Math Formulas

Math Formulas

Teachtopia Presents: Math Formulas You Need to Know

Teachtopia Presents

Math Formulas You Need to Know

A student-friendly reference sheet covering essential formulas from elementary school through college.

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Elementary

Elementary School Formulas

Core arithmetic, measurement, fractions, and basic geometry.

See the shape behind the formulaConnect area and perimeter rules to rectangles, squares, and triangles.

Perimeter of a Rectangle

P = 2l + 2w

l = length, w = width

Example: l = 8, w = 3 → P = 2(8) + 2(3) = 22 units

Area of a Rectangle

A = l × w

Example: l = 7, w = 4 → A = 7 × 4 = 28 square units

Area of a Square

A = s²

s = side length

Example: s = 5 → A = 5² = 25 square units

Perimeter of a Square

P = 4s

Example: s = 6 → P = 4(6) = 24 units

Area of a Triangle

A = ½bh

b = base, h = height

Example: b = 10, h = 6 → A = ½(10)(6) = 30 square units

Fraction of a Number

part = fraction × whole

Example: ¾ of 20 → ¾ × 20 = 15

Average

average = sum ÷ number of values

Example: 6, 8, 10 → average = 24 ÷ 3 = 8

Distance

d = r × t

distance = rate × time

Example: r = 50 mph, t = 3 hr → d = 50 × 3 = 150 miles

Percent

percent = (part ÷ whole) × 100

Example: 18 out of 24 → (18 ÷ 24) × 100 = 75%

Middle School

Middle School Formulas

Ratios, percentages, geometry, probability, and introductory algebra.

From numbers to graphsSlope, circles, probability, and the Pythagorean theorem become easier when you can picture them.

Slope

m = (y₂ − y₁) ÷ (x₂ − x₁)

Example: (2, 3) and (6, 11) → m = (11 − 3) ÷ (6 − 2) = 2

Slope-Intercept Form

y = mx + b

Example: m = 2, b = 3 → y = 2x + 3. If x = 4, y = 11

Pythagorean Theorem

a² + b² = c²

Example: a = 3, b = 4 → 3² + 4² = 25 → c = 5

Circumference of a Circle

C = 2πr

Example: r = 5 → C = 2π(5) = 10π ≈ 31.4 units

Area of a Circle

A = πr²

Example: r = 4 → A = π(4²) = 16π ≈ 50.3 square units

Volume of a Rectangular Prism

V = lwh

Example: l = 5, w = 3, h = 2 → V = 5 × 3 × 2 = 30 cubic units

Simple Interest

I = Prt

P = principal, r = rate, t = time

Example: P = $1,000, r = 5%, t = 2 → I = 1000(.05)(2) = $100

Probability

P(event) = favorable outcomes ÷ total outcomes

Example: rolling a 6 on a fair die → P = 1 ÷ 6 ≈ 16.7%

Percent Change

% change = (change ÷ original) × 100

Example: 50 increases to 60 → (10 ÷ 50) × 100 = 20% increase

High School

High School Formulas

Algebra, geometry, trigonometry, statistics, and exponential growth.

Algebra meets geometryQuadratics, coordinate geometry, trigonometry, and sequences all describe visible patterns.

Quadratic Formula

x = (−b ± √(b² − 4ac)) ÷ 2a

Example: x² − 5x + 6 = 0 → a=1, b=−5, c=6 → x = 2 or 3

Distance Formula

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Example: (1, 2) and (4, 6) → d = √(3² + 4²) = 5

Midpoint Formula

M = ((x₁+x₂)/2, (y₁+y₂)/2)

Example: (2, 4) and (8, 10) → M = (5, 7)

Circle Equation

(x − h)² + (y − k)² = r²

Example: center (2, −1), r = 3 → (x − 2)² + (y + 1)² = 9

Trigonometric Ratios

sin θ = opp/hyp cos θ = adj/hyp tan θ = opp/adj

Example: opposite = 3, adjacent = 4, hypotenuse = 5 → sin θ = 3/5, cos θ = 4/5, tan θ = 3/4

Compound Interest

A = P(1 + r/n)^(nt)

Example: $1,000 at 6% compounded monthly for 2 years → A = 1000(1+.06/12)²⁴ ≈ $1,127.16

Exponential Growth / Decay

y = a(1 ± r)^t

Example: 500 grows 10% per year for 3 years → y = 500(1.10)³ = 665.5

Arithmetic Sequence

aₙ = a₁ + (n − 1)d

Example: 4, 7, 10, …; a₁=4, d=3 → a₅ = 4 + 4(3) = 16

Geometric Sequence

aₙ = a₁r^(n − 1)

Example: 3, 6, 12, …; a₁=3, r=2 → a₅ = 3(2⁴) = 48

College

College Math Formulas

Common formulas from calculus, statistics, linear algebra, and finance.

Change, accumulation, and dataCalculus and statistics use formulas to describe motion, rates of change, areas, and distributions.

Derivative Definition

f′(x) = lim[h→0] (f(x+h) − f(x)) / h

Example: f(x)=x² → [((x+h)²−x²)/h] → 2x as h → 0

Power Rule

d/dx(xⁿ) = nxⁿ⁻¹

Example: d/dx(x⁵) = 5x⁴

Product Rule

(fg)′ = f′g + fg′

Example: f=x², g=x³ → (x²x³)′ = (2x)(x³) + (x²)(3x²) = 5x⁴

Quotient Rule

(f/g)′ = (f′g − fg′) / g²

Example: f=x², g=x+1 → derivative = [2x(x+1) − x²] / (x+1)²

Chain Rule

d/dx f(g(x)) = f′(g(x))g′(x)

Example: d/dx[(3x+1)²] = 2(3x+1)(3) = 6(3x+1)

Fundamental Theorem of Calculus

∫ₐᵇ f(x)dx = F(b) − F(a)

Example: ∫₀² x dx = [x²/2]₀² = 2

Standard Score

z = (x − μ) / σ

Example: x=85, μ=75, σ=5 → z = (85−75)/5 = 2

Sample Mean

x̄ = (Σx) / n

Example: 4, 6, 8, 10 → x̄ = 28/4 = 7

Matrix Multiplication

Cᵢⱼ = Σ AᵢₖBₖⱼ

Example: [[1,2],[3,4]] × [[5,6],[7,8]] → [[19,22],[43,50]]

Study tip: Don’t only memorize formulas. Learn what each variable means, when the formula applies, and practice with one example of each.