Calculator Guides & How-To Articles
the math behind a calculator makes you a smarter decision-maker. Our guides explain the formulas, assumptions, and real-world context that power every calculation on ThisCalc. Whether you are evaluating a mortgage, planning retirement savings, tracking health metrics, or solving everyday math problems, these articles break down the concepts in plain language with worked examples.
Each guide links directly to the relevant calculator so you can apply what you learn immediately. We also publish side-by-side comparisons that weigh the pros and cons of common financial and health decisions - renting versus buying, fixed versus variable rates, BMI versus body fat percentage, and more. Bookmark this page and check back as we add new guides regularly.
Guides & Articles
Zipf's Law: Why the 2nd Word Is Half as Common
Zipf's law explained: why "the" appears twice as often as "of," how the 1/rank rule reaches cities and websites, the competing theories, and where it breaks.
Metcalfe's Law: Double the Network, Quadruple Value
Metcalfe's Law explained: why a network's value scales as n squared, the n(n-1)/2 connection math, the fax-machine effect, and the famous n log n criticism.
Hex Grid Distance: The Cube-Coordinate Trick
How cube coordinates (q+r+s=0) turn hex grid distance into (|dq|+|dr|+|ds|)/2, why strategy games use three axes, and the classic orientation bug to avoid.
Brainerd-Robinson Coefficient: Dating Sites by Pottery
How archaeologists use the Brainerd-Robinson similarity coefficient (0-200) to order sites in time by their pottery proportions. Formula, history, and the shared-type gotcha.
RAID Rebuild Risk: Why RAID 5 Fails While Healing
Why a degraded RAID 5 rebuild reads every surviving drive and can abort on a single URE. The 10^14 math, rebuild-time formulas, and why RAID 6 survives it.
Relativistic Rocket Equation & Why Starships Need Antimatter
How the Tsiolkovsky rocket equation goes relativistic, why chemical and fusion fuel caps out near 0.1c, and why only antimatter delivers sane starship fuel ratios.
Ham Radio Range: Why a Handheld Hits a Far Repeater
Why a 5-watt handheld reaches a faraway repeater: the radio-horizon formula, why range scales with the square root of antenna height, and a worked example.
How Thick Does Ice Need to Be? Frozen-Lake Physics
Gold's formula P = 50 x T squared explains ice load limits: why 8 inches of clear ice holds about 3,200 lb and strength grows with thickness squared.
The Coriolis Effect: Hurricanes vs. Your Sink
The Coriolis effect explained: why hurricanes spin counterclockwise up north, clockwise down south, never form on the equator, and why the bathtub myth is false.
The Pandolf Equation: Energy Cost of a Ruck March
How the US Army's 1977 Pandolf equation predicts the watts burned during a ruck march from body mass, pack load, speed, grade, and a terrain factor.
How Dead Reckoning Navigation Works (No GPS)
How dead reckoning works: track position from a known start using heading, speed, and time (distance = speed x time), why errors compound, and its GPS-backup role.
The Doppler Effect Explained: Why Sirens Shift Pitch
How the Doppler effect works: wavefront bunching makes a passing siren drop in pitch, and the same physics powers radar, ultrasound, and cosmic redshift.
Dunbar's Number: Why You Cap Out Near 150
Dunbar's number says your brain can hold about 150 stable relationships. Explore the neocortex science, the nested 5-15-50-150 layers, and the live debate.
The Lindy Effect: Why Old Things Keep Lasting
What is the Lindy effect? The idea that non-perishable things like books and ideas age in reverse, plus the Lindy's deli origin, Taleb, and Pareto math.
O'Neill Cylinders: How Spinning Cities Fake Gravity
How O'Neill cylinder artificial gravity works: spin creates centripetal force at the inner wall to mimic 1g, why bigger radius means slower, gentler rotation.
The Physics of Scuba Gas: Partial Pressures
How partial pressure works in scuba diving: Dalton's law explains why nitrogen turns narcotic and oxygen turns toxic with depth, with the math behind every gas mix.
Sous Vide Pasteurization: The Time and Temperature Science
The science of sous vide pasteurization: why time and temperature together kill bacteria, so 140F held long matches an instant 165F. Baldwin's tables explained.
D&D Advantage Probability: Is It Really a +5?
Advantage means rolling 2d20 and keeping the higher. The exact math: it peaks at +5 when you need an 11, stays +4 to +5 across DC 8-14, and fades at the edges.
The Maillard Reaction: Why Browned Food Tastes Amazing
The Maillard reaction explained: how amino acids and reducing sugars brown food above 140C, why a dry surface matters, and how it differs from caramelization.
How Much Bigger Is a Crater Than Its Asteroid?
A crater is typically 10 to 25 times wider than the asteroid that made it. See real crater-to-impactor ratios for Barringer and Chicxulub, plus Pi-group scaling.
How the Alcubierre Warp Drive Works
A plain-English guide to the Alcubierre warp drive: how it warps spacetime to beat light without breaking relativity, the exotic-matter catch, and the energy numbers.
What Is a Kugelblitz? A Black Hole Made of Light
A kugelblitz is a hypothetical black hole made from pure light, not matter. Explore Wheeler's 1955 idea, the Schwarzschild math, and the 2024 paper ruling it out.
How Time of Death Is Estimated: The Henssge Nomogram
How forensic pathologists estimate time of death using the Henssge nomogram: body-cooling science, the Marshall-Hoare two-exponential curve, and why it works.
How a Hohmann Transfer Orbit Works
How a Hohmann transfer orbit works: the two-burn, minimum-fuel maneuver that stretches a circular orbit into an ellipse, with real delta-v numbers and the vis-viva math.
Bi-Elliptic Transfer: The Maneuver That Beats Hohmann
When the final-to-initial orbit radius ratio exceeds about 11.94, a three-burn bi-elliptic transfer uses less total delta-v than the "optimal" Hohmann. Here is why.
How Much Does Time Slow Down Near Light Speed?
How much does time slow down near light speed? At 0.9c time runs ~2.3x slower, ~7x at 0.99c. See the Lorentz factor, the twin paradox, and real proof.
How the Elo Rating System Works
How the Elo rating system works: the expected-score formula, why a 100-point gap means 64% odds, the K-factor dial, and why upsets pay big in chess and games.
The Kelly Criterion: The Math of How Much to Bet
How the Kelly Criterion sizes a bet when you have an edge: the f=(bp-q)/b formula, a worked coin example, why it maximizes growth, plus fractional Kelly.
Coffee Extraction Explained: Ratio, TDS & the 18-22% Rule
Coffee extraction percentage and the golden ratio explained: the SCA 1:18 standard, brew strength (TDS) versus extraction yield, the formula, and why 18-22% is best.
How to Calculate an Asteroid Impact's Energy in Megatons
Calculate an asteroid impact's energy in megatons of TNT: get mass from diameter and density, apply KE = 1/2 m v squared, scale by angle, then convert.
Mortgage Recast vs Refinance vs Prepay
Mortgage recast vs refinance vs lump-sum prepay: the real dollar math, the rate rule that picks your winner, and which loans can't recast at all.
The 20/4/10 Rule: How Much Car Can I Afford?
The 20/4/10 rule car affordability guide: 20% down, 4-year max term, all car costs under 10% of income. See the per-salary ceiling and the hidden trap.
How Much to Save Per Month to Reach a Goal
Learn how much to save per month to reach a goal with one simple formula, then see how a savings rate shrinks the number, with a table of worked examples.
Should I Pay Off Debt or Invest First?
Should I pay off debt or invest first? Use a simple rate-vs-return rule, with clear thresholds and a worked example, to decide where your next dollar goes.
How Long to Save $1 Million: The Timeline Grid
How long to save $1 million depends on your monthly contribution and return. See the exact monthly amount needed at 10, 20, 30, and 40 years.
How Much Will My Bonus Be Taxed? The Honest Answer
Wondering how much will my bonus be taxed? Learn why withholding looks so high, the flat 22% rule vs. your real tax, and what you actually keep after taxes.
How Much to Set Aside for 1099 Taxes
Wondering how much to set aside for 1099 taxes? Save 25-35% of net profit. Here is how to build that number from SE tax plus your federal bracket.
How to Calculate Quarterly Estimated Taxes
Learn how to calculate quarterly estimated taxes, the 2026 due dates, both safe-harbor rules, and how to avoid the underpayment penalty as a freelancer.
Marginal vs Effective Tax Rate Explained
Marginal vs effective tax rate, explained with a clear worked example. Learn why a raise never lowers your take-home and which rate to use for which decision.
How Many Calories to Lose 1 Pound a Week
How many calories to lose 1 pound a week? Cut 500/day from your TDEE. See the deficit-to-rate table, safe calorie floors, and why a stall is normal.
How to Get Rid of PMI: The Two Thresholds
How to get rid of PMI: the 20% request milestone vs the 22% automatic-cancellation rule, a worked $300k example, and three ways to drop it faster.
Extra to Pay Off a 30-Year Mortgage in 15 Years
How much extra to pay off a 30-year mortgage in 15 years? See the monthly principal by balance and rate, plus the six figures in interest you can save.
How Much Does 1% Affect Your Mortgage Payment?
How much does 1 percent affect mortgage payment? See the dollar impact on monthly P&I, lifetime interest, and your buying power, with a worked example.
How Long Will $1 Million Last in Retirement?
How long will $1 million last in retirement? See how spending, withdrawal rate, Social Security, and inflation change the answer from under 10 to 30+ years.
Max Out 401(k) or Roth IRA First? The Right Order
Should you max your 401(k) or Roth IRA first? Follow the contribution waterfall: capture the match, then the Roth IRA, then max the 401(k). With a 2026 example.
How Much Will My 401(k) Be Worth at Retirement?
Wondering how much your 401(k) will be worth at retirement? Learn the three drivers, see a worked $100k-salary example, and project your own balance free.
How Much Do I Need to Retire at 55?
How much do I need to retire at 55? Learn the savings target, why 55 costs more than 65, and how to bridge the gap before Medicare and penalty-free withdrawals.
Is Overtime Taxed More in 2026? The Real Answer
Is overtime taxed more in 2026? No. Learn why your check looked smaller, how marginal rates really work, and the new No Tax on Overtime deduction explained.
Does Side Gig Income Raise My Tax Bracket?
Does side gig income raise your tax bracket? See how 1099 income stacks on W-2 wages, why self-employment tax stings, and how much to set aside, with the math.
How Increasing Your 401(k) Affects Take-Home Pay
Wondering how increasing 401k affects take-home pay? A $1 pre-tax contribution cuts your paycheck by only ~$0.75, not $1. See the percent-to-dollars math.
How to Calculate Mortgage Payments: A Complete Guide
Learn how to calculate monthly mortgage payments with the P&I formula, understand principal vs interest, and see how rates affect your costs.
Understanding Compound Interest: The Complete Guide
Learn how compound interest works, the formula behind it, how compounding frequency matters, and the Rule of 72 with real-world examples.
How to Calculate BMI: Formula, Ranges & What It Means
Learn the BMI formula for metric and imperial units, understand BMI ranges, and discover what your number really means for your health.
How to Calculate Percentages: 5 Easy Methods With Examples
Master five practical methods for calculating percentages, from basic formulas to mental math shortcuts, with worked examples for each.
How Much House Can I Afford? Rules, Formulas & Calculator
Use the 28/36 rule, front-end and back-end DTI ratios, and our calculator to find out how much home you can realistically afford.
How Much Should You Have Saved for Retirement by Age?
See retirement savings benchmarks by age, learn the 4% rule, and calculate how much you need to retire comfortably at every life stage.
How to Save for a House: Down Payment Guide
A step-by-step guide to saving for your first home, from setting a down payment target to choosing the right savings strategy.
Debt Payoff Strategies: Snowball vs Avalanche
Compare the debt snowball and avalanche methods with real examples. Find the best approach to become debt-free faster.
Home Renovation Cost Guide: Room by Room
Estimate renovation costs for every room in your home with material quantities, labor rates, and budget-saving tips.
How to Calculate Body Composition: BMI, Body Fat & More
Understand BMI, body fat percentage, and ideal weight formulas. Compare measurement methods and learn what the numbers mean.
Financial Independence (FIRE): Calculate Your Number
Learn the FIRE formula, calculate your financial independence number, and build a savings plan to retire early.
Understanding Investment Returns: ROI, CAGR, and Key Metrics Explained
Learn how to measure investment returns with ROI, CAGR, time-weighted returns, and risk-adjusted metrics like the Sharpe ratio.
GPA Guide: Understanding Scales, Weighted vs Unweighted, and Planning for College
Learn how GPA scales work, the difference between weighted and unweighted GPA, how colleges evaluate your GPA, and strategies to improve.
Running Pace Calculator Guide: Training Zones, VO2 Max, and Race Pacing
Learn how running pace works, how to train in different zones, what VO2 max means for performance, and how to build a race day pacing strategy.
How Tax Brackets Work: Marginal Rates, Deductions, and Effective Tax Rate
Understand how the progressive tax system works, the difference between marginal and effective tax rates, and how to estimate your tax liability.
Comparisons
Side-by-side analysis to help you make informed decisions. View all comparisons →
Renting vs Buying a Home
Compare the true costs of renting vs buying a home. Side-by-side analysis of monthly costs, equity, flexibility, and long-term wealth building.
Simple vs Compound Interest
Understand the difference between simple and compound interest with clear examples. See how compounding frequency affects growth.
Fixed vs Variable Interest Rate
Compare fixed and variable interest rates for mortgages and loans. Learn when each rate type saves you money and the risks involved.
BMI vs Body Fat Percentage
Compare BMI and body fat percentage as health metrics. Learn what each measures, their accuracy limitations, and which gives a better picture.
Leasing vs Buying a Car
Compare leasing vs buying a car with real cost breakdowns. Analyze monthly payments, total cost of ownership, and flexibility.
401(k) vs IRA
Compare 401(k) and IRA retirement accounts side by side. Contribution limits, tax advantages, employer matching, and which to prioritize.
Debt Snowball vs Avalanche
Compare the debt snowball and avalanche methods. See which pays off debt faster, which saves more interest, and which keeps you motivated.
15-Year vs 30-Year Mortgage
Compare 15-year and 30-year mortgage terms. See how monthly payments, total interest, and equity building differ between each option.
Metric vs Imperial
Compare the metric and imperial measurement systems. Understand the key differences, conversion factors, and which countries use each.
Calories vs Macros
Compare calorie counting and macro tracking for diet management. Learn the pros, cons, and which approach fits your fitness goals.