Math Formulas

Comprehensive collection of mathematical formulas grouped by category.

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Algebraic Formulas

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Difference of Squares

a^{2} - b^{2} = (a - b) (a + b)

Perfect Square (+)

{(a + b)}^{2} = a^{2} + 2 a b + b^{2}

Perfect Square (-)

{(a - b)}^{2} = a^{2} - 2 a b + b^{2}

Sum of Cubes

a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})

Difference of Cubes

a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})

Cube of Sum

{(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}

Cube of Difference

{(a - b)}^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

functions

Series & Sums

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Sum of first n natural numbers

\sum_{i = 1}^{n} i = \frac{n (n + 1)}{2}

Sum of first n squares

\sum_{i = 1}^{n} i^{2} = \frac{n (n + 1) (2 n + 1)}{6}

Sum of first n cubes

\sum_{i = 1}^{n} i^{3} = {(\frac{n (n + 1)}{2})}^{2}

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Exponents & Roots

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Product Rule

a^{m} \cdot a^{n} = a^{m + n}

Quotient Rule

\frac{a^{m}}{a^{n}} = a^{m - n}

Power of a Power

{(a^{m})}^{n} = a^{m n}

Power of a Product

{(a b)}^{n} = a^{n} b^{n}

Zero Exponent

a^{0} = 1

Negative Exponent

a^{- n} = \frac{1}{a^{n}}

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Geometry Areas

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Rectangle Area

A = l \times w

Triangle Area

A = \frac{1}{2} b h

Circle Area

A = π r^{2}

Circle Circumference

C = 2 π r

Trapezoid Area

A = \frac{1}{2} (a + b) h

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Trigonometry

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Sine Definition

\sin θ = \frac{Opposite}{Hypotenuse}

Cosine Definition

\cos θ = \frac{Adjacent}{Hypotenuse}

Tangent Definition

\tan θ = \frac{Opposite}{Adjacent}

Pythagorean Identity

\sin^{2} θ + \cos^{2} θ = 1

Sine Rule

Relates the sides of a triangle to the sines of its angles.

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2 R

Cosine Rule

Generalization of the Pythagorean theorem for any triangle.

c^{2} = a^{2} + b^{2} - 2 a b \cos C

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Coordinate Geometry

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Section Formula

P = (\frac{m x_{2} + n x_{1}}{m + n}, \frac{m y_{2} + n y_{1}}{m + n})

Slope of a Line

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

Point-Slope Form

y - y_{1} = m (x - x_{1})

Distance of Point from Line

d = \frac{| A x_{1} + B y_{1} + C |}{\sqrt{A^{2} + B^{2}}}

Angle Between Lines

The angle θ between two lines with slopes m₁ and m₂.

\tan θ = | \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |

Area of Triangle

Area of a triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃).

Area = \frac{1}{2} | x_{1} (y_{2} - y_{3}) + x_{2} (y_{3} - y_{1}) + x_{3} (y_{1} - y_{2}) |

Centroid of Triangle

The intersection of the three medians of the triangle.

C = (\frac{x_{1} + x_{2} + x_{3}}{3}, \frac{y_{1} + y_{2} + y_{3}}{3})

Two-Point Form

\frac{y - y_{1}}{x - x_{1}} = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

Slope-Intercept Form

y = m x + b

Intercept Form

\frac{x}{a} + \frac{y}{b} = 1

Dist. Between Parallel Lines

Distance between Ax + By + C₁ = 0 and Ax + By + C₂ = 0.

d = \frac{| C_{1} - C_{2} |}{\sqrt{A^{2} + B^{2}}}