Basic Math Formulas

Most of the math you use outside school comes down to a short list of formulas: percentages, fractions, averages, area and volume, a little algebra, and a few money formulas. This page collects the basic math formulas in one place, with a worked example under each one so you can see how it is used.

Where we have a calculator for a formula, it is linked, so you can check your own numbers with the working shown.

Order of Operations

When a sum has more than one operation, work in this order:

Parentheses (brackets)
Exponents (powers and roots)
Multiplication and Division, from left to right
Addition and Subtraction, from left to right

In the US this is taught as PEMDAS. In the UK it is BODMAS or BIDMAS. They are the same rule.

Example: 8 + 2 × 5 = 8 + 10 = 18, not 50. Multiplication comes before addition.

Percentage Formulas

Percentage: (Part ÷ Whole) × 100
Part: (Percentage ÷ 100) × Whole
Percentage change: (New − Original) ÷ Original × 100

Example: 36 out of 48 is 36 ÷ 48 × 100 = 75%.

Our step-by-step guide on how to calculate percentage covers every type of percentage question. To check an answer, use the percentage calculator or, for increases and decreases, the percentage change calculator.

Fraction Formulas

Add or subtract: a/b + c/d = (ad + bc) ÷ bd
Multiply: a/b × c/d = ac ÷ bd
Divide: a/b ÷ c/d = a/b × d/c (flip the second fraction and multiply)

Example: 2/3 + 1/4 = (8 + 3) ÷ 12 = 11/12.

Always simplify the answer by dividing the top and bottom by their greatest common factor. The fraction calculator does this and shows the common denominator it used. To turn a decimal into a fraction, use the decimal to fraction calculator.

Ratio and Proportion

A ratio compares two amounts, written a : b. Two ratios are in proportion when a/b = c/d, and you can solve for a missing value by cross-multiplying.

Example: a recipe uses 3 cups of flour for 4 people. How much for 20 people? 3/4 = x/20, so 4x = 60 and x = 15 cups.

Averages: Mean, Median, Mode and Range

For the numbers 4, 7, 7, 9, 13:

Mean (the usual “average”): add them and divide by how many there are. 40 ÷ 5 = 8.
Median: the middle value when they are in order. 7.
Mode: the value that appears most often. 7.
Range: the largest minus the smallest. 13 − 4 = 9.

The median is often a better guide than the mean when one value is far bigger or smaller than the rest, such as house prices or salaries. An average calculator can find all four at once.

Exponents and Roots

a^m × a^n = a^(m + n)
a^m ÷ a^n = a^(m − n)
(a^m)n = a^(m × n)
a^0 = 1
a^(−n) = 1 ÷ a^n
√a = a^(1/2), and the nth root of a = a^(1/n)

Example: 2³ × 2⁴ = 2⁷ = 128.

To simplify a square root, take out any perfect square factors: √72 = √(36 × 2) = 6√2. The square root calculator gives both the decimal answer and the simplified radical. For powers, use the exponent calculator.

Area Formulas

Shape	Area formula
Square	side²
Rectangle	length × width
Triangle	½ × base × height
Parallelogram	base × height
Trapezoid (trapezium)	½ × (a + b) × height
Circle	π × radius²

Example: a circle with a radius of 3 cm has an area of π × 3² ≈ 28.27 cm².

Perimeter and Circumference

Square: 4 × side
Rectangle: 2 × (length + width)
Circle (circumference): 2 × π × radius, or π × diameter

Example: fencing a 12 m × 8 m garden needs 2 × (12 + 8) = 40 m of fence.

Volume Formulas

Shape	Volume formula
Cube	side³
Rectangular box (cuboid)	length × width × height
Cylinder	π × radius² × height
Cone	⅓ × π × radius² × height
Sphere	⁴⁄₃ × π × radius³

Example: a cylinder 5 cm tall with a radius of 2 cm holds π × 2² × 5 ≈ 62.83 cm³.

The Pythagorean Theorem

For a right-angled triangle with shorter sides a and b and longest side (hypotenuse) c:

a² + b² = c²

Example: if the two shorter sides are 6 and 8, then c = √(36 + 64) = √100 = 10.

Builders use the 3-4-5 version of this to check that a corner is square.

Basic Algebra Formulas

Slope of a line: m = (y₂ − y₁) ÷ (x₂ − x₁)
Equation of a straight line: y = mx + b, where m is the slope and b is where the line crosses the y-axis
Distance between two points: d = √((x₂ − x₁)² + (y₂ − y₁)²)
Quadratic formula: for ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) ÷ 2a

Example: x² − 5x + 6 = 0 has a = 1, b = −5 and c = 6. That gives x = (5 ± √(25 − 24)) ÷ 2 = (5 ± 1) ÷ 2, so x = 3 or x = 2.

Factors and Multiples

A prime number has exactly two factors: 1 and itself.
Prime factorization writes a number as a product of primes. 360 = 2³ × 3² × 5.
The greatest common factor (GCF) is the largest number that divides two numbers exactly.
The least common multiple (LCM) is the smallest number both divide into.

Example: 12 = 2² × 3 and 18 = 2 × 3². The GCF takes the lowest power of each shared prime: 2 × 3 = 6. The LCM takes the highest power of each prime: 2² × 3² = 36.

The prime factorization calculator shows the factor tree for any number.

Money Formulas

Simple interest: I = P × r × t, where P is the amount, r is the yearly rate as a decimal and t is the time in years.
Compound interest: A = P × (1 + r/n)^(n × t), where n is the number of times interest is added each year.
Sales tax: Total = Price × (1 + tax rate)
Discount: Sale price = Price × (1 − discount rate)
Markup: Selling price = Cost × (1 + markup rate)

Example: simple interest. $1,000 at 5% for 3 years earns 1,000 × 0.05 × 3 = $150.

Example: compound interest. The same $1,000 at 5% compounded once a year for 3 years grows to 1,000 × 1.05³ = $1,157.63, which is $7.63 more than simple interest.

Check these with the sales tax calculator, the discount calculator and the markup calculator.

Speed, Distance and Time

Speed = Distance ÷ Time
Distance = Speed × Time
Time = Distance ÷ Speed

Example: a 150-mile drive at an average of 60 mph takes 150 ÷ 60 = 2.5 hours, or 2 hours 30 minutes.

In physics, velocity also includes direction, and constant acceleration uses v = u + at. The velocity calculator covers both.

Temperature Conversion

Celsius to Fahrenheit: °F = °C × 9/5 + 32
Fahrenheit to Celsius: °C = (°F − 32) × 5/9

Example: 180 °C is 180 × 9/5 + 32 = 356 °F, which is why recipes often round it to 350 °F.

Put the Formulas to Work

See how these formulas show up in real life in our guide to everyday math examples, or browse all math calculators to check your answers with the working shown.