Calculator Guides and Formulas
Every calculator on this site already shows its formula next to the result. These guides are the longer version: where a method comes from, what its terms actually mean, and how to read an answer that looks wrong. They are written for the moment when the number is not the problem.
There are 85 of them, grouped below by what they are about. Every group names the calculators that run the same arithmetic, and you can go the other way from the full calculator library when you would rather start from the tool.
Guide or calculator
The split is simpler than it looks. If you have the numbers and want the answer, open the calculator. If you have the answer and have to defend it, reproduce it by hand in an exam, or rebuild it in a spreadsheet, open the guide. Most guides here pair with a calculator that runs the same arithmetic, and each one links to it, so starting on the wrong side of the pair costs you one click rather than a wasted search.
Why some topics have several guides
Standard deviation has six, which looks like repetition and is not. One explains what the statistic measures, one gives the formula, one separates the sample and population versions that use different divisors, one covers the spreadsheet functions people actually reach for, one names the symbol, and one sets variance beside it. Those are six different questions that arrive at the same topic, and answering them on one page would bury five of them.
Algebra vocabulary
The words for the parts of an expression, and the words for what operations produce. In 5x + 3 the term is 5x, the coefficient is 5, the variable is x and the constant is 3 - four names for four things, and most of the confusion around them is confusion about their neighbours. These are grouped apart from the algebra FORMULA guides on purpose: a reader looking up what a coefficient is has a different question from one looking up the quadratic formula. Product, sum, factor and array sit here too, as the words for what the operations produce and for the picture that shows it - an array of 24 dots IS the factor pairs of 24, drawn.
What Is a Term in Math? Definition and How to Count Terms
A term is a piece of an expression separated by plus or minus signs. Covers counting terms, like terms, monomials and binomials, and term vs factor.
What Is a Coefficient in Math? Definition and Examples
A coefficient is the number multiplying a variable. Covers finding it, the invisible coefficient of 1, negative coefficients, and coefficient vs constant vs term.
What Is a Constant in Math? Definition and Examples
A constant is a value that does not change. Covers constants in expressions, named constants like pi and e, constant vs variable, and the constant term in a polynomial.
What Is a Variable in Math? Definition and Examples
A variable is a symbol standing for a value that can change. Covers why letters are used, variable vs constant, independent and dependent variables, and conventions.
What Is a Function in Math? Definition, Notation and Examples
A function assigns exactly one output to each input. Covers the defining rule, the vertical line test, f(x) notation, domain and range, and function vs relation vs equation.
What Is Domain in Math? How to Find the Domain of a Function
The domain is the set of inputs a function accepts. Covers the three restrictions that exclude values, all three notations, domain vs range, and domain in context.
What Is a Product in Math? Definition and Examples
A product is the result of multiplying numbers together. Covers product vs sum, difference and quotient, worked examples with fractions and negatives, and how products relate to factors.
What Is a Sum in Math? Definition and Examples
A sum is the result of adding numbers together. Covers sum vs product, sigma notation, the 1-to-n shortcut, worked examples with fractions and series, and how SUM behaves in a spreadsheet.
What Is an Array in Math? Definition and Examples
An array is objects arranged in equal rows and columns. Covers how arrays model multiplication and division, why they prove commutativity, arrays as factor pairs, and array vs matrix vs area model.
What Is a Factor in Math? Definition, Examples and How to Find Them
A factor divides a number exactly with no remainder. Covers how to list every factor, prime factorisation, the greatest common factor, and factor vs multiple.
Tools that run this arithmetic: Simplify Calculator, Polynomial Calculator, Factor Calculator, Domain Calculator.
Statistics and spread
Measuring how far a data set spreads from its centre, and placing one value inside that spread. The sample and population distinction runs through most of these, and it changes the answer rather than the wording.
What Is Standard Deviation?
Learn what standard deviation means, what it tells you, how to interpret it, and how it relates to variance, averages, and spread in a data set.
Standard Deviation Formula
Learn the standard deviation formula, the difference between sample and population standard deviation, and how to calculate standard deviation step by step.
Sample vs Population Standard Deviation
Learn the difference between sample and population standard deviation, when to use each formula, and how they relate to STDEV.S and STDEV.P in Excel.
Variance vs Standard Deviation
Learn the difference between variance and standard deviation, how they are related, and when to use each one to understand spread in a data set.
How to Calculate Standard Deviation in Excel
Learn how to calculate standard deviation in Excel using STDEV.S and STDEV.P, when to use each formula, and how to check your result with a calculator.
Standard Deviation Symbol
Learn the standard deviation symbols used in statistics, including σ for population standard deviation, s for sample standard deviation, μ, x̄, n, and n − 1.
Z Score Formula
Learn the z score formula, how to calculate a z score step by step, and what positive, negative, and zero z scores mean in statistics.
How to Find Range in Statistics: Formula, Steps, and Examples
Learn how to calculate the statistical range of a dataset, see step-by-step examples, and understand how range compares to standard deviation and IQR.
What Is Mode in Math? Definition, Calculation, and Examples
Learn how to find the mode of a dataset, understand unimodal, bimodal, and multimodal distributions, and compare mean, median, and mode.
What Is Mean in Math? Definition, Formula and Examples
What mean means in math: the sum of the values divided by how many there are. Includes the formula, worked examples, mean vs median vs mode, and when the mean misleads.
What Is Median in Math? Definition and How to Find It
The median is the middle value of a sorted data set. Covers odd and even counts, the position formula, median vs mean, and why income is reported as a median.
What Is Range in Math? Both Meanings, Explained
Range means two different things in math: maximum minus minimum in statistics, and the set of outputs for a function. Both explained, with worked examples and range vs domain.
Percent Error Formula
Learn the percent error formula, why absolute value is used, and how to calculate percent error step by step with chemistry and physics examples.
Tools that run this arithmetic: Standard Deviation Calculator – Variance & SD, Variance Calculator, Z Score Calculator.
Percentages and change
The three questions that all get called a percentage: what share one number is of another, what a value becomes after a rise or fall, and how big the move between two values was. Mixing up the second and third is the error these guides exist to prevent.
How to Calculate a Percentage
Step-by-step guide to calculating percentages. Learn the three main percentage formulas, see worked examples for finding percent of a number, percent change, and tips and discounts.
How to Calculate Percentage Increase
Learn how to calculate percentage increase, use the percentage increase formula, and check your answer with a simple calculator.
Percentage Increase Formula
Learn the percentage increase formula, what each part means, and how to use it with simple examples for prices, pay, traffic, and other increases.
Percent Change Formula
Learn the percent change formula, how to calculate percentage increase and decrease, and see step-by-step worked examples.
Tools that run this arithmetic: Percentage Calculator – 3 Ways to Calculate %, Percentage Increase Calculator – Percent Change, Percent Decrease Calculator.
Algebra, sequences and number theory
Solving and rearranging rather than measuring. Sequences appear three times here because the explicit and recursive forms describe the same list in incompatible ways, and picking the wrong one is what makes the nth term come out wrong.
Quadratic Formula
Learn the quadratic formula, what each part means, how the discriminant works, and how to solve quadratic equations step by step.
Slope Formula
Learn the slope formula, what each part means, and how to find the slope of a line between two points step by step.
Point-Slope Formula
Learn how to use the point-slope formula (y - y1 = m(x - x1)) to find the equation of a line, see worked examples, and check when to apply it.
Arithmetic Sequence Formula
Learn the arithmetic sequence formula to calculate the nth term (a_n) and sum of a sequence, with step-by-step worked examples.
Explicit Formula
Learn how to use explicit formulas in sequences, compare explicit vs recursive formulas, and solve nth-term math problems step-by-step.
Recursive Formula
Learn how recursive formulas work, how they define sequences using previous terms, the difference between recursive and explicit formulas, and step-by-step math examples.
Combination Formula
Learn how to use the combination formula (nCr) in statistics, understand the difference between combinations and permutations, and check worked examples.
Change of Base Formula
Learn the change of base formula (log_b(x) = log(x)/log(b)) to evaluate logarithms with any base using common or natural logs, with worked examples.
What Is a Factor in Math? Definition, Finding Factor Pairs, and Trees
Learn what factors are, the difference between factors and multiples, how to list factor pairs, and how to find the Greatest Common Factor (GCF).
What Is a Prime Number? Definition, Examples, and the Sieve of Eratosthenes
Learn what prime numbers are, the difference between prime and composite numbers, why 1 is neither, and how to find primes using the Sieve of Eratosthenes.
What Is a Rational Number? Definition, Examples, and Properties
Understand rational numbers, how they compare to irrational numbers, how to write them as ratios, and the difference between terminating and repeating decimals.
What Is e in Math? Euler's Number Explained
e is Euler's number, roughly 2.718281828459045. Covers where it comes from, e vs pi, how to find it on a calculator, worked continuous-growth examples, and its link to the natural log.
What Is Standard Form in Math? Both Meanings Explained
Standard form means scientific notation in the UK and Ax + By = C for a line in the US. Covers both, with conversion steps, worked examples and how to tell which one a question means.
Tools that run this arithmetic: Quadratic Formula Calculator, Slope Calculator, Factor Calculator.
Geometry vocabulary
The words for shapes and for what you are allowed to do to them. Congruent and similar are the pair most often swapped, and the difference is one line: same angles makes shapes similar, same angles AND same lengths makes them congruent. A translation is the transformation that guarantees congruence, since sliding a figure changes nothing but where it sits. Ray, line and segment differ only in how many endpoints they have, which is why the ray guide settles them on the one test that is checkable rather than definitional: a segment has a midpoint and a ray cannot. These sit apart from the formula guides below for the same reason the algebra words do - looking up what congruent means is a different question from looking up the area of a trapezium.
What Does Congruent Mean in Math? Definition and Rules
Congruent means same size and same shape. Covers congruent vs similar, the five triangle congruence rules, why AAA and SSA fail, a worked SAS example, and congruence in modular arithmetic.
What Is a Ray in Math? Definition and Examples
A ray is part of a line with one endpoint that continues forever in one direction. Covers ray vs line vs line segment, how to name a ray, why order matters, and worked angle and number-line examples.
What Is a Translation in Math? Definition and Examples
A translation slides every point of a shape the same distance in the same direction. Covers the (x + a, y + b) rule, translation vs rotation, reflection and dilation, a worked coordinate example, and why a translation is a vector.
What Is a Quadrilateral? Definition, Types, and Formulas
A complete guide to quadrilaterals: definition, classifications (square, rectangle, parallelogram, trapezoid, rhombus), properties, and area/perimeter formulas.
Tools that run this arithmetic: Right Triangle Calculator, Midpoint Formula Calculator, Vector Calculator.
Geometry and measurement
Lengths, areas and volumes, plus the coordinate-geometry pair that most courses teach together. The square-footage guide is the practical end of the same arithmetic, written for a room rather than an exam question.
Geometry Formulas
A complete reference list of geometry formulas for 2D and 3D shapes, including area, perimeter, volume, surface area, coordinate geometry, and right triangles.
Area Formula
Learn the area formulas for squares, rectangles, triangles, circles, trapezoids, and parallelograms, with step-by-step worked examples for each shape.
Surface Area Formula
Learn how to calculate the surface area of cubes, rectangular prisms, cylinders, spheres, and cones, with worked examples for each shape.
Perimeter Formula
Learn how to calculate the perimeter of squares, rectangles, triangles, and circles (circumference), with step-by-step worked examples.
Volume Formula
Learn the volume formulas for cubes, boxes, cylinders, spheres, cones, and pyramids, with worked examples, cubic-unit conversions, and how resizing a dimension changes the result.
Circumference Formula
Learn the circumference formula, how to calculate the distance around a circle using radius or diameter, and check worked examples.
Arc Length Formula
Learn the arc length formula, how to calculate the length of a curve along a circle's edge in radians or degrees, and see worked examples.
Distance Formula
Learn the distance formula, how it works with coordinates, and how to find the distance between two points step by step.
Midpoint Formula
Learn the midpoint formula, what each coordinate means, and how to find the midpoint between two points step by step.
How to Find the Area of a Triangle: Formulas, Examples, and Math Rules
Master the triangle area formula (Area = ½ × base × height). Learn how to identify bases and perpendicular heights, solve step-by-step examples, and use Heron's formula.
How to Calculate Square Feet: A Practical Guide for Homes and Rooms
Learn how to measure and calculate square footage for rooms, yards, and irregular spaces, with worked examples and material ordering tips.
Tools that run this arithmetic: Area Calculator – Find the Area of a Shape, Volume Calculator – Rectangular Prism, Cylinder, Sphere & Cone, Distance Formula Calculator, Square Footage Calculator – Sq Ft for Any Room.
Physics and chemistry
The formulas that carry units, where an answer can be arithmetically right and physically meaningless. Velocity and acceleration are set out separately because one is a rate and the other is a rate of a rate.
Velocity Formula
Learn the velocity formula, how to calculate speed and direction, what each variable means, and check worked kinematic examples.
Acceleration Formula
Learn the acceleration formula, what each variable means, how to use it with units like m/s², and step-by-step worked examples.
Kinetic Energy Formula
Learn the kinetic energy formula, what each variable means, how to use it with SI units like Joules, and review worked physics examples.
Work Formula
Learn the work formula (W = F × d), how to calculate work at an angle with cosine, SI units (Joules), and step-by-step physics worked examples.
Empirical Formula
Learn how to calculate the empirical formula of a chemical compound from mass or percent composition with a step-by-step worked example.
Tools that run this arithmetic: Velocity Calculator, Acceleration Calculator, Molecular Weight Calculator.
Grades, GPA and admissions
Reading a mark you have been given, and the scoring behind the exams that produce them. The AP and admissions pages are index pages rather than articles, collecting the score calculators for a whole exam season.
How to Calculate GPA
Complete guide to calculating your GPA on the 4.0 scale. Learn the formula, see worked examples for weighted and unweighted GPA, and convert letter grades to grade points.
What Is a Good GPA?
A complete guide to what counts as a good GPA in high school and college. See national averages, college admissions benchmarks, scholarship thresholds, and graduate school requirements.
How to Calculate Your Final Grade
Step-by-step guide to calculating your final class grade. Learn how to handle weighted assignments, figure out what you need on a final exam, and convert points to percentages.
When Do AP Scores Come Out in 2026? (July 6 Release Date)
2026 AP scores release on Monday, July 6, 2026 starting at 8 AM Eastern Time. Full schedule, how to access your scores, and what to expect after the College Board exam.
AP Score Calculators
Free AP score calculators for every major AP exam. Predict your 1-5 score for APUSH, AP Calc AB/BC, AP Bio, AP Chem, AP Lang, AP Lit, AP Gov, AP World, AP Stats, and AP Precalc using the official section weightings.
College Admissions Calculators
Free score calculators for every major college admissions test. Predict your SAT (400-1600), ACT (1-36), PSAT/NMSQT with National Merit Selection Index, and AP exam scores. Used by students nationwide.
Chemistry Calculators
Free chemistry calculators for molar mass, molarity, stoichiometry, dilutions, pH, limiting reactants, and more. Each one shows the formula and works through the steps.
Tools that run this arithmetic: GPA Calculator, Final Grade Calculator – What You Need to Pass, AP Score Calculator.
Borrowing and interest
How a monthly payment is actually assembled, and why two loans quoted at the same rate can cost different amounts. The APR and interest-only comparisons are the two that change decisions rather than just explaining notation.
How to Calculate a Loan Payment
Learn how to calculate a monthly loan payment using loan amount, APR, and loan term. See the loan payment formula, worked example, and extra payment impact.
How to Calculate a Mortgage Payment
Learn how to calculate a mortgage payment with principal, interest, taxes, insurance, PMI, and HOA dues. See the formula and worked example.
How to Calculate a Car Payment
Learn how to calculate a monthly car payment with vehicle price, down payment, trade in value, sales tax, APR, loan term, and fees.
How to Calculate Loan Interest
Learn how to calculate loan interest using simple interest, interest only payments, and amortizing loan examples.
APR vs Interest Rate
Learn the difference between APR and interest rate, why APR can be higher than the stated rate, and which number to use when comparing loans.
Simple Interest vs Compound Interest
Learn the difference between simple interest and compound interest, how each formula works, and when to use each calculator.
Interest Only Loan Formula
Learn the interest only loan formula, how to calculate interest only payments, and why the principal balance does not go down during the interest only period.
Interest Only vs Amortizing Loan
Compare interest only payments and amortizing loan payments, including how each payment works, what happens to the principal balance, and when to use each calculator.
What Is Annual Income? Gross vs. Net Salary Explained
Learn what annual income means, the difference between gross and net income, common pre-tax and post-tax deductions, and how to convert hourly wages to salary.
Tools that run this arithmetic: Loan Payment Calculator – Amortization Tool, Mortgage Calculator, Car Payment Calculator.
Energy expenditure and body measures
What the standard body-baseline estimates are actually estimating. BMR and TDEE are separated deliberately: one is what you burn at rest, the other is that figure scaled by how you spend the day, and the two get quoted interchangeably.
What Is TDEE?
Learn what TDEE means, what it includes, how it differs from BMR, and how to use your estimated daily calories for weight loss, maintenance, or muscle gain.
How to Calculate TDEE
Learn how to calculate TDEE using BMR and activity level, what the formula means, and how to use your result for weight loss, maintenance, or muscle gain.
BMR vs TDEE
Learn the difference between BMR and TDEE, how each one is calculated, and which number matters more for weight loss, maintenance, or muscle gain.
How to Calculate BMI
Step-by-step guide to calculating Body Mass Index (BMI). Learn the metric and imperial formulas, see worked examples, and understand BMI categories from underweight to obesity.
Tools that run this arithmetic: TDEE Calculator, BMR Calculator, BMI Calculator.
Writing, dates and file tools
The corners of the site that are not arithmetic. Sentence structure, counting forward to a date, opening a file format Windows will not preview, and the letter-to-number tradition behind the gematria tools.
What Is a Compound Sentence? Rules, Coordinating Conjunctions, and Semicolons
Learn the definition of a compound sentence, how to join independent clauses using FANBOYS or semicolons, and how to avoid common comma splices.
What Is 90 Days From Today? How to Calculate Date Offsets
Learn how to calculate exactly 90 days from today or any starting date, count across different months, and check common calendar deadline rules.
How to Open HEIC Files on Windows (Free, No Software)
HEIC files won't open on Windows by default. Here's how to view and convert iPhone HEIC photos to JPG or PDF for free, right in your browser, with no software to install.
PDF Tools
Free online PDF tools to convert, rotate, crop, split, delete, reorder pages, and add page numbers to your PDF files. Processed 100% locally.
What Is Gematria? A Simple Guide to How Gematria Works
Plain-language explanation of gematria: what it is, how letter-to-number ciphers work in English, Hebrew, and Greek, and what the numbers can and can't tell you.
Tools that run this arithmetic: Word Counter, Date Calculator, Gematria Calculator.
Where these lead
The calculators behind the guides
Each guide explains a method that a tool on this site already runs. These are the sections those tools live in.
- Education where most of these guides point: the statistics, algebra, geometry and grade tools they explain.
- Money the payment and interest calculators behind the borrowing guides.
- Health the BMR, TDEE and BMI tools the energy guides are written against.
- Home where the square-footage guide leads, since most home projects are priced from an area.
- Writing counters and readability, next to the sentence-structure guide.
- Everyday Life date arithmetic, which the ninety-days guide works through by hand.
- Gematria Tools the gematria calculators the tradition guide describes.
- PDF Tools the browser-side PDF and file tools two of these guides cover.
Questions
About these guides
How Resources fits in
Each calculator already shows its formula on the same page as the result. Resources is for the longer explanations that do not fit in a single panel: what a method actually means, where it comes from, and how to read the output sensibly.