Mathematics Formulas

A = l × w

Multiply length by width.

e.g. 8 cm by 5 cm → A = 40 cm².

A = ½ × b × h

Half of base times perpendicular height.

e.g. b = 10, h = 6 → A = 30.

A = πr²

Pi times the square of the radius.

e.g. r = 7 → A = 49π ≈ 154.

a² + b² = c²

For right-angled triangles only.

e.g. 3² + 4² = 5².

I = (P × R × T) ÷ 100

Interest = principal × rate × time, divided by 100.

e.g. P=1000, R=5, T=2 → I = 100.

x = (−b ± √(b² − 4ac)) ÷ (2a)

Solves any quadratic equation.

e.g. x² − 5x + 6 = 0 → x = 2 or x = 3.

d = √((x₂ − x₁)² + (y₂ − y₁)²)

Uses the Pythagorean theorem in the plane.

e.g. (0,0) and (3,4) → d = 5.

M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2)

Average of the coordinates.

e.g. (2,4) and (6,8) → M = (4,6).

m = (y₂ − y₁) ÷ (x₂ − x₁)

Change in y divided by change in x.

e.g. (1,2) and (3,6) → m = 2.

logₐ(xy) = logₐ(x) + logₐ(y)

Brings multiplication down to addition.

e.g. log₂(8) = log₂(2) + log₂(4) = 1 + 2 = 3.

aₙ = a + (n − 1)d

Add (n−1) copies of the common difference to the first term.

e.g. a=3, d=2, n=5 → a₅ = 11.

Sₙ = (n ÷ 2)(2a + (n − 1)d)

Number of terms times the average of the first and last term.

e.g. a=3, d=2, n=5 → S₅ = 35.

det = ad − bc

Products of the main diagonal minus the other diagonal.

e.g. [[4,2],[3,1]] → det = −2.

u · v = uₓvₓ + uᵧvᵧ

Sum of the products of corresponding components.

e.g. u=(2,3), v=(4,−1) → u·v = 5.

d/dx [xⁿ] = nxⁿ⁻¹

Bring the exponent down, then reduce it by one.

e.g. d/dx [x³] = 3x².

∫ xⁿ dx = xⁿ⁺¹ ÷ (n + 1) + C

Inverse of the differentiation power rule.

e.g. ∫ x² dx = x³ ÷ 3 + C.

limₓ→a f(x) = f(a)

Works when the function is continuous at the point.

e.g. limₓ→2 (x² + 1) = 5.

mean = (Σ xᵢ) ÷ n

Sum of values divided by the count.

e.g. {4,6,8} → mean = 6.

P(E) = favourable ÷ total outcomes

Always between 0 and 1.

e.g. P(rolling a 4) on a die = 1/6.