Free printable references

Downloadable Formula Sheets

Four subjects, with clear formulas, conditions and worked examples.

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Four pages. Under 260 KB per PDF. No sign-up.

Choose your formula sheet

Download only the subject you need, or open its preview before saving. Every sheet has six selected references, definitions beside the symbols, conditions that affect the answer and a complete three-step example. The online preview is also useful when you want to enlarge the text on a phone.

Sheet 1 · One page · Six references

Finance formula sheet

Interest, borrowing and growth

Compare a savings model, a regular loan payment or growth between two values.

A4 PDF US Letter PDF Editable LaTeX

Same content in both paper sizes. PDF files are under 160 KB each.

Preview formulas, conditions and worked example

Constant-rate educational models. No fees, taxes, inflation adjustment or irregular cash flows are included. A nominal interest rate is not necessarily a legally disclosed APR.

Compound growth

A=P\left(1+\frac{r}{m}\right)^{mt}

$P$ is the initial balance; $r$ is the annual nominal rate as a decimal; $m$ is compounding periods per year; $t$ is years. No later deposits or withdrawals.

Simple interest

I=Prt,\qquad A=P+I

$I$ is interest on the unchanged principal $P$ . Use an annual decimal rate $r$ and years $t$ ; this model does not compound interest.

Effective annual yield

y=\left(1+\frac{r}{m}\right)^m-1

Use annual nominal decimal rate $r$ and $m$ compounding periods per year. $y$ is the decimal annual yield; interest stays in the account.

Level loan payment

M=\frac{Pi}{1-(1+i)^{-N}}

$P$ is principal; $M$ is each payment. Use rate $i>0$ per payment period and $N\ge1$ end-of-period payments. For $i=0$ , use $M=P/N$ .

Present value

PV=\frac{F}{(1+i)^N}

$F$ is one future amount after $N$ periods; $i>-1$ is the discount rate per period. This does not value a series of separate cash flows.

Compound annual growth

g=\left(\frac{V_f}{V_0}\right)^{1/t}-1

$V_0>0$ and $V_f>0$ are beginning and ending values; $t>0$ is years. $g$ is a decimal rate. Intermediate cash flows are excluded.

Worked example: Two years of monthly compounding

Start with $P=1000$ currency units at a nominal annual rate of $5\%$ , compounded monthly for two years. There are no additional cash flows.

Write the annual rate as a decimal and count the compounding periods. $r=0.05,\quad m=12,\quad mt=24$
Substitute without rounding the periodic rate. $A=1000\left(1+\frac{0.05}{12}\right)^{24}$
Round the final balance to two decimal places. $A\approx1104.94,\qquad I\approx104.94$

The unrounded balance is used to find interest. Monthly deposits would require a different model; use the linked calculator for that scenario.

Sources used for this sheet

Check a savings scenario Read the subject glossary

Sheet 2 · One page · Six references

Math formula sheet

Percentages, coordinates, geometry and spread

Keep common classroom relationships beside your working, with their conditions visible.

A4 PDF US Letter PDF Editable LaTeX

Same content in both paper sizes. PDF files are under 160 KB each.

Preview formulas, conditions and worked example

A compact selection, not a complete course formula list. The meaning of a symbol is local to its card. Use consistent length units before calculating areas or distances.

Percentage of a whole

a=\frac{p}{100}W

$p$ is the percentage coefficient, $W$ is the whole and $a$ is the part. For $25\%$ , enter $p=25$ , not $0.25$ .

Relative percent change

\Delta p=\frac{V_1-V_0}{V_0}\times100\%

Use the starting value $V_0>0$ as the baseline. $V_1$ is the final value. A zero baseline is undefined; percentage points are a different comparison.

Slope through two points

m=\frac{y_2-y_1}{x_2-x_1}

The points are $(x_1,y_1)$ and $(x_2,y_2)$ . Require $x_2\ne x_1$ ; a vertical line has undefined slope. Keep subtraction order consistent.

Area of a circle

A=\pi r^2

$r$ is the radius, not the diameter. $A$ uses square units. With diameter $d$ , first use $r=d/2$ . Retain $\pi$ until the final rounding.

Pythagorean theorem

a^2+b^2=c^2

For a right triangle, $a$ and $b$ are its legs and $c$ is the hypotenuse opposite the right angle. This relationship does not apply to every triangle.

Sample standard deviation

s=\sqrt{\frac{\sum_{j=1}^{n}(x_j-\bar{x})^2}{n-1}}

$x_j$ are observations, $\bar{x}$ is their mean and $n\ge2$ is the count. For the entire population, use denominator $n$ and population mean instead.

Worked example: Find a slope and check its sign

Find the slope through $(2,3)$ and $(8,15)$ . Use the same point order for the horizontal and vertical changes.

Subtract the first coordinates from the second coordinates. $\Delta x=8-2=6,\quad\Delta y=15-3=12$
Divide the vertical change by the nonzero horizontal change. $m=\frac{12}{6}=2$
Check that the slope reproduces the vertical change. $m\Delta x=2\times6=12=\Delta y$

Both coordinates increase, so a positive slope makes sense. Swapping both points preserves the slope; swapping only one subtraction changes it incorrectly.

Sources used for this sheet

Explore the full math formula library Read the subject glossary

Sheet 3 · One page · Six references

Unit conversion sheet

Exact factors and temperature offsets

Check the dimension, conversion direction and unit definition before substituting a value.

A4 PDF US Letter PDF Editable LaTeX

Same content in both paper sizes. PDF files are under 160 KB each.

Preview formulas, conditions and worked example

Factors use the international inch, foot and avoirdupois pound. Squared quantities need squared length factors. Binary storage prefixes are distinct from decimal SI prefixes.

Inches to centimetres

1\,\mathrm{in}=2.54\,\mathrm{cm}

Exact international-inch definition. Multiply an inch value by this factor to obtain centimetres; divide to reverse the conversion.

Square feet to square metres

1\,\mathrm{ft}^{2}=0.09290304\,\mathrm{m}^{2}

Exact, from $1\,\mathrm{ft}=0.3048\,\mathrm{m}$ . Square the length factor for area. This is not the historical US survey foot.

Litres to cubic centimetres

1\,\mathrm{L}=1000\,\mathrm{cm}^{3}

Also $1\,\mathrm{mL}=1\,\mathrm{cm}^{3}$ exactly. Volume does not determine mass without a density for the material and its conditions.

Fahrenheit to Celsius

T_C=\frac{5}{9}(T_F-32)

$T_F$ and $T_C$ are numerical temperature readings on the Fahrenheit and Celsius scales. For a temperature difference, use $\Delta T_C=5\Delta T_F/9$ without the offset.

Pounds of mass to kilograms

1\,\mathrm{lb}=0.45359237\,\mathrm{kg}

Exact for the international avoirdupois pound of mass. A pound-force is a force unit and requires a different conversion.

Binary storage to bytes

1\,\mathrm{GiB}=2^{30}\,\mathrm{B}

Exactly $1\,073\,741\,824$ bytes. By contrast, $1\,\mathrm{GB}=10^9\,\mathrm{B}$ . A byte contains eight bits; keep the unit label with the number.

Worked example: Convert an area without a linear-factor error

Convert $12\,\mathrm{ft}^{2}$ to square metres using the international foot. Display the final result to four decimal places.

Square the exact length conversion factor. $(0.3048)^2=0.09290304$
Apply that area factor and retain all intermediate digits. $12\times0.09290304=1.11483648$
Attach the square unit and round the final display. $12\,\mathrm{ft}^{2}\approx1.1148\,\mathrm{m}^{2}$

The exact factor does not add precision to a measured input. Four decimal places here are a requested display choice, not a claim about measurement uncertainty.

Sources used for this sheet

Check an area conversion Read the subject glossary

Sheet 4 · One page · Six references

Scientific constants sheet

Defining constants and standard gravity

Carry the unit and definition with a reference value when setting up science calculations.

A4 PDF US Letter PDF Editable LaTeX

Same content in both paper sizes. PDF files are under 160 KB each.

Preview formulas, conditions and worked example

The first five values are exact SI defining constants. Standard gravity is an exact conventional reference, not a measurement of local gravitational acceleration. Measured inputs still carry uncertainty.

Speed of light in vacuum

c=299\,792\,458\,\mathrm{m\,s^{-1}}

Exact SI defining constant. This is the speed in vacuum, not the propagation speed through an arbitrary material.

Planck constant

h=6.62607015\times10^{-34}\,\mathrm{J\,s}

Exact SI defining constant. Preserve the exponent and the joule-second unit when using an energy-frequency relationship.

Elementary charge

e=1.602176634\times10^{-19}\,\mathrm{C}

Exact, positive magnitude of the elementary charge. An electron has charge $-e$ ; the sign depends on the particle.

Avogadro constant

N_A=6.02214076\times10^{23}\,\mathrm{mol^{-1}}

Exact SI defining constant. Specify the counted entity, such as atoms, molecules or ions, when connecting amount of substance and entity count.

Boltzmann constant

k_B=1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}

Exact SI defining constant. Absolute temperature in relationships using this constant is measured in kelvin, not Celsius readings.

Standard acceleration of gravity

g_n=9.80665\,\mathrm{m\,s^{-2}}

Exact conventional value. For local weight use the appropriate local $g$ ; standard gravity does not describe every location or elevation.

Worked example: Weight under the standard-gravity convention

Use a mass of $2.000\,\mathrm{kg}$ with standard gravity. Report the force to four significant figures, matching the stated mass precision.

Choose the stated convention and distinguish mass from force. $W=mg_n$
Multiply the mass by the conventional acceleration. $W=2.000\times9.80665=19.6133\,\mathrm{N}$
Round the force, not the defining reference value. $W\approx19.61\,\mathrm{N}$

This is a standard-gravity example. A local weight calculation needs local acceleration, and a precision statement should follow the actual measurement uncertainty.

Sources used for this sheet

Convert force units Read the subject glossary

How to download, print and use a sheet

Step 1
Choose the task and paper size
Pick one subject or the complete pack. Download A4 for A4 paper, or US Letter for Letter paper. Open the preview if you want to inspect the formulas and conditions first; no account or email address is needed.
Step 2
Print a readable reference
Open the PDF in a PDF viewer and select the matching paper size, portrait orientation and Actual Size or 100% scaling. Turn off browser-added headers and footers when printing the webpage. Check the preview before printing; Fit may slightly resize the sheet to suit your printer's printable area.
Step 3
Apply, check and retain the units
Identify the unknown, check the formula's conditions and translate the known values into consistent units. Substitute without rounding intermediate values, then round the final result appropriately. Compare with the worked example or linked tool and label the answer so a later reader can interpret it.

For handwritten working, pair the sheet with printable square graph paper. Keep the reference separate from your substitutions so you can check which values came from the question and which came from a definition.

Four checks before using a formula

Match rates to periods
A yearly rate and a monthly payment are different time bases. Convert the rate as the model requires and count the matching periods. For $5\%$ , the decimal rate is $0.05$ ; the percentage coefficient is $5$ . Read which convention the formula uses.
Check the formula's conditions
A vertical line does not have a finite slope. The Pythagorean relationship needs a right triangle. A sample standard deviation uses a different denominator from a population calculation. The note beside each formula is part of the method, not an optional detail.
Keep dimensions and definitions
Area factors are squared length factors. Temperature readings need an offset, but temperature differences do not. Mass and force are different quantities. Write units at each stage and check the exact meaning of a storage prefix before choosing a factor.
Round only the final result
Retain the exact factor or full calculator precision through the working. Then choose decimal places or significant figures that suit the question. An exact defining constant does not make a measured input exact or justify extra digits in a final answer.

What these reference sheets are designed for

These sheets help you move from a remembered formula to a checkable calculation. Each page is deliberately focused: six relationships or values, enough context to interpret them, and one example that shows substitution and a final check. A short reference works best when it stays beside your own working, rather than replacing the reasoning that connects a question to an equation.

For students, the math sheet supports everyday percentage, coordinate, geometry and statistics questions. For teachers, a printed copy can accompany a lesson or a worked solution, and the editable LaTeX source lets you replace the example for your class. For practical measurement tasks, the conversion and constants sheets keep unit definitions close to the calculation. The finance sheet makes compounding and payment assumptions visible before you compare numerical outcomes.

The selections are educational references, not an exhaustive syllabus or an official exam-board booklet. Check your course requirements and permitted-materials policy before using a sheet in an assessment. Finance models omit the lender-specific fees and timing that can change a real statement; scientific examples require the conditions and measurement precision stated in your own problem. The linked tools and subject glossaries provide broader explanations when a compact page is not enough.

You may save the files for offline study and print copies for personal or classroom use. Open the LaTeX document in an editor to adapt the text or formulas, then compile it and inspect the new pagination. The provided PDFs already have fixed A4 and Letter layouts, selectable text, source links and a revision date, making it easier to identify which version you used.

Formula-sheet questions

What is included in each free formula-sheet download?

Each subject PDF contains six selected formulas or reference values, definitions and conditions beside each entry, one original example with three worked steps, source links and a link back to the live resource. The four-page pack combines all four subjects. These are focused reference sheets, not a complete syllabus, a workbook of practice questions or an official examination formula booklet.

Can I download and use the sheets without an account?

Yes. The PDF and LaTeX links lead directly to files, with no account, email form or checkout. Save a file once and use it offline; embedded links to source publishers and calculators require an internet connection. You may print copies for personal study or classroom use. The formulas remain selectable text rather than screenshots, so you can zoom without enlarging a bitmap of the page.

Should I choose A4 or US Letter, and will printing change the formulas?

Choose the file that matches the paper in your printer. A4 measures $210\times297\,\mathrm{mm}$ ; US Letter measures $8.5\times11\,\mathrm{in}$ . Both versions contain the same formulas and examples. Use portrait orientation and check the print preview. Actual Size preserves the designed typography; Fit can help a printer with larger unprintable margins, but it scales the page rather than changing the mathematical content.

What can I edit in the LaTeX source?

The source download is a complete .tex document, not a screenshot or a list of pasted equations. You can change wording, examples, spacing and paper size in a LaTeX editor. It uses common LaTeX packages and defaults to A4; the opening comment explains the Letter setting. Recompile after editing and check the conditions, symbols and pagination. Your edited source may paginate differently from the provided PDFs.

Why do the finance formulas sometimes differ from a bank's result?

The sheet uses simplified, constant-rate models with explicitly stated compounding and payment timing. A lender or savings provider may use calendar-day rules, fees, periodic rounding, changing rates or cash flows that the formula does not include. An advertised or legally disclosed APR is not automatically the nominal interest rate required by a model. Check the product terms and choose the linked calculator's applicable inputs before comparing results.

Are all the constants and conversion factors exact?

The listed unit definitions and binary prefix factors are exact for the specified units. The five SI defining constants are exact, while standard gravity is an exact conventional reference rather than a local measurement. That does not make a measured input exact. Preserve unit labels, distinguish mass from force and decide final precision from the measurements or the problem's requested rounding, not from the number of printed digits alone.

Can I use these formula sheets in an examination?

Only if your instructor or examination provider permits them. Toolarithm's sheets are independent teaching references and are not endorsed by an examination board. Check the permitted materials and required notation before taking any sheet into an assessment. For practice, use the conditions and worked example to understand why a formula applies, then try a fresh problem without relying on the sheet as an answer key.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

Equations are checked by substituting worked examples into the stated relationship and comparing the result with the interactive implementation.

Read the editorial methodology

Dates and sources

Updated October 31, 2026.

Choose your formula sheet

Finance formula sheet

Compound growth

Simple interest

Effective annual yield

Level loan payment

Present value

Compound annual growth

Worked example: Two years of monthly compounding

Sources used for this sheet

Math formula sheet

Percentage of a whole

Relative percent change

Slope through two points

Area of a circle

Pythagorean theorem

Sample standard deviation

Worked example: Find a slope and check its sign

Sources used for this sheet

Unit conversion sheet

Inches to centimetres

Square feet to square metres

Litres to cubic centimetres

Fahrenheit to Celsius

Pounds of mass to kilograms

Binary storage to bytes

Worked example: Convert an area without a linear-factor error

Sources used for this sheet

Scientific constants sheet

Speed of light in vacuum

Planck constant

Elementary charge

Avogadro constant

Boltzmann constant

Standard acceleration of gravity

Worked example: Weight under the standard-gravity convention

Sources used for this sheet

How to download, print and use a sheet

Choose the task and paper size

Print a readable reference

Apply, check and retain the units

Four checks before using a formula

Match rates to periods

Check the formula's conditions

Keep dimensions and definitions

Round only the final result

What these reference sheets are designed for

Formula-sheet questions

What is included in each free formula-sheet download?

Can I download and use the sheets without an account?

Should I choose A4 or US Letter, and will printing change the formulas?

What can I edit in the LaTeX source?

Why do the finance formulas sometimes differ from a bank's result?

Are all the constants and conversion factors exact?

Can I use these formula sheets in an examination?

Who maintains this page

Ownership and review

Methodology

Dates and sources