How to use
This page is the map. Nine sections of the site, four colours that always mean the same thing, and one set of habits about units that everything else here is built on. Read it once and the rest of the site stops needing to explain itself.
Nothing below is a picture of the site. Every heading, note, warning and worked step you see is the real component, rendered here. If I change how an answer looks, this page changes with it.
1What each part is for
Nine doors, written as the moment you would reach for each one rather than what it technically is.
- Formula Solvers — you know every number but one. Enter what you have, leave the unknown blank, and the algebra rearranges itself around it. 933 formulas, 3,148 rearrangements, so most formulas solve for any of their variables and not just the one on the left.
- Unit Converters — the number is right and the unit is wrong. One value goes in and comes back in every unit of that quantity at once, which is faster than guessing which one you wanted.
- Units — you are not certain the two units in front of you are the same thing. 891 pages covering 911 units: what each one is, whether the conversion is exact or measured, and which of the several definitions in circulation this site uses.
- Constants — you need a value and you need to be able to say where you got it. 361 of them, each with its unit, its uncertainty and its source.
- Fluids — the property moves with temperature and a single pinned number will not do. 16 fluids with density, viscosity, specific heat and thermal conductivity as curves, not as one row of a table.
- Steam — the gauge reads psig and the calculation wants absolute. Saturated steam by pressure and by temperature, superheat, and both pressures side by side in every row.
- Periodic Table — you need a molar mass. All 118 elements, and each one hands its atomic weight straight to the solvers that want it.
- Examples — the problem has parts (a), (b) and (c), and each part feeds the next. 64 worked problems you can edit: change a given and every step downstream recalculates.
- Calculator — the arithmetic is just arithmetic. A long tape that shows its work, the constants library one search away, and answers you can hand to any solver.
Two more, not in the nav because they are views rather than sections: categories group the solvers by subject, and sets group them the way a course or a trade would.
2The colours mean things
Four signals, and they never trade places. The palette is one hue rotated three ways: the red is HYDRONIC’s, the blue and the green are that same red turned around the wheel.
reda notice, or a door. The brand wears it, the nav underlines where you are with it, and the headings that hand a value to another tool are set in it.
You are looking at two of those right now: the raised e in up in the masthead, and the underline sitting under Guide in the nav, which is how every page on this site says where you are. The fe monogram in section 3 carries the same red on its raised letter. Here is the third.
- the calculator — as a number to keep working with
It appears above the links that carry a number somewhere else. The heading is the notice; the links under it stay blue, because they are the click.
The √ opening each item back in section 1 is red too, and so is the one in that card. It is a radical set in the maths face, not a checkmark: on a site about formulas, the mathematical tick is the right tick.
One red is held back from all of that. The error box is a softer, deeper red that means only one thing: there is no answer, and here is why.
Fired here by filling every field. Leaving one blank is how you tell the solver which variable to find.
blueyou can click it. Every link, the Calculate button, the chips in the jump list at the top of this page, and the frame around a finished answer.
Shown, not wired: this one is disabled on purpose. The real button lives at the foot of every solver form, and pressing Enter in any field does the same thing.
The answer frame is blue for a reason worth naming: it is the same blue as the button that produced it, so the payoff visibly belongs to the press. Section 3 shows a real one.
amberread this before you carry on. It never blocks anything and never changes a number.
This efficiency comes out above 100 % — a motor cannot put out more power than it draws, so one of the readings is off: shaft power is usually estimated from torque and speed, and input power from a meter, and the two are rarely measured with equal care. Re-check both before rating the motor, or size the motor larger if the load genuinely needs this output.
This one is real: 10 kW out of 9 kW in, solved at build time. The answer stands at 111 %; the note says what a number like that is telling you.
Amber also carries the input hints, which ask a question about something you typed rather than correcting it. Section 5 has a live one.
greya citation. Where a number came from, or why a symbol is spelled the way it is.
h_f1, h_f2 and h_fg2 filled from the steam tables · saturated at 128 psig
It sits above a form that arrived pre-filled, and it disappears the moment you type over the last of those values.
Green is deliberately unused. The tokens exist in the stylesheet and nothing on the site paints with them. Green is the colour every calculator on earth reaches for to mean “the answer,” which is exactly why it is kept empty here: the answer already has the blue frame, and a fifth signal would only dilute the four that carry meaning.
3How to read an answer
A solver does not just hand back a number. It shows the work in order, and the order is fixed: rearrange, substitute, simplify, convert, justify the unit, answer. Steps three, four and five each appear only when they have something to say, so the shortest reveal on the site runs to three steps and the longest to six. The example below is a six.
Hydrostatic pressure: water at 1,000 kg/m³, thirty feet of it, answered in kilopascals. Metric density and an imperial depth in the same calculation is the case this whole site was built for.
Open that calculation in the real solver and it will arrive with the same numbers in the same units. That link is not a special case: every solver keeps its current state in the address bar, so the URL is always a copy of what is on screen.
Step 1, and why the label matters
The first label says which variable the solver went looking for. It is rendered as live maths rather than text for a specific reason: the labels are upper-cased in CSS, and “for m” turned into “FOR M” is a mass relabelled as a molar mass. Same for the unit in the step above the answer. Capitalisation is physics here, not typography.
“Up to 5 significant figures”
Up to, not exactly. Trailing zeros are dropped, so a value that lands on ninety is shown as 90 rather than padded out to look more precise than it is. The substitution steps above the answer deliberately carry more digits than the answer does, so you can see for yourself that the rounding happened at the end and not in the middle.
What actually moves between steps
The figure you read is rounded. The value that travels is not. Every carry on this site moves at full precision: a worked example passes its step answers along in SI with every digit intact, a “use this answer in” link carries the full number, and flipping metric to imperial converts your typed values through SI rather than through what was on screen. Four chained steps, each rounded to what the screen happens to show, can accumulate percent-level error behind a result that still looks authoritative. That is a trap I would rather the site simply never set.
Under the answer, most solvers print the same result in every unit of that quantity. That table is not decoration: it is the fastest way to sanity-check an answer, because a number that is wrong is usually obviously wrong in some unit you have a feel for.
4Exact versus measured constants
Every constant page carries a Status row, and it says one of two things. Either the value is exact by definition, or it was measured and the page tells you how well.
| Speed of light in vacuum | 299,792,458 m/s |
| Status | Exact by definition — no uncertainty |
| Newtonian constant of gravitation | 6.6743e-11 m³/(kg·s²) |
| Status | Measured: ± 1.50e-15 m³/(kg·s²) (0.000022 relative) |
Read straight from the constants library, formatted by the same code the constant pages use.
What “exact by definition” means
It means the number is not a measurement at all. In 2019 the SI was rebuilt on seven constants whose values were fixed by decree: the speed of light, the Planck constant, the elementary charge, the Boltzmann constant, the Avogadro constant, the caesium hyperfine frequency and the luminous efficacy of 540 THz radiation. They stopped being things we measure and became the things we measure everything else against. The metre is now defined by , which is why that figure has no error bar and never will.
The badge reaches a little wider than those seven. Anything fixed by agreement rather than by experiment carries it too: standard gravity at 9.80665 m/s², set by the CGPM in 1901 and unchanged since, or the inch, which is exactly 25.4 mm because six standards bodies said so in 1959. Constants derived exactly from exact ones inherit it — the Faraday constant is the Avogadro constant times the elementary charge, and both of those are now definitions. 61 of the library’s 361 constants are exact in one of those ways.
Why a constant is never shown rounded
A rounded exact value is a contradiction on the page. The index once rendered the speed of light as 299,792,460 because it formatted everything to eight significant figures, directly under a sentence calling it exact. Constants now print at their full round-trip precision, always: whatever you see is the whole stored value. Rounding on this site belongs to answers, not to reference data.
What the uncertainty figure is telling you
The ± is a standard uncertainty in the same unit as the value: one standard deviation, not a worst case and not a tolerance band. The bracketed figure beside it is the relative standard uncertainty, which is how metrologists actually compare constants. G sits at 0.000022 relative, about 22 parts per million, and it is the least precisely known of the fundamental constants by a wide margin. That is the honest reason a gravitational answer here cannot be quoted past a few figures no matter how many the solver prints. Every constant also names its source on the same card, and most of them read CODATA 2022.
The speed of light and the gravitational constant are the two pages to compare if you want to see the difference in full.
5Units discipline
One rule underneath everything: a value carries its unit everywhere. Every input has a unit select beside it, every shared link encodes the unit next to the number, and the tape records the unit with the result. A number passed around without its unit is the single failure this site was built to stop.
The metric / imperial toggle at the top of a solver moves the selects and converts your typed numbers with them. It does not swap the unit underneath a number you already entered, which would silently reinterpret 40 litres as 40 gallons and change the answer without anyone touching a field.
Gauge versus absolute
The gauge on the header and the number in the calculation are almost never the same number. The steam pages give you a real toggle for it, and every table there prints both pressures side by side. Elsewhere, the unit pages warn you.
Watch out: Almost every psi reading in the field is gauge pressure, psig, meaning pressure above ambient. Absolute pressure is psia. A tyre at 32 psig is at 46.7 psia. Gas law calculations, compressor work and boiling point predictions all need absolute pressure, and feeding a gauge reading into them is one of the most common errors in process work.
The same box, the same words, as on the unit page itself.
Flipping a steam field between absolute and gauge never silently reinterprets a number you typed, for the same reason the metric toggle does not: those are two different claims about the world, and a fitter reading a gauge and an engineer reading a table each mean their own one. The psi page has the rest of it.
A temperature and a temperature difference are not the same quantity
This is the affine point, and it catches people constantly. A temperature converts with a scale factor and an offset. A difference converts with the factor alone, because the offsets cancel. So:
| 100 °F, as a temperature | 37.78 °C |
| 100 F°, as a rise or a drop | 55.56 C° |
Same two numbers, same two scales, and the answers differ by more than 17 degrees. So this site keeps them as two separate quantity types with two separate sets of units, and it writes them differently: the degree sign goes after the scale letter for an interval. 30 °C is a temperature. 30 C° is a rise of thirty. Kelvin needs no mark either way, since an interval of 1 K is just 1 K.
Type a Celsius reading into a field asking for absolute kelvin and the solver asks about it rather than correcting you. It never blocks the calculation: 25 K is a perfectly good temperature if you work with liquid helium, and a reader who means what they typed is right and the site is not.
25 K is -248.15 °C. If you meant 25 °C, switch the unit — this field wants an absolute temperature, not a change in temperature.
On a solver this sits indented under the field it is asking about. Here there is no field above it, so on a wide screen the indent has nothing to line up with.
Why some symbols shout and others whisper
psi is never PSI. kWh is never KWH. And on a water treatment page, gpd and gpm stay lower case while their own million-fold multiple is written in capitals. That is not an inconsistency in the site, and where a reader would reasonably suspect a typo the unit page says so in its own grey note.
Why it is written MGD: Capitals, while gpd, gpm and gph beside it stay lower case — not an inconsistency but two different habits meeting. The small ones are read as abbreviations; MGD is spelled out letter by letter, and AWWA, the EPA and the standard texts all print it that way. There is a safety reason to keep it shouting on a treatment page: lower-case mgd is one slip from mg/d, milligrams per day, which is a real dosing unit in the same plant and five orders of magnitude away from a plant rating.
Grey, not amber: it is a citation about the page in front of you, not a hazard in the arithmetic.
The MGD page carries that note in full, along with the trap underneath it: MGD means million US gallons per day here, and in British-influenced practice the same three letters have meant imperial gallons, a figure twenty percent larger.
6Making work flow between tools
Real problems are not one calculation. The site is built so a number never has to be retyped, and so that wherever it lands it still knows where it came from.
Send it onward
Solve anything and a red hand-off heading appears under the learning zone listing where that answer can go next: the calculator, and the solvers that take this quantity as an input. The reference pages push the other way. An element sends its molar mass, and a steam state sends its enthalpies as a set, so an hf from one pressure can never end up beside an hfg from another.
The tape is one shared history
The calculator’s tape is not the calculator’s alone. Every solve anywhere on the site writes to it: the formula, your inputs with their units, and the answer. Solve on a solver page, open the calculator, and ANS is that answer. Lines arriving from a solver are marked with a , and a line stamps itself with a when you send its value out again, so the tape reads as a ledger of where numbers went rather than a list of sums.
The little calculator in the corner of an answer box is that door. It is where the = key sits on a real calculator, which is the joke and also the reason it is there.
The provenance note, and when it expires
A form that arrives pre-filled says so, in grey, above the inputs. It names which variables it filled and what state they were read at. Two places send that note today: an element handing over its atomic weight, and the steam tables handing over a state.
The note is a claim about numbers on screen, so it has to stop being true the moment those numbers stop being those numbers. Edit one and the note drops that variable. Edit the last of them and the note goes entirely, and the link you copy stops crediting a source for numbers that are no longer its own.
There is a middle state that matters more than either end, and it is the one place a provenance note is allowed to turn amber:
h_f1, h_f2 and h_fg2 filled from the steam tables · saturated at 128 psig
h_f1 and h_fg2 still hold values from the steam tables · saturated at 128 psig — h_f2 has been changed, and figures read together at one state usually only hold together. Check these still describe the same state.
Values read together at one state usually only hold together. Replacing one of a set is exactly the mismatch the sending page promised to prevent, so the note stops being a citation and becomes a caution.
A value that knows its formula but not its variable
Send a result from the calculator into a solver and it arrives half-addressed: the calculator knows the number and the destination page, but not which input you meant. An amber bar at the top of the form asks you to finish the address, and reminds you to check that row’s unit before you do.
Examples are worked chains you can edit
An example is an ordered run of solvers where each answer becomes a later input. It is laid out like an exam paper: the givens as data rows, the parts as a lettered “determine (a), (b)” list, then the steps, then a closing sentence that reads the answer back in the scenario’s own terms.
Carried values are read-only and labelled with the step they came from, because a carried value is by definition whatever the earlier step produced and editing it would break the chain rather than explore it. Literal givens are editable, and changing one recalculates everything downstream.
Change a given and the closing statement disappears until you reset, because it was written about the original numbers.
Every chain is also a test. Its expected final answer is worked by hand and stored with it, and on multi-part problems each part’s last step is pinned the same way, so a wrong intermediate that happens to cancel out still fails the suite.
7Printing
Four pages are built to leave the screen: the steam tables, the steam diagrams, the glycol freeze points, and every fluid property page. Each carries a Print or save as PDF button, sitting next to whatever control decides what gets printed.
The button is only a trigger. It hands the page to your browser’s own print dialogue, which is also its save-as-PDF, so Ctrl+P does exactly the same thing.
What comes out is a plate, not a screenshot of a web page. Landscape, Letter, one chart per sheet. Letter is stated rather than left to the printer’s locale, and that is not fussiness: A4 is 18 mm wider, so a plate laid out to fill A4 loses its right edge on the paper this site’s readers actually load.
Each sheet is self-contained. The chart takes the lower left, and the numbers that belong to it stack in a panel down the right side, so a plate scissored off and pinned to a wall still carries its own data. Site chrome, navigation, the sidebar and the on-screen controls all drop out. A letterhead with the page’s full URL takes the first sheet, because a printed table is a snapshot and the live one moves with the library.
Every chart carries the watermark. It is drawn inside the SVG rather than laid over it in CSS, which means it survives all three ways a chart leaves this site: printing, right-click-save, and a screenshot. Browsers drop CSS backgrounds on paper and never drop SVG text.
8Why you can trust it, and where you cannot
The whole site started with a table of classmates holding four different answers to the same problem and no way to settle it. So the testing is not an afterthought here, and it has one rule at its centre.
Checked against numbers worked by hand
Every solver is anchored to expected values derived by hand from textbooks and reference data, never against numbers the code produced. That distinction is the whole point. A test written by running the code and recording what came out proves only that the code still does what it did yesterday, including the parts that were wrong yesterday. An anchor is authored independently of the formula it checks, so it catches wrong physics: a missing one-half, an inverted ratio, the wrong constant.
2,091 anchors run against 933 formulas on every test pass, replayed through the real solve pipeline with the real unit conversions rather than a simplified stand-in, so an anchor cannot pass while the page disagrees with it.
A coverage gate, so silence is not a pass
Counting anchors is not enough on its own: a formula with no anchor at all produces no failures, which looks identical to a formula that passed. So the suite checks coverage separately. Any formula in the catalogue without a single anchor fails the run by name, the day it lands. That gate was added because coverage really had regressed once, and the uncovered formulas included the demo on the site’s own front door.
A second harness works from the other direction. It draws random values, runs a formula forward and then back through each of its rearrangements, and requires the original number to reappear. Where an inverse genuinely cannot round-trip, because the forward relation is many-to-one, the formula has to say so in writing and the harness verifies the branch it returns is a true inverse instead. Wrong algebra still fails.
Guards refuse the impossible, notes flag the suspicious
2,993 of the site’s 3,148 rearrangements carry a domain guard. Ask for a loop with no temperature drop and the solver refuses in the red box and explains why, because a loop with no drop moves no heat at any flow. That is the red lane: no answer.
Suspicious is handled differently. 17 rearrangements carry a design note instead, and a design note ships with the answer rather than replacing it. An efficiency past 100 % is a motor to re-measure or a motor to upsize, not an arithmetic error, and the amber note in section 2 is the real one. Mathieu’s rule, in his own words: getting answers over 100 % is okay if there is a note that something needs to change in the design.
Where the numbers themselves come from
The steam tables are computed, not transcribed. Every value is produced from IAPWS-IF97 when the page is built, and the engine is checked against the standard’s own published values plus identities the standard never prints. A steam table typed in by hand is a steam table with a typo in it, and the typo gets found by whoever sizes a trap from the wrong row. The constants library cites its sources per constant, mostly CODATA 2022. Fluid properties are correlations with declared validity ranges, and the glycol page tabulates the published ASHRAE anchors themselves rather than samples of the curve drawn through them.
Now the limits, plainly
- This is an education site. A tested calculation is not a stamped calculation, and nothing here has been reviewed by an engineer of record.
- An anchor proves a formula, not your problem. The arithmetic being right says nothing about whether it was the right formula, whether your inputs describe the situation, or whether the assumptions behind the relation hold at your conditions.
- Correlations have edges. Fluid properties are fitted curves with stated ranges, and past those ranges they are extrapolation dressed as data. Where the published sources genuinely disagree, as they do for glycol above 60 % by volume, the page says so instead of picking a winner quietly.
- Displayed precision is not accuracy. 5 figures on screen is a reading convenience. A gravitational answer cannot be better than G is, and no rounding rule on this page can fix that.
- Test coverage moves. The numbers above are true of the build you are reading. New formulas land, and the gate is what keeps them from landing unchecked.
If a result matters commercially, treat it as the discussion copy and reproduce the calculation yourself before relying on it. That is not lawyering: it is what I would do with anyone else’s number, including my own from fifteen years ago. The fine print at the foot of every page on this site, including this one, says the same thing.
stands behind its math, and is not responsible for how the results are used. That line lives in the footer below, and it means exactly what it says. The About page has the rest of the story.