Formula Sheet

Interactive Formula Sheet

Formula Sheet

Interactive formula bank for mathematics and science

Algebra Formulas

Quadratic Formula

x = [-b ± √(b² - 4ac)] / 2a

Solution for equations of the form ax² + bx + c = 0

Arithmetic Sequence

aₙ = a₁ + (n - 1)d

nth term of an arithmetic sequence

Geometric Sequence

aₙ = a₁ × rⁿ⁻¹

nth term of a geometric sequence

Sum of Arithmetic Series

Sₙ = n/2 × (a₁ + aₙ)

Sum of first n terms of arithmetic sequence

Sum of Geometric Series

Sₙ = a₁(1 - rⁿ)/(1 - r)

Sum of first n terms of geometric sequence (r ≠ 1)

Exponent Rules

aᵐ × aⁿ = aᵐ⁺ⁿ

Product of powers with same base

Logarithm Properties

logₐ(mn) = logₐ m + logₐ n

Product rule for logarithms

Distance Formula

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

Distance between two points in coordinate plane

Geometry Formulas

Pythagorean Theorem

a² + b² = c²

For right triangles, where c is the hypotenuse

Area of Circle

A = πr²

Area of a circle with radius r

Circumference of Circle

C = 2πr

Circumference of a circle with radius r

Area of Triangle

A = (1/2)bh

Area of a triangle with base b and height h

Area of Rectangle

A = l × w

Area of a rectangle with length l and width w

Volume of Cube

V = s³

Volume of a cube with side length s

Volume of Cylinder

V = πr²h

Volume of a cylinder with radius r and height h

Surface Area of Sphere

SA = 4πr²

Surface area of a sphere with radius r

Trigonometry Formulas

Pythagorean Identity

sin²θ + cos²θ = 1

Fundamental trigonometric identity

Sine Rule

a/sinA = b/sinB = c/sinC

For any triangle with sides a, b, c and angles A, B, C

Cosine Rule

a² = b² + c² - 2bc cosA

For any triangle with sides a, b, c and angle A

Tangent Identity

tanθ = sinθ / cosθ

Definition of tangent function

Reciprocal Identities

cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ

Reciprocal trigonometric functions

Area of Triangle (Trig)

A = (1/2)ab sinC

Area using two sides and included angle

Angle Sum Identity

sin(A+B) = sinA cosB + cosA sinB

Sine of sum of two angles

Double Angle Formula

sin(2θ) = 2 sinθ cosθ

Double angle formula for sine

Calculus Formulas

Power Rule

d/dx(xⁿ) = nxⁿ⁻¹

Derivative of x raised to a power

Product Rule

d/dx(uv) = u'v + uv'

Derivative of product of two functions

Quotient Rule

d/dx(u/v) = (u'v - uv')/v²

Derivative of quotient of two functions

Chain Rule

d/dx[f(g(x))] = f'(g(x)) × g'(x)

Derivative of composite functions

Derivative of sin x

d/dx(sin x) = cos x

Derivative of sine function

Derivative of cos x

d/dx(cos x) = -sin x

Derivative of cosine function

Power Rule for Integration

∫xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)

Integration of power functions

Fundamental Theorem

∫ₐᵇ f(x) dx = F(b) - F(a)

Where F'(x) = f(x)

Physics Formulas

Newton's Second Law

F = ma

Force equals mass times acceleration

Kinetic Energy

KE = (1/2)mv²

Energy of motion

Potential Energy

PE = mgh

Gravitational potential energy

Ohm's Law

V = IR

Voltage equals current times resistance

Density Formula

ρ = m/V

Density equals mass divided by volume

Work Formula

W = Fd cosθ

Work equals force times displacement times cosine of angle

Power Formula

P = W/t

Power equals work divided by time

Speed Formula

v = d/t

Speed equals distance divided by time

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