Math Formulas

Comprehensive collection of mathematical formulas grouped by category.

percent

Fractions & Percentages

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function

Algebraic Formulas

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Difference of Squares

$$ a^2 - b^2 = (a-b)(a+b) $$

Perfect Square (+)

$$ (a+b)^2 = a^2 + 2ab + b^2 $$

Perfect Square (-)

$$ (a-b)^2 = a^2 - 2ab + b^2 $$

Sum of Cubes

$$ a^3 + b^3 = (a+b)(a^2 - ab + b^2) $$

Difference of Cubes

$$ a^3 - b^3 = (a-b)(a^2 + ab + b^2) $$

Cube of Sum

$$ (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 $$

Cube of Difference

$$ (a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $$

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

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

Sum of first n cubes

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

superscript

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

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

Power of a Product

$$ (ab)^n = a^n b^n $$

Zero Exponent

$$ a^0 = 1 $$

Negative Exponent

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

shapes

Geometry Areas

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Rectangle Area

$$ A = l \times w $$

Triangle Area

$$ A = \frac{1}{2} bh $$

Circle Area

$$ A = \pi r^2 $$

Circle Circumference

$$ C = 2 \pi r $$

Trapezoid Area

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

architecture

Trigonometry

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Sine Definition

$$ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} $$

Cosine Definition

$$ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} $$

Tangent Definition

$$ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} $$

Pythagorean Identity

$$ \sin^2 \theta + \cos^2 \theta = 1 $$

Sine Rule

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

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

Cosine Rule

Generalization of the Pythagorean theorem for any triangle.

$$ c^2 = a^2 + b^2 - 2ab \cos C $$

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Coordinate Geometry

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Section Formula

$$ P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) $$

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

$$ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} $$

Angle Between Lines

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

$$ \tan \theta = \left| \frac{m_1 - m_2}{1 + m_1m_2} \right| $$

Area of Triangle

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

$$ \text{Area} = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| $$

Centroid of Triangle

The intersection of the three medians of the triangle.

$$ C = \left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right) $$

Two-Point Form

$$ \frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1} $$

Slope-Intercept Form

$$ y = mx + b $$

Intercept Form

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

Dist. Between Parallel Lines

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

$$ d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} $$