Basic Math Formulas

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Find here a comprehensive list of basic math formulas commonly used when doing basic math computation

Basic Math Formulas

The formulas used most in school math, each with an example. Search, or pick a topic.

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78 formulas

1. Arithmetic

Rule	Formula	Example
Addition	a + b = sum	7 + 5 = 12
Subtraction	a − b = difference	12 − 5 = 7
Multiplication	a × b = product	6 × 4 = 24
Division	a ÷ b = quotient, b ≠ 0	24 ÷ 4 = 6
Division with a remainder	dividend = divisor × quotient + remainder	23 = 4 × 5 + 3, so 23 ÷ 4 = 5 R 3
Commutative property	a + b = b + a a × b = b × a	3 + 5 = 5 + 3 = 8
Associative property	(a + b) + c = a + (b + c) (a × b) × c = a × (b × c)	(2 + 3) + 4 = 2 + (3 + 4) = 9
Distributive property	a(b + c) = ab + ac	3(4 + 5) = 12 + 15 = 27
Order of operations (PEMDAS)	Parentheses → Exponents → Multiplication and Division (left to right) → Addition and Subtraction (left to right)	2 + 3 × 4² = 2 + 3 × 16 = 2 + 48 = 50

2. Exponents and roots

Rule	Formula	Example
Power	aⁿ = a × a × … × a (n factors)	2⁴ = 2 × 2 × 2 × 2 = 16
Product of powers	a^m × aⁿ = a^{m + n}	2³ × 2² = 2⁵ = 32
Quotient of powers	a^m ÷ aⁿ = a^{m − n}	5⁴ ÷ 5² = 5² = 25
Power of a power	(a^m)ⁿ = a^mn	(3²)³ = 3⁶ = 729
Zero exponent	a⁰ = 1, a ≠ 0	7⁰ = 1
Negative exponent	$a^{- n} = \frac{1}{a^{n}}$	$2^{- 3} = \frac{1}{2^{3}} = \frac{1}{8}$
Square root	√a = b means b² = a, b ≥ 0	√49 = 7, because 7² = 49
Root of a product	$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$	√36 = √4 × √9 = 2 × 3 = 6

3. Fractions

Rule	Formula	Example
Adding fractions	$\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$	$\frac{1}{2} + \frac{1}{3} = \frac{3 + 2}{6} = \frac{5}{6}$
Subtracting fractions	$\frac{a}{b} - \frac{c}{d} = \frac{a d - b c}{b d}$	$\frac{1}{2} - \frac{1}{3} = \frac{3 - 2}{6} = \frac{1}{6}$
Multiplying fractions	$\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}$	$\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}$
Dividing fractions	$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a d}{b c}$	$\frac{1}{2} \div \frac{1}{3} = \frac{1}{2} \times \frac{3}{1} = \frac{3}{2}$
Equivalent fractions	$\frac{a}{b} = \frac{a \times k}{b \times k}$	$\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}$
Simplifying a fraction	$\frac{a}{b} = \frac{a \div k}{b \div k}$ use k = GCF of a and b	$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$
Improper fraction to mixed number	$\frac{a}{b} = q \frac{r}{b}$ q = whole-number part of a ÷ b, r = remainder	$\frac{7}{3} = 2 \frac{1}{3}$
Mixed number to improper fraction	$w \frac{a}{b} = \frac{w \times b + a}{b}$	$2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$

The addition and subtraction formulas use bd as a common denominator. You can also use the least common denominator; either way, simplify the answer at the end.

4. Ratios, decimals and percents

Rule	Formula	Example
Proportion (cross products)	$\frac{a}{b} = \frac{c}{d} \Rightarrow a \times d = b \times c$	$\frac{2}{3} = \frac{x}{12} \Rightarrow 3 x = 24, x = 8$
Unit price	unit price = total price ÷ number of units	$6.00 ÷ 4 = $1.50 each
Percent to decimal	divide by 100	35% = 35 ÷ 100 = 0.35
Decimal to percent	multiply by 100	0.8 × 100 = 80%
Finding the percent	$percent = \frac{part}{whole} \times 100$	$\frac{15}{60} \times 100 = 25%$
Finding the part	$part = \frac{percent × whole}{100}$	$\frac{20 \times 150}{100} = 30$
Finding the whole	$whole = \frac{part × 100}{percent}$	$\frac{30 \times 100}{20} = 150$
Percent increase	$\frac{new − original}{original} \times 100$	50 → 60: $\frac{10}{50} \times 100 = 20%$
Percent decrease	$\frac{original − new}{original} \times 100$	80 → 60: $\frac{20}{80} \times 100 = 25%$

In these formulas the percent is a plain number, such as 25 for 25%.

5. Consumer math

Calculation	Formula	Example
Discount	discount = list price × discount rate	$80 × 0.25 = $20
Sale price	sale price = list price − discount	$80 − $20 = $60
Sale price in one step	sale price = list price × (1 − discount rate)	$80 × 0.75 = $60
Discount rate	discount rate = discount ÷ list price	$20 ÷ $80 = 0.25 = 25%
Sales tax	sales tax = price × tax rate	$60 × 0.08 = $4.80
Total cost	total cost = price + sales tax	$60 + $4.80 = $64.80
Tip	tip = cost of the meal × tip rate	$40 × 0.15 = $6
Commission	commission = total sales × commission rate	$2,000 × 0.05 = $100
Profit	profit = revenue − cost	$500 − $320 = $180
Simple interest	I = P × r × t	$1,000 × 0.05 × 3 = $150
Compound interest	$A = P {(1 + \frac{r}{n})}^{n t}$	$1,000 at 5% once a year for 2 years: 1,000 × 1.05² = $1,102.50

Rates are written as decimals: 20% is 0.20. P = principal (money borrowed or saved), r = yearly interest rate, t = time in years, n = number of times interest is added each year, A = final amount.

6. 2D shapes: area and perimeter

Shape	Area	Perimeter
Rectangle	A = l × w	P = 2l + 2w
Square	A = s²	P = 4s
Triangle	A = ½ × b × h	P = a + b + c
Parallelogram	A = b × h	P = 2a + 2b
Trapezoid	A = ½(b₁ + b₂)h	P = a + b₁ + b₂ + c
Rhombus	A = ½ × d₁ × d₂	P = 4s
Circle	A = πr²	C = 2πr = πd

l = length, w = width, s = side, b = base, h = height (measured straight up from the base), a and c = the other sides, d₁ and d₂ = diagonals, r = radius, d = diameter = 2r, π ≈ 3.14.

7. Angles and triangles

Rule	Formula	Example
Pythagorean theorem	a² + b² = c² (c = hypotenuse)	3² + 4² = 9 + 16 = 25 = 5²
Angles of a triangle	∠A + ∠B + ∠C = 180°	50° + 60° + 70° = 180°
Angles of a polygon	sum of interior angles = (n − 2) × 180°	hexagon: (6 − 2) × 180° = 720°
Complementary angles	x + y = 90°	30° and 60°
Supplementary angles	x + y = 180°	110° and 70°

8. 3D shapes: volume and surface area

Shape	Volume	Surface area
Cube	V = s³	SA = 6s²
Rectangular prism (box)	V = l × w × h	SA = 2(lw + lh + wh)
Any prism	V = B × h	SA = 2B + ph
Triangular prism	V = ½ × b × h × l	SA = bh + (s₁ + s₂ + s₃)l
Cylinder	V = πr²h	SA = 2πr² + 2πrh
Cone	V = ⅓πr²h	SA = πr² + πrl
Pyramid	V = ⅓ × B × h	SA = B + ½pl (regular pyramid)
Sphere	V = ⁴⁄₃πr³	SA = 4πr²

B = area of the base, p = perimeter of the base, h = height, l = slant height (cone and pyramid) or length (triangular prism). For the triangular prism, b and h belong to the triangle and s₁, s₂, s₃ are its sides.

9. Coordinate graphing

Rule	Formula	Example
Slope	$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$	(1, 2) and (3, 8): $\frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$
Slope-intercept form	y = mx + b	y = 3x − 1: slope 3, crosses the y-axis at −1
Distance between two points	$d = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$	(0, 0) and (3, 4): $\sqrt{9 + 16} = 5$
Midpoint	$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2})$	(2, 4) and (6, 10): (4, 7)

10. Basic statistics

Measure	How to find it	Example
Mean (average)	$\frac{a_{1} + a_{2} + \dots + a_{n}}{n}$	4, 8, 9: (4 + 8 + 9) ÷ 3 = 7
Median	the middle value when the data is in order; with an even count, the average of the two middle values	3, 5, 9, 12: (5 + 9) ÷ 2 = 7
Mode	the value that appears most often (there can be more than one, or none)	2, 3, 3, 7: mode = 3
Range	range = largest value − smallest value	2, 3, 3, 7: 7 − 2 = 5

11. Distance, rate, time and temperature

Rule	Formula	Example
Distance	d = r × t	60 mph × 3 h = 180 miles
Rate (speed)	r = d ÷ t	150 km ÷ 2 h = 75 km/h
Time	t = d ÷ r	120 miles ÷ 40 mph = 3 h
Celsius to Fahrenheit	$°F = \frac{9}{5} \times °C + 32$	20 °C: 36 + 32 = 68 °F
Fahrenheit to Celsius	$°C = \frac{5}{9} \times (°F - 32)$	212 °F: $\frac{5}{9} \times 180 = 100$ °C

Basic math test

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Special factoring formulas

Special math constant

Pascal's triangle

Basic Math Formulas

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