Tutorials
Short reads that explain the math behind each calculator — formulas, pitfalls, and why the numbers work.
How to Calculate BMI
Mass over height squared — and what the number does not mean
Body mass index (BMI) is a screening ratio: body mass divided by the square of height. This tutorial walks through the SI formula, a 70 kg / 1.75 m worked example, and WHO adult category bands you can try with sliders. BMI is not a diagnosis of health, body fat, or disease.
Read article → Fitness · 5 min readHow to Calculate BMR
Mifflin–St Jeor calories at rest — and what the number is not
Basal metabolic rate (BMR) estimates the calories your body uses at rest. This tutorial walks through the Mifflin–St Jeor formulas, a 70 kg / 175 cm / age 30 worked example, and a playground that substitutes your numbers. BMR is an estimate of resting energy, not a meal plan or a medical assessment.
Read article → Date Time · 5 min readHow to Calculate Age from Dates
Full years, then leftover months and days — calendar age, not a health score
Calendar age is the elapsed time from a birth date to an as-of date, written as completed years, then remaining months, then remaining days. This tutorial walks through that breakdown and a two-date playground. Age here is civil arithmetic, not a medical assessment, not a diagnosis, and not biological age.
Read article → Personal Finance · 5 min readHow to Read an Amortization Schedule
Each row splits one payment into interest, principal, and a new balance
An amortization schedule lists every period of a fixed installment loan: interest on the current balance, principal from the rest of the payment, and the balance that remains. This tutorial walks through how to read those columns, a $20,000 / 6.5% / 5-year monthly worked example, and a playground that shows first-year principal versus interest with an extra-payment toggle. Figures are illustrative; fees and taxes are omitted.
Read article → Finance · 5 min readHow to Calculate APY from APR
Effective yearly yield after compounding — APY = (1 + r/m)^m − 1
APY converts a nominal APR and a compounding frequency into an effective yearly rate. This tutorial walks through APY = (1 + r/m)^m − 1, a 5% APR / monthly worked example, and a playground that turns the frequency dial so you can see the effective-yield gap. Figures are illustrative; fees and taxes are omitted.
Read article → Geometry · 5 min readHow to Calculate Area of Common Shapes
Rectangle, triangle, and trapezoid — count the square units inside
Area is the measure of a region in the plane: how many unit squares fit inside the boundary. This tutorial walks through A = ℓ w for a rectangle (the Area calculator), A = ½ b h for a triangle, and A = ½ (a + b) h for a trapezoid, an 8 × 5 worked example, and a playground that redraws the labeled figure as you switch shapes.
Read article → Statistics · 5 min readHow to Calculate an Average (Mean)
Sum divided by count — the balance point of a list
The arithmetic mean is the sum of the values divided by how many values there are. This tutorial walks through x̄ = (1/n) Σ xᵢ, a 1, 2, 2, 3, 4, 5 worked example, and a tiny-set playground that places the mean as a fulcrum on a number line. It is a teaching sketch, not a full statistics package.
Read article → Fitness · 5 min readHow to Estimate Body Fat
Navy circumferences — and what the percentage is not
The U.S. Navy method estimates body-fat percentage from tape measurements and height. This tutorial walks through the sex-specific log formulas, a 175 cm / 84 cm waist / 38 cm neck worked example, and a category scale you can try with sliders. Navy %BF is a circumference estimate, not a diagnosis or a DEXA scan.
Read article → Business · 5 min readHow to Calculate Break-Even
Units and revenue that cover fixed costs — contribution per unit
Break-even is the sales volume where contribution exactly covers fixed costs. This tutorial walks through Q = F / (P − V), a $12,000 / $40 / $15 worked example, and a playground that turns fixed cost, price, and variable cost into units, revenue, and a contribution ladder. Figures are illustrative; fees and taxes are omitted.
Read article → Finance · 5 min readHow to Calculate CAGR
A constant annual rate that connects start to end
Compound annual growth rate (CAGR) smooths a multi-year path into one constant yearly rate. This tutorial walks through (V_end / V_start)^{1/t} − 1, a $10,000 / $16,105 / 5-year worked example, and a playground that builds the smoothed growth staircase as you change start, end, and years. Figures are illustrative; fees and taxes are omitted.
Read article → Fitness · 5 min readHow to Calculate Daily Calories (TDEE)
BMR times an activity factor — and what the number is not
Total daily energy expenditure (TDEE) estimates the calories you use in a typical day: resting metabolism scaled by how active you are. This tutorial walks through Mifflin–St Jeor BMR, the activity multipliers used on this site, and a 70 kg / 175 cm / age 30 worked example. TDEE is a planning estimate, not a meal plan or a medical assessment.
Read article → Fitness · 5 min readHow to Calculate a Calorie Deficit
TDEE minus a planning offset — and why the weekly change is approximate
A calorie deficit is a planning gap between estimated daily energy use (TDEE) and intake. This tutorial walks through Mifflin–St Jeor TDEE, a 500 kcal/day offset, and the ~7700 kcal/kg rule of thumb for weekly change. The result is a screening estimate, not a diet prescription, a guaranteed loss rate, or medical advice.
Read article → Geometry · 5 min readHow to Calculate Circle Area and Circumference
π r² and 2 π r — radius is the only input
A circle is fully determined by its radius r. Area is the shaded disk π r²; circumference is the boundary length 2 π r. This tutorial walks through both formulas, a radius-3 worked example, and a playground that substitutes your r into the π expressions.
Read article → Personal Finance · 5 min readHow to Calculate Compound Interest
Interest on interest — A = P(1 + r/m)^{mt}
Compound interest reinvests earned interest into the balance, so later periods earn interest on interest. This tutorial walks through A = P(1 + r/m)^{mt}, a $5,000 / 4% / 10-year monthly worked example, and a playground that compares compound growth with simple interest. Figures are illustrative; fees and taxes are omitted.
Read article → Chemistry · 5 min readHow to Use C1V1 = C2V2
Dilution conserves moles of solute — solve V2 from C1, V1, and C2
A dilution keeps the amount of solute fixed while the solution volume changes. That conservation is C1 V1 = C2 V2. This tutorial walks through the identity, a C1 = 1, V1 = 50, C2 = 0.1 worked example matching the Dilution Calculator, and beakers that rebalance as you change concentration or volume.
Read article → Sales · 5 min readHow to Calculate a Discount
Percent off the labeled price — then what you actually pay
A percent discount reduces a labeled original price by a fraction of 100. This tutorial walks through amount saved and sale price, using the same $100 at 20% off defaults as the Discount Calculator. Sales tax, coupons stacked on coupons, and “compare at” prices are outside this identity.
Read article → Other · 5 min readHow to Calculate a Pregnancy Due Date
Naegele’s rule: last menstrual period plus 280 days — educational only
Naegele’s rule estimates a due date as 280 days (40 weeks) after the first day of the last menstrual period (LMP). This tutorial walks through that calendar arithmetic and a strip from LMP to week 40. It is an educational estimate, not a clinical due date, not a diagnosis, and not medical advice. Ultrasound dating often differs.
Read article → Math · 5 min readHow to Work with Fractions
Add, subtract, multiply, divide — then reduce by the gcd
A fraction a/b is a ratio of integers with a nonzero denominator. This tutorial walks through the four arithmetic operations, a 3/9 + 1/6 worked example that reduces to 1/2, and a playground with two fractions, an operation toggle, and unit bars. The Fraction Calculator on this site uses the same identities.
Read article → Physics · 5 min readHow Free Fall Distance Grows with Time
s = ½ g t² from rest — distance grows with time squared
In free fall from rest, ignoring air resistance, the distance fallen is half g times time squared. This tutorial walks through s = ½ g t² and v = g t, a 2 s / 9.80665 m/s² worked example matching the Free Fall Calculator, and a height-versus-time sketch that substitutes your t.
Read article → Finance · 5 min readHow to Calculate Future Value
A present amount grown at a constant period rate — FV = PV(1 + r)^n
Future value compounds a single present amount over a fixed number of periods. This tutorial walks through FV = PV(1 + r)^n, a $1,000 / 5% / 10-period worked example, and a playground that traces the growth curve as you change PV, r, and n. Figures are illustrative; fees and taxes are omitted.
Read article → Math · 5 min readHow to Calculate GPA
Credit-weighted letter grades — quality points over credits
Grade point average (GPA) is a credit-weighted mean of course grades on a points scale. This tutorial walks through the usual 4.0 unweighted conversion, a three-course worked example, and a playground that multiplies each letter by its credits. Schools differ on plus/minus values and on whether an A+ is 4.0 or 4.3 — match the scale your transcript uses.
Read article → Fitness · 5 min readHow to Calculate Heart Rate Training Zones
220 − age, five bands — and why the estimate is not a diagnosis
Heart-rate training zones are fractions of an estimated maximum, often 220 minus age. This tutorial walks through that estimate, five intensity bands, and the optional Karvonen formula when a resting heart rate is entered. The numbers are a coarse training sketch, not medical advice and not a diagnosis of fitness or risk.
Read article → Fitness · 5 min readHow Ideal Weight Formulas Work
Devine IBW from height — and what the number is not
Ideal body weight (IBW) formulas estimate a reference mass from height and sex. This tutorial walks through the Devine equations used on this site, a 175 cm worked example, and a height slider that places the estimate on a weight band. Devine IBW is a dosing-era estimate, not a healthy-weight target or medical advice.
Read article → Finance · 5 min readHow to Calculate Inflation Impact
What a fixed cash amount buys after prices rise
Inflation scales prices by (1 + i)^t and erodes the purchasing power of cash left uninvested. This tutorial walks through the inflation factor, a $100 / 3% / 10-year worked example, and a playground that shows what $100 buys as years and the rate change. Figures are illustrative; fees and taxes are omitted.
Read article → Personal Finance · 5 min readHow to Calculate a Loan Payment
Level installments that amortize principal — and what they omit
An amortizing loan spreads principal and interest into equal period payments. This tutorial walks through the standard annuity formula, a $20,000 / 6.5% / 5-year monthly worked example, and a playground that stacks principal against total interest as you change term and rate. Figures are illustrative; fees and taxes are omitted.
Read article → Fitness · 5 min readHow to Calculate Protein and Macros
Grams from calories and g/kg protein — a planning split, not a diet
Macronutrients convert calorie shares into grams: protein and carbohydrate at 4 kcal/g, fat at 9 kcal/g. This tutorial sets protein from body weight (g/kg), then splits the remaining calories between carbs and fat. The split is a teaching estimate, not medical advice, not a diagnosis, and not a meal plan.
Read article → Sales · 5 min readHow to Calculate Margin vs Markup
Same profit, different denominators — price versus cost
Gross margin is profit as a share of selling price (revenue). Markup is the same profit as a share of cost. This tutorial walks through both, using the Margin Calculator defaults of revenue $100 and cost $60. The two percents are not interchangeable.
Read article → Statistics · 5 min readMean vs Median vs Mode
Three summaries of the same list — and why they can disagree
Mean is the arithmetic average, median is the middle of the ordered list, and mode is the most frequent value. This tutorial walks through all three on 1, 2, 2, 3, 4, 5, then a playground that plots three markers on one sorted strip. It is a teaching sketch, not a full statistics package.
Read article → Chemistry · 5 min readHow to Calculate Moles
n = m / M — mass and moles share molar mass as the bridge
A mole counts 6.02214076×10²³ entities. Mass and moles are linked by molar mass: n = m / M. This tutorial walks through that identity, an 18 g / 18.015 g·mol⁻¹ worked example matching the Mole Calculator, and a mass↔mole bridge that substitutes your numbers.
Read article → Personal Finance · 5 min readHow to Calculate a Mortgage Payment
Level monthly installments on a long-term home loan — and what they omit
A mortgage is a long-term amortizing loan secured by property. This tutorial walks through the standard annuity formula, a $350,000 / 6.75% / 30-year monthly worked example, and a playground that breaks one payment into principal versus interest. Figures are illustrative; fees and taxes are omitted.
Read article → Fitness · 5 min readHow to Calculate Running Pace
Time divided by distance — and what the clock is not
Running pace is elapsed time divided by distance, usually written as minutes and seconds per kilometer or mile. This tutorial walks through that ratio, a 5 km / 25 min worked example, and dials that drive a pace clock plus even-split preview. Pace is a training arithmetic, not a fitness prescription or medical advice.
Read article → Math · 5 min readHow to Calculate a Percentage
Part over whole, scaled to 100 — three common setups
A percentage compares a part to a whole on a scale of 100. This tutorial walks through the three problems the Percentage Calculator solves: what is P% of X, X as a percent of Y, and percent change from A to B. The worked example uses the same 15% of 80 defaults as the calculator.
Read article → Math · 5 min readHow to Calculate Percentage Change
New minus old, divided by old — the sign is the story
Percentage change answers how far a number moved relative to where it started, not what share it is of a whole. This tutorial walks through (new − old) / old × 100, a 100 → 125 worked example (a 25% increase), and a signed bar that shows increase versus decrease. It is not the same question as “what is P% of X.”
Read article → Finance · 5 min readHow to Calculate Present Value and Annuity Payments
Discount a lump sum — or a stream of level payments — back to today
Present value asks what a future cash amount is worth now at a constant period rate. This tutorial walks through PV = FV(1 + r)^(−n) for a lump sum, the ordinary-annuity formula for level end-of-period payments, a $1,000 / 5% / 10-period worked example, and a playground that compares the two as you move r and n. Figures are illustrative; fees and taxes are omitted.
Read article → Geometry · 5 min readHow to Use the Pythagorean Theorem
c = √(a² + b²) — the hypotenuse of a right triangle
The Pythagorean theorem relates the three sides of a right triangle: the square of the hypotenuse equals the sum of the squares of the two legs. This tutorial walks through the formula, a 3-4-5 worked example, and a playground that substitutes your legs into c = √(a² + b²).
Read article → Math · 5 min readHow to Solve a Quadratic Equation
ax² + bx + c = 0 — discriminant and the quadratic formula
A quadratic equation is a degree-2 polynomial set equal to zero. This tutorial walks through the quadratic formula, a 1x² − 3x + 2 worked example with roots 1 and 2, and a playground that slides a, b, and c onto a parabola sketch. When the discriminant is negative there are no real roots — the parabola misses the x-axis.
Read article → Finance · 5 min readHow to Calculate ROI
Net gain over cost — a simple percent return
Return on investment (ROI) is net gain divided by what you spent, as a percent. This tutorial walks through (Final − Cost) / Cost × 100, a $1,000 / $1,350 worked example, and a playground that stacks gain against cost as the live percent updates. Figures are illustrative; fees and taxes are omitted.
Read article → Finance · 5 min readHow to Use the Rule of 72
Years to double ≈ 72 ÷ rate% — next to the exact compound formula
The Rule of 72 estimates how long a balance takes to double at a constant annual rate: divide 72 by the rate in percent. This tutorial walks through that shortcut, the exact doubling time ln(2)/ln(1 + r), an 8% worked example, and a playground that plots the rule against the exact formula as you move the rate. Figures are illustrative; fees and taxes are omitted.
Read article → Finance · 5 min readHow to Convert Salary to Hourly Wage
Annual pay divided by scheduled hours in the year — and the reverse
An equivalent hourly wage is annual salary divided by the hours you are scheduled to work in a year. This tutorial walks through hourly = salary ÷ (hours/week × weeks/year), a $52,000 / 40-hour / 52-week worked example ($25/hour), and a playground that converts both ways. Overtime, benefits, and tax withholding are not in this identity.
Read article → Sales · 5 min readHow to Calculate Sales Tax and VAT
Add a rate to a net price — or pull the tax out of a gross total
Sales tax and value-added tax (VAT) are a percentage of the net (pre-tax) price. This tutorial walks through adding tax at checkout and extracting tax from a tax-included sticker, using the same $49.99 at 8.25% defaults as the Sales Tax Calculator. Jurisdiction, exemptions, and rounding at the register sit outside this identity.
Read article → Personal Finance · 5 min readHow to Calculate Simple Interest
Linear growth I = Prt — and how it differs from compounding
Simple interest grows in a straight line with time: I = Prt. This tutorial walks through that formula, a $1,000 / 5% / 3-year worked example, and a playground that places simple interest beside compound growth on the same principal, rate, and years. Figures are illustrative; fees and taxes are omitted.
Read article → Math · 5 min readHow to Calculate the Slope of a Line
Rise over run between two points
Slope measures how steep a line is: the vertical change divided by the horizontal change between two distinct points. This tutorial walks through m = (y₂ − y₁) / (x₂ − x₁), a (1, 2) to (5, 4) worked example with slope 1/2, and a playground that draws the line and the rise/run triangle. A vertical line has undefined slope because the run is zero.
Read article → Statistics · 5 min readHow to Calculate Standard Deviation
Spread around the mean — and why n−1 is not the same as n
Standard deviation measures how spread out values are around the mean. This tutorial walks through the sample (n−1) and population (n) formulas, a 1, 2, 2, 3, 4, 5 worked example, and a tiny dataset playground that plots the spread. It is a teaching sketch, not a full statistics package.
Read article → Math · 5 min readHow to Calculate the GPA You Need This Semester
Solve for the term average that hits a cumulative target
If you already have a cumulative GPA and a target, the remaining credits have to carry the difference. This tutorial rearranges the GPA definition to solve for the term average you need, works a 3.2 → 3.5 example, and lets you slide current GPA, credits done, target, and remaining credits. The result can sit above 4.0 — which means the target is out of reach on a standard unweighted scale.
Read article → Sales · 5 min readHow to Calculate a Tip
A percent of the bill, added on top — not a tax
A tip (gratuity) is a percentage of the bill added for service. This tutorial walks through tip = bill × rate and total = bill + tip, using the same $50 at 18% defaults as the Tip Calculator. Customary rates vary; the arithmetic does not decide who you tip or whether tax is in the base.
Read article → Units · 5 min readHow to Convert Units
Multiply by a factor so unwanted labels cancel
Unit conversion restates the same physical quantity with a different label. For length and mass you hop through a shared base, or you write conversion factors as fractions so units cancel like algebra. This tutorial walks through that factor-label method, a 12 inch → centimeter → meter ladder, and a playground that strikes the canceled labels as you change the amount.
Read article → Geometry · 5 min readHow to Calculate Volume of 3D Shapes
Box, cylinder, and sphere — how much space is inside
Volume is the measure of space a solid occupies: how many unit cubes fit inside. This tutorial walks through V = ℓ w h for a rectangular box (the Volume calculator), V = π r² h for a cylinder, and V = ⁴⁄₃ π r³ for a sphere, a 4 × 3 × 2 worked example, and a playground that sketches the solid as you switch shapes.
Read article → Statistics · 5 min readHow to Calculate a Z-Score
How far a value sits from the mean, in standard-deviation units
A z-score restates a raw value as a signed distance from a mean, measured in standard deviations. This tutorial walks through z = (x − μ) / σ, a 115 / 100 / 15 worked example, and a bell-curve playground that moves a marker as you change μ, σ, and x. The curve is a teaching sketch of a normal model, not a full statistics tool.
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