180° | All three angles in any triangle |
| Quadrilateral angles | Sum = 360° | All four angles in any quadrilateral |
| Polygon angles | Sum = (n-2) × 180° | n=number of sides |
| Complementary angles | Sum = 90° | Two angles that add to a right angle |
| Supplementary angles | Sum = 180° | Two angles that add to a straight line |
| Exterior angle | Exterior = sum of opposite interior angles | For triangles |

## Digital & Computing

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Bits to Bytes | 1 Byte = 8 bits | Basic digital storage unit |
| Kilobyte (KB) | 1 KB = 1,024 bytes | Binary: 2¹⁰ bytes |
| Megabyte (MB) | 1 MB = 1,024 KB ≈ 1 million bytes | 2²⁰ bytes exactly |
| Gigabyte (GB) | 1 GB = 1,024 MB ≈ 1 billion bytes | 2³⁰ bytes exactly |
| Terabyte (TB) | 1 TB = 1,024 GB ≈ 1 trillion bytes | 2⁴⁰ bytes exactly |
| Bit rate | Mbps = megabits per second | Internet speed; 8 Mbps = 1 MB/s |

## Astronomy & Space

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Light year | ~9.46 trillion km | Distance light travels in one year |
| Astronomical Unit (AU) | ~150 million km | Average Earth-Sun distance |
| Parsec | 3.26 light years | Astronomical distance measurement |
| Earth circumference | ~40,075 km (at equator) | About 24,900 miles |
| Moon distance | ~384,400 km from Earth | Average; varies with orbit |
| Sun distance | ~150 million km (1 AU) | Average Earth-Sun distance |

## Human Body

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Normal body temp | 37°C = 98.6°F | Average; varies by person and time |
| Resting heart rate | 60-100 beats/min (adult) | Athletes often lower (40-60) |
| Maximum heart rate | ~220 - age | Rough estimate for exercise |
| Blood volume | ~5 liters (adult) | About 7% of body weight |
| Water intake | ~2-3 liters/day | Varies by activity and climate |

## Practical Estimates

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Walking speed | ~5 km/h or 3 mph | Average comfortable pace |
| Running speed | ~10 km/h or 6 mph | Jogging pace |
| Bicycle speed | ~15-20 km/h (casual) | Varies greatly with fitness/terrain |
| Typing speed | ~40 words/min (average) | 60-80 wpm is good; 100+ is excellent |
| Reading speed | ~200-250 words/min | Average adult; varies widely |
| Car fuel economy | ~7-10 L/100km (good) | Or 25-35 mpg (US) |

## Algebra Basics

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Quadratic formula | x = [-b ± √(b²-4ac)] / 2a | Solves ax² + bx + c = 0 |
| Slope | m = (y₂-y₁)/(x₂-x₁) | Rise over run |
| Line equation | y = mx + b | m=slope, b=y-intercept |
| Point-slope form | y - y₁ = m(x - x₁) | Using known point (x₁,y₁) |
| Distance between points | d = √[(x₂-x₁)² + (y₂-y₁)²] | 2D Pythagorean theorem |
| Midpoint | M = [(x₁+x₂)/2, (y₁+y₂)/2] | Average of coordinates |

## Logarithms & Exponents

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Exponent rules | x^a × x^b = x^(a+b) | Also: x^a / x^b = x^(a-b) |
| Power of power | (x^a)^b = x^(ab) | Multiply exponents |
| Log definition | log_b(x) = y means b^y = x | Base b logarithm |
| Log rules | log(xy) = log(x) + log(y) | Product rule |
| Log rules | log(x/y) = log(x) - log(y) | Quotient rule |
| Log rules | log(x^n) = n × log(x) | Power rule |
| Change of base | log_b(x) = log(x) / log(b) | Convert to common logs |

## Physics Constants (additional)

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Planck's constant (h) | 6.626 × 10⁻³⁴ J·s | Quantum mechanics fundamental |
| Avogadro's number | 6.022 × 10²³ mol⁻¹ | Particles per mole |
| Gas constant (R) | 8.314 J/(mol·K) | Ideal gas law |
| Boltzmann constant (k) | 1.381 × 10⁻²³ J/K | Per particle gas constant |
| Electron charge (e) | 1.602 × 10⁻¹⁹ C | Elementary charge |

## Financial Calculations

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Tip calculation | Tip = Bill × % (as decimal) | 20% tip = Bill × 0.20 |
| Discount price | Sale = Original × (1 - discount%) | 30% off = Original × 0.70 |
| Sales tax | Total = Price × (1 + tax rate) | 8% tax = Price × 1.08 |
| Markup | Selling price = Cost × (1 + markup%) | 50% markup = Cost × 1.50 |
| Profit margin | Margin % = (Revenue - Cost)/Revenue × 100 | Net profit / revenue |
| Break-even | Units = Fixed costs / (Price - Variable cost) | When profit = 0 |

## pH & Chemistry

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| pH definition | pH = -log[H⁺] | Measure of acidity |
| Neutral pH | pH = 7 (pure water) | <7 is acidic, >7 is basic |
| Molar mass calculation | Sum atomic masses | From periodic table |
| Molarity | M = moles / liters | Concentration measure |
| Ideal gas law | PV = nRT | P=pressure, V=volume, n=moles, R=constant, T=temp |

## Sound & Acoustics

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Decibel (dB) | dB = 10 log(I/I₀) | Logarithmic intensity scale |
| Hearing threshold | 0 dB | Quietest sound humans can hear |
| Conversation | ~60 dB | Normal speaking volume |
| Hearing damage | >85 dB (prolonged) | >120 dB causes immediate damage |
| Frequency (pitch) | f = 1/T | Frequency = 1 / period |
| Wavelength | λ = v/f | Speed / frequency |

## Pressure & Atmosphere

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Atmospheric pressure | 101,325 Pa = 1 atm = 14.7 psi | At sea level |
| Pressure formula | P = F/A | Force per unit area |
| Depth pressure | P = ρgh | ρ=density, g=gravity, h=depth |
| Boyle's Law | P₁V₁ = P₂V₂ | At constant temperature |
| PSI to kPa | kPa = psi × 6.895 | 1 psi ≈ 6.9 kPa |
| Tire pressure | ~32-35 psi (typical car) | 220-240 kPa |

## Building & Construction

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Board feet | BF = (thickness × width × length) / 144 | Thickness & width in inches, length in feet |
| Concrete volume | Cubic yards = (L × W × H) / 27 | L, W, H in feet |
| Paint coverage | ~350-400 sq ft per gallon | One coat; varies by surface |
| Roof pitch | Pitch = rise / run | Often expressed as "4:12" (4" rise per 12" run) |
| Stair rise/run | Rise: 7-8", Run: 10-11" | Building code typical |

## Gardening & Agriculture

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Area to acres | Acres = sq ft / 43,560 | 1 acre = 43,560 sq ft |
| Hectare | 1 hectare = 10,000 m² = 2.47 acres | Metric land measure |
| Fertilizer NPK | Three numbers = N-P-K ratio | Nitrogen-Phosphorus-Potassium percentages |
| Watering depth | 1" of water = 0.62 gallons/sq ft | Typical garden watering |
| Growing season | Days between last spring & first fall frost | Varies by location |

## Navigation & Maps

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Map scale | Real distance = Map distance × Scale | 1:50,000 means 1 cm = 0.5 km |
| Compass bearings | 0°/360° = North, 90° = East, 180° = South, 270° = West | Measured clockwise from North |
| Great circle distance | Use haversine formula | Shortest path on sphere |
| Latitude degree | ~111 km (69 miles) | Varies slightly, but roughly constant |
| Longitude degree | Varies: ~111 km × cos(latitude) | Converges at poles |

## Miscellaneous Useful

| Concept | Formula/Value | Notes |
|---------|---------------|-------|
| Fibonacci sequence | F(n) = F(n-1) + F(n-2) | 0, 1, 1, 2, 3, 5, 8, 13, 21... |
| Prime numbers start | 2, 3, 5, 7, 11, 13, 17, 19, 23, 29... | Only divisible by 1 and themselves |
| Roman numerals | I=1, V=5, X=10, L=50, C=100, D=500, M=1000 | Subtract when smaller before larger |
| Military time | 13:00 = 1:00 PM | Add 12 to PM hours (except 12 PM) |
| Knuckle month trick | Knuckles = 31 days, valleys = 30 or fewer | Except February |
| Pregnancy duration | ~40 weeks = 280 days | From last menstrual period |

---

This reference is designed for quick lookup when you need formulas without access to internet or textbooks!