Tutorials
Short reads that explain the math behind each calculator — formulas, pitfalls, and why the numbers work.
Business
Chemistry
How 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 → 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 →Date Time
Finance
How 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 → 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 → 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 → 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 → 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 → 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 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 → 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 →Fitness
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 read FeaturedHow 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 → 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 → 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 → 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 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 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 → 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 → 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 →Geometry
How to Calculate Surface Area to Volume Ratio
S/V for a sphere, cube, or box — then read it as inverse length
Surface-to-volume ratio is the outside divided by the inside: S/V. This tutorial uses the three solids the Surface Area to Volume Ratio Calculator actually takes — sphere, cube, and rectangular box — works the default sphere of radius 3 to S/V = 1, and lets you switch shapes in a playground.
Read article → Geometry · 7 min read FeaturedHow to Find the Centroid of a Right Triangle at (b/3, h/3)
From the right-angle vertex, G sits at one-third of each leg
If a right triangle has its right angle at the origin, the centroid is not a mystery point you have to construct from scratch. From that right-angle vertex, G = (b/3, h/3): one-third of the way along the base and one-third of the way up the height. This tutorial derives that shortcut from the three-vertex average, works b = 6 and h = 4, and opens the same points in the Centroid of a Triangle Calculator.
Read article → Geometry · 6 min readHow Surface Area to Volume Ratio Affects Cooling and Heating
Heat crosses S; thermal mass sits in V — so the rate tracks S/V
Heat leaves (or enters) through the surface and is stored in the interior. For a similar solid, the cooling rate per degree tracks S/V, which falls as the object grows. This tutorial uses cubes the Surface Area to Volume Ratio Calculator can actually type — 1 cm versus 2 cm — not a cylinder or a person-shaped mesh.
Read article → Geometry · 6 min readHow Surface Area to Volume Ratio Changes with Size
Similar solids: S grows as L², V as L³, so S/V falls as 1/L
If you scale a cube or a sphere without changing its shape, surface area grows with the square of the length and volume with the cube. The ratio S/V therefore falls as 1/L. This tutorial doubles a 1-unit cube to side 2 and 4, checks the same 1/r story on a sphere, and uses the Surface Area to Volume Ratio Calculator on those rows.
Read article → Geometry · 6 min readHow the Right-Triangle Centroid Relates to the Three-Vertex Formula
G = (b/3, h/3) is the vertex average after the right angle sits at the origin
The right-triangle shortcut G = (b/3, h/3) is not a second formula. It is the three-vertex average G = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3) after you put the right angle at the origin. This tutorial shows that algebra, why the calculator’s default triangle is not that special case, and when to type the three points instead of dividing the legs by three.
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 → 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 → 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 → Geometry · 6 min readHow to Check a Right-Triangle Centroid with Medians (2:1 Ratio)
G divides each median 2:1; on a right triangle at the origin that recovers (b/3, h/3)
A median joins a vertex to the midpoint of the opposite side. The centroid G is where the three medians meet, two-thirds of the way from each vertex toward the opposite midpoint. On a right triangle with the right angle at the origin, that 2:1 split on the hypotenuse median is exactly G = (b/3, h/3).
Read article → Geometry · 6 min readHow to Compare the Surface Area to Volume Ratio of a Cube and a Sphere
Same diameter is a length trick; equal volume is the shape question
A cube has S/V = 6/a and a sphere has S/V = 3/r. Set the cube’s side equal to the sphere’s diameter and both ratios collapse to 6/L — that is a size match, not a shape verdict. Hold volume fixed and the sphere wins (lowest S/V). This tutorial uses Sphere and Cube on the Surface Area to Volume Ratio Calculator, then stretches a box at the same V.
Read article → Geometry · 8 min readHow to Find the Centroid When the Right Angle Is Not at the Origin
Do not treat (b/3, h/3) as absolute coordinates — translate from the right-angle vertex, or average the three vertices
The right-triangle shortcut G = (b/3, h/3) is measured from the right-angle vertex, not from (0, 0) on the worksheet. If that vertex sits at (xᵣ, yᵣ) and the legs run with the axes, add (b/3, h/3) to (xᵣ, yᵣ). The reliable fallback is the three-vertex average on the Centroid of a Triangle Calculator.
Read article → Geometry · 6 min readHow to Get Centroid Coordinates of a Right Triangle from Legs b and h
Two leg lengths become three vertices, then G = (b/3, h/3)
If you only have the two legs of a right triangle, you can still get centroid coordinates. Place the right angle at the origin, turn b and h into the vertices (0, 0), (b, 0), and (0, h), then G = (b/3, h/3). A worked b = 6, h = 4 gives G = (2, 4/3). The hypotenuse is not an input, and the units of the legs become the units of G.
Read article → Geometry · 6 min readHow to Measure the Centroid from the Right-Angle Vertex
G = (b/3, h/3) is an offset from the right angle, not a screen coordinate
Homework often says the centroid is (b/3, h/3) measured from the right-angle vertex. That phrase names a local frame: put the right angle at the origin, then G sits b/3 along one leg’s direction and h/3 along the other. This tutorial walks that measurement, a b = 6, h = 4 example, and why sliding the triangle changes the absolute coordinates even though the offsets from the right angle stay the same.
Read article → Geometry · 7 min readHow to Plot the Centroid of a Right Triangle on the Coordinate Plane
Axes, legs b and h, then G = (b/3, h/3) from the right-angle vertex
Homework that says “plot the centroid” wants a point on the plane, not only a formula. Place the right angle at the origin, draw legs b and h along the axes, then mark G = (b/3, h/3) from that vertex — the same average the Centroid of a Triangle Calculator reports from the three vertices.
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 → Geometry · 7 min readWhy Smaller Cells Have a Higher Surface Area to Volume Ratio
The membrane is S; the cytoplasm is V — S/V falls as the cell grows
A cell swaps food, gases, and waste across its membrane. That membrane is surface; the living interior is volume. As a similar blob grows, S/V falls as 1/L, so each unit of interior gets less gate. This tutorial uses cubes and spheres the Surface Area to Volume Ratio Calculator actually takes, not a cartoon organelle the form cannot draw.
Read article →Math
How 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 · 10 min read FeaturedHow to Calculate Your GPA for Free Online
College credits, a 4.0 scale, and this site’s free GPA tools
You can calculate a college GPA for free on this site without creating an account or uploading a transcript. This walkthrough is for university credit hours on a 4.0 scale: gather what your registrar actually uses, run one term in the Semester GPA Calculator, convert assignment percents when you still have scores instead of letters, check whether a course goal is still reachable on the final, and plan a cumulative target. The pages are teaching checks, not an official record.
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 → 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 → 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 → 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 → 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 → 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 →Other
Personal Finance
How 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 → 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 → 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 → 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 → 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 →Physics
How to Calculate Angular Resolution with the Rayleigh Criterion
θ = 1.22 λ / D for a circular aperture — then convert to arcseconds
Two point sources are just resolved when the first dark ring of one Airy pattern sits on the peak of the other. For a circular aperture that split is θ = 1.22 λ / D. This tutorial works 550 nm through a 10 cm opening to 6.71×10⁻⁶ rad ≈ 1.38″, the same defaults as the Angular Resolution Calculator.
Read article → Physics · 6 min readHow Aperture Diameter Sets Diffraction-Limited Resolution
At fixed λ, θ scales as 1/D — double the pupil, halve the angle
The Rayleigh angle is θ = 1.22 λ / D. Hold wavelength still and double the aperture, and the diffraction limit halves. This tutorial keeps 550 nm fixed, compares 10 cm with 20 cm, and uses the Angular Resolution Calculator on the same pair.
Read article → Physics · 6 min readHow Atmospheric Seeing Limits Real-World Angular Resolution
When the 1–2″ seeing disk is larger than 1.22 λ / D
Diffraction-limited θ = 1.22 λ / D is a pupil floor, not a promise. From the ground, atmospheric seeing is often 0.5–2″ and swallows a sharp Airy core. This tutorial compares an 8-inch Rayleigh floor of 0.68″ with a 1.5″ seeing disk and the Angular Resolution Calculator.
Read article → Physics · 5 min readHow End Corrections Affect Velocity Stack Sizing
An open mouth behaves a little longer than the wall — L_eff ≈ L_phys + kD
An open pipe reflects slightly outside the physical lip, so the acoustic length is a bit longer than the wall you can measure. This tutorial uses the teaching correction L_eff = L_phys + kD, a 50 mm mouth at k = 0.3, and the stack split after a 350 mm runner. The Velocity Stack Calculator leaves the correction off until you turn it on.
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 → Physics · 5 min readHow Intake Air Temperature Changes Your Velocity Stack Estimate
C ≈ 331.3 √(1 + T/273.15) — warmer air, longer Ls at the same RPM
The wave in the runner travels at the speed of sound in the intake air. Hotter air raises C, and Ls scales with C, so a stack sized in a cold garage sits a few percent long once the bay is hot. This tutorial uses C ≈ 331.3 √(1 + T/273.15), the 0 °C versus 40 °C swing at 6000 rpm, and a temperature slider. The Velocity Stack Calculator uses the same C(T).
Read article → Physics · 6 min readHow to Calculate Velocity Stack Length for a Target RPM
Ls = (C × 60 × S) / (4 × N × O) — a quarter-wave starting length
A velocity stack is the flared end of an intake tract. This tutorial walks through the quarter-wave teaching form Ls = (C × 60 × S) / (4 × N × O), a 6000 rpm / four-stroke / third-harmonic worked example that matches the Velocity Stack Calculator defaults, and a playground that substitutes your RPM and order.
Read article → Physics · 5 min readHow to Choose Harmonic Order for Street vs Race Intake Stacks
Odd orders 1, 3, 5, 7 — longer for mid-range, shorter for high RPM
Harmonic order O is the lever that turns one target RPM into a family of lengths. This tutorial picks a street-style O = 3 at 6000 rpm versus a shorter race-style O = 5 at 9000 rpm using Ls = (C × 60 × S) / (4 × N × O), the same identity as the Velocity Stack Calculator. The number is a starting length, not a dyno curve.
Read article → Physics · 5 min readHow to Compare Odd Harmonics in a Velocity Stack Length Table
Orders 1, 3, 5, 7 at one C and S — length falls as 1/O
One RPM produces four odd-harmonic lengths, not one magic trumpet. This tutorial reads the 1 / 3 / 5 / 7 table at 6000 rpm and C = 343 m/s, then lets you slide RPM so the rows update. The Velocity Stack Calculator prints the same table. Use it to compare long versus short, not to skip a dyno.
Read article → Physics · 6 min readHow to Compare Rayleigh and Dawes Limits for Double Stars
1.22 λ / D versus the empirical 116 / D(mm) visual split
Rayleigh’s circular-aperture rule is θ = 1.22 λ / D. Dawes’ limit is a tighter visual recipe for double stars, about 116 / D arcseconds when D is in millimetres. This tutorial compares both at 100 mm and 550 nm using the Angular Resolution Calculator for the Rayleigh number.
Read article → Physics · 5 min readHow to Convert Angular Resolution from Radians to Arcseconds
θ″ = 206265 θ — the astronomy unit for a tiny Rayleigh angle
Rayleigh’s formula returns radians. Star charts, seeing reports, and double-star catalogs use arcseconds. Multiply θ by 206265. This tutorial converts the 10 cm / 550 nm result 6.71×10⁻⁶ rad into 1.38″ with the Angular Resolution Calculator.
Read article → Physics · 5 min readHow to Convert Between Tuned RPM and Velocity Stack Length
N = (C × 60 × S) / (4 × Ls × O) — the same identity, flipped
If the tract is already on the engine, the useful question is which RPM that length is a quarter-wave of. This tutorial rearranges Ls = (C × 60 × S) / (4 × N × O) to solve for N, works a 400 mm four-stroke example, and points at the Velocity Stack Calculator’s length and RPM modes. Neither direction is a dyno.
Read article → Physics · 6 min readHow to Estimate the Diffraction Limit of a Camera or Lens
Entrance pupil D = f / N, then the same θ = 1.22 λ / D
A camera lens is still a circular aperture. The entrance pupil is D = f / N, the focal length divided by the f-number. Plug that D into θ = 1.22 λ / D. This tutorial works a 50 mm f/1.8 at 550 nm to about 4.98″ and the Angular Resolution Calculator.
Read article → Physics · 5 min readHow to Measure Total Intake Tract Length Including the Velocity Stack
Ls is valve to mouth — the stack is only the leftover after the runner
A velocity stack is the flared end of a longer acoustic path. This tutorial shows how to measure that path from the intake valve to the open mouth, subtract the runner you already have, and isolate the trumpet with L_stack ≈ L_eff − L_existing. The Velocity Stack Calculator reports that split; a tape on the engine is still the measurement.
Read article → Physics · 5 min readHow to Size Velocity Stacks for ITBs Without Treating Dyno Results as Optional
Same quarter-wave budget per throttle — the dyno is the measurement
Individual throttle bodies share a formula with a single carburetor trumpet, not a permission to skip the rollers. This tutorial uses the quarter-wave length as a per-ITB starting budget, calls out plenums and airboxes that break the open-end story, and treats a dyno pull as the result, not an optional extra. The Velocity Stack Calculator does not replace that pull.
Read article → Physics · 6 min readHow Wavelength Affects Telescope Angular Resolution
At fixed D, θ scales with λ — visible is sharp, radio is coarse
The Rayleigh angle is θ = 1.22 λ / D. Hold the aperture still and double the wavelength, and the diffraction limit doubles. This tutorial keeps a 25 m dish fixed, compares 550 nm with 21 cm, and points the same identity at the Angular Resolution Calculator.
Read article →Sales
How 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 → 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 → 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 → 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 →Statistics
How 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.
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 → 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 → 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.
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