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Guide · 13 lessons

Learn the math behind fitness and sleep calculators

These are short guides, about 5 minutes each, to the formulas behind the calculators. Each one walks through the math with a worked example and lists its sources with a last-checked date. A quick check at the end lets you test what you read, and a link takes you straight to the matching calculator. Your progress stays in this browser.

Every number is shown with its formula and its source, so you can check the math yourself.

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Module 1 Energy and calories

How BMR, TDEE, calorie tables and macro grams are calculated, and where the equations and multipliers come from.

  1. Mifflin-St Jeor vs Harris-Benedict: how the BMR equations differBoth equations predict resting energy use from weight, height, age and sex. They were fitted to different groups of people decades apart, so the coefficients differ and Harris-Benedict usually lands a little higher.6 min read✓ Read
  2. Activity multipliers explained: where 1.2 to 1.9 come fromTDEE is BMR times an activity factor. The five calculator steps are widely used rounded multipliers. The FAO/WHO/UNU report measures the same idea as a physical activity level (PAL) and puts most lifestyles between 1.40 and 2.40.6 min read✓ Read
  3. Calorie math: percentage vs fixed changes from maintenanceA percentage scales with the size of the maintenance estimate; a fixed number does not. The same 500 kcal is a small share of a large estimate and a large share of a small one. The 3,500-kcal shortcut ignores how energy use changes over time.6 min read✓ Read
  4. Macro grams from calories: Atwater factors and the AMDREach gram of protein or carbohydrate counts as 4 kcal and each gram of fat as 9 kcal. Multiply calories by a percentage, divide by the factor, and you have grams. The AMDR gives the ranges of percentages published for adults.6 min read✓ Read

Module 2 Body measurements

The math behind BMI and the U.S. Navy body fat method, the published cutoffs, and what each one can't tell you.

  1. The BMI formula and its limitsBMI is weight divided by height squared. It began as a population statistic in 1832 and got its name in 1972. WHO and CDC use it to screen adults, but it does not separate fat from muscle or bone.6 min read✓ Read
  2. The U.S. Navy body fat method: from tape measure to percentageTwo 1984 Navy reports fitted log equations that predict body density from a few tape measurements and height. Siri's 1961 equation then turns density into a fat percentage. The reports put the typical error at about 3.5 to 4.4 percentage points.6 min read✓ Read

Module 3 Protein and water

How published intake benchmarks for protein and total water turn into grams and liters for your numbers.

  1. Protein per kg: the RDA vs the ISSN rangeThe RDA of 0.8 g per kg per day is set to cover nearly all healthy adults. The ISSN range of 1.4 to 2.0 g per kg is a sports-nutrition benchmark for people who exercise. They answer different questions, so they give different numbers.6 min read✓ Read
  2. Water intake reference values: EFSA vs US/CanadaBoth bodies publish adequate intakes of total water, which counts drinks and the water in food. EFSA's 2010 values are 2.5 L for men and 2.0 L for women; the 2004 US/Canada values are 3.7 L and 2.7 L. They are reference values, not a glass count.6 min read✓ Read

Module 4 Training math

Heart rate zones, max heart rate formulas, running pace and race estimates, and one rep max equations, step by step.

  1. Karvonen vs percent of max: two ways to set heart rate zonesPercent of max multiplies your maximum heart rate. The Karvonen method takes a percentage of the gap between resting and maximum, then adds the resting rate back. The same percentage gives a higher number with Karvonen.6 min read✓ Read
  2. Max heart rate: 220 − age vs 208 − 0.7 × age220 − age traces back to 1970s reviews and was never fitted to a large data set. Tanaka's 208 − 0.7 × age came from a meta-analysis of 18,712 people. The two agree at age 40 and drift apart above and below it. Either can miss a given person by about 10 bpm.6 min read✓ Read
  3. Pace, time and distance: the running math, plus Riegel's race estimatesPace is time divided by distance; speed is the inverse. Converting between kilometers and miles is one multiplication. Riegel's formula scales a race time to a new distance with an exponent near 1.06, and it was built for efforts of about 3.5 to 230 minutes.6 min read✓ Read
  4. One rep max: how the Epley and Brzycki formulas workEpley adds one thirtieth of the weight per rep; Brzycki divides by a factor that shrinks with each rep. They give the same answer at exactly 10 reps. Below that Epley reads higher, above it Brzycki does, and both get less reliable past about 10 reps.6 min read✓ Read

Module 5 Sleep

How sleep cycle timing is counted, why 90 minutes is an average, and what the published sleep-hour numbers say.

  1. Sleep cycle math: where the 90-minute rule comes fromThe calculator subtracts the time it takes to fall asleep, then steps back in 90-minute blocks. Real cycles last about 90 to 110 minutes and change through the night, so the times are a guide, not an alarm setting. NINDS puts most adults at 7 to 9 hours.6 min read✓ Read

These guides explain published formulas and reference values. They are not medical advice, and the formulas aren't made for pregnancy, children or medical conditions. Each guide shows the date it was last checked.

Educational calculators based on published formulas. Not medical advice, diagnosis or treatment, and not made for pregnancy, children or medical conditions. Health disclaimer · Terms of use.