Mathematics Formula Reference

Quick reference guide for essential mathematical formulas, complete with examples and variable explanations to support your learning.

Basic Mathematics

Essential formulas for basic mathematics

Area of Rectangle

Formula
A = length × width

Example:

A = 5 × 3 = 15 square units

Variables:

A:Area
length:Length
width:Width

Area of Circle

Formula
A = πr²

Example:

A = π × 4² = 50.27 square units

Variables:

A:Area
r:Radius
π:Pi (≈ 3.14159)

Pythagorean Theorem

Formula
a² + b² = c²

Example:

3² + 4² = 5² → 9 + 16 = 25

Variables:

a:Side A
b:Side B
c:Hypotenuse

Distance Formula

Formula
d = √[(x₂-x₁)² + (y₂-y₁)²]

Example:

d = √[(4-1)² + (6-2)²] = √[9 + 16] = 5

Variables:

d:Distance
x₁:X coordinate 1
y₁:Y coordinate 1
x₂:X coordinate 2
y₂:Y coordinate 2

Percentage Formulas

Essential formulas for percentage formulas

Percentage of a Number

Formula
Result = (Percentage ÷ 100) × Number

Example:

25% of 80 = (25 ÷ 100) × 80 = 20

Variables:

Result:Final result
Percentage:Percentage value
Number:Original number

Percentage Change

Formula
Change% = [(New - Old) ÷ Old] × 100

Example:

From 50 to 60: [(60-50) ÷ 50] × 100 = 20%

Variables:

Change%:Percentage change
New:New value
Old:Original value

Find the Whole

Formula
Whole = Part ÷ (Percentage ÷ 100)

Example:

If 20 is 25% of what? 20 ÷ (25÷100) = 80

Variables:

Whole:Complete amount
Part:Partial amount
Percentage:Percentage value

Find the Percentage

Formula
Percentage = (Part ÷ Whole) × 100

Example:

15 out of 60: (15 ÷ 60) × 100 = 25%

Variables:

Percentage:Percentage result
Part:Partial amount
Whole:Complete amount

Algebra

Essential formulas for algebra

Slope Formula

Formula
m = (y₂ - y₁) ÷ (x₂ - x₁)

Example:

m = (8 - 2) ÷ (4 - 1) = 6 ÷ 3 = 2

Variables:

m:Slope
x₁:X coordinate 1
y₁:Y coordinate 1
x₂:X coordinate 2
y₂:Y coordinate 2

Quadratic Formula

Formula
x = [-b ± √(b² - 4ac)] ÷ 2a

Example:

For x² + 5x + 6 = 0: x = [-5 ± √(25-24)] ÷ 2 = -2 or -3

Variables:

x:Solution
a:Coefficient of x²
b:Coefficient of x
c:Constant term

Point-Slope Form

Formula
y - y₁ = m(x - x₁)

Example:

Point (2,3), slope 4: y - 3 = 4(x - 2)

Variables:

y:Y coordinate
x:X coordinate
m:Slope
x₁:Known X
y₁:Known Y

Trigonometry

Essential formulas for trigonometry

Sine Function

Formula
sin(θ) = opposite ÷ hypotenuse

Example:

sin(30°) = 0.5

Variables:

sin(θ):Sine of angle
θ:Angle
opposite:Opposite side
hypotenuse:Hypotenuse

Cosine Function

Formula
cos(θ) = adjacent ÷ hypotenuse

Example:

cos(60°) = 0.5

Variables:

cos(θ):Cosine of angle
θ:Angle
adjacent:Adjacent side
hypotenuse:Hypotenuse

Tangent Function

Formula
tan(θ) = opposite ÷ adjacent

Example:

tan(45°) = 1

Variables:

tan(θ):Tangent of angle
θ:Angle
opposite:Opposite side
adjacent:Adjacent side

Law of Cosines

Formula
c² = a² + b² - 2ab×cos(C)

Example:

c² = 3² + 4² - 2(3)(4)×cos(90°) = 25

Variables:

a:Side A
b:Side B
c:Side C
C:Angle opposite side C

Statistics

Essential formulas for statistics

Mean (Average)

Formula
Mean = Sum of all values ÷ Number of values

Example:

Mean of [2,4,6,8] = (2+4+6+8) ÷ 4 = 5

Variables:

Mean:Average value

Standard Deviation

Formula
σ = √[Σ(x - μ)² ÷ N]

Example:

For data set [2,4,6], σ = 1.63

Variables:

σ:Standard deviation
x:Individual value
μ:Mean
N:Number of values

Probability

Formula
P(A) = Number of favorable outcomes ÷ Total outcomes

Example:

Rolling a 6: P(6) = 1 ÷ 6 = 0.167

Variables:

P(A):Probability of event A

Using These Formulas

Quick Tips:

  • • Always identify which formula applies to your problem
  • • Substitute known values into the formula
  • • Solve for the unknown variable step by step
  • • Check your answer by substituting back

Practice with Calculators:

  • • Use our Math Calculator for basic computations
  • • Scientific Calculator for advanced functions
  • • Verify your manual calculations
  • • Explore different values to understand patterns