List of all Mathematics formulas intermediate to advanced level.

Comprehensive Mathematics Formula Guide
Class 6th to class 12th and college level.

You must visit these two links for more advance formulas:

• Differential Equations Formulas
• Trigonometry Formulas

Calculus:

Rational Numbers Property
Commutative Property	Addition	${\displaystyle a+b=b+a}$	For any rational numbers a and b.
	Subtraction	${\displaystyle a–b\ne b–a}$	For any rational numbers a and b.
	Multiplication	${\displaystyle (a\times b)=(b\times a)}$	For any rational numbers a and b.
	Division	${\displaystyle (a÷b)\ne (b÷a)}$	For any rational numbers a and b.
Associative Property	Addition	${\displaystyle (a+b)+c=a+(b+c)}$	For any rational numbers a, b, and c.
	Subtraction	${\displaystyle (a–b)–c\ne a–(b–c)}$	For any rational numbers a, b, and c.
	Multiplication	${\displaystyle (a\times b)\times c=a\times (b\times c)}$	For any rational numbers a, b, and c.
	Division	${\displaystyle (a÷b)÷c\ne a÷(b÷c)}$	For any rational numbers a, b, and c.
Distributive Property		${\displaystyle a\times (b+c)=(a\times b)+(a\times c)}$	For any three rational numbers a, b and c.

Number Formation

Number Formation
Number	Mathematical Representation	Example
One Digit Number	${\displaystyle a=a}$	${\displaystyle a=5}$ ${\displaystyle 5=5}$
Two Digit Number	${\displaystyle ab=10a+b}$	${\displaystyle ab=15}$ ${\displaystyle a=1}$ ${\displaystyle b=5}$ ${\displaystyle 15=10\times 1+5}$
Three Digit Number	${\displaystyle abc=100a+10b+c}$	${\displaystyle abc=251}$ ${\displaystyle a=2}$ ${\displaystyle b=5}$ ${\displaystyle c=1}$ ${\displaystyle 251=100\times 2+10\times 5+1}$
Four Digit Number	${\displaystyle abcd=1000a+100b+10c+d}$	${\displaystyle abcd=3121}$ ${\displaystyle a=3}$ ${\displaystyle b=1}$ ${\displaystyle c=2}$ ${\displaystyle d=1}$ ${\displaystyle 3121=1000\times 3+100\times 1+10\times 2+1}$
${\displaystyle ...\infty }$	${\displaystyle ...\infty }$	So on...

Laws of Exponents Property

Laws of Exponents Property
Number	Mathematical Representation
${\displaystyle x^0}$	${\displaystyle x^0=1}$
${\displaystyle {\left(1\right)}^n}$	${\displaystyle {\left(1\right)}^n=1}$
${\displaystyle x^{-n}}$	${\displaystyle x^{-n}=\frac{1}{x^n}}$
${\displaystyle {\left(x^n\right)}^m}$	${\displaystyle {\left(x^n\right)}^m=x^{nm}}$
${\displaystyle \frac{x^n}{x^m}}$	${\displaystyle \frac{x^n}{x^m}=x^{n-m}}$
${\displaystyle \frac{x^n}{y^n}}$	${\displaystyle \frac{x^n}{y^n}={\left(\frac{x}{y}\right)}^n}$
${\displaystyle x^n\times y^n}$	${\displaystyle x^n\times y^n={\left(xy\right)}^n}$
${\displaystyle {\left(\frac{x}{y}\right)}^{-n}}$	${\displaystyle {\left(\frac{x}{y}\right)}^{-n}={\left(\frac{y}{x}\right)}^n}$

Algebraic Identity

Algebraic Identity Property
Expression	Identity
${\displaystyle {(a+b)}^2}$	${\displaystyle {(a+b)}^2=a^2+b^2+2ab}$
${\displaystyle {(a–b)}^2}$	${\displaystyle {(a–b)}^2=a^2+b^2–2ab}$
${\displaystyle (a+b)(a–b)}$	${\displaystyle (a+b)(a–b)=a^2–b^2}$
${\displaystyle (x+a)(x+b)}$	${\displaystyle (x+a)(x+b)=x^2+(a+b)x+ab}$
${\displaystyle (x+a)(x–b)}$	${\displaystyle (x+a)(x–b)=x^2+(a–b)x–ab}$
${\displaystyle (x–a)(x+b)}$	${\displaystyle (x–a)(x+b)=x^2+(b–a)x–ab}$
${\displaystyle (x–a)(x–b)}$	${\displaystyle (x–a)(x–b)=x^2–(a+b)x+ab}$
${\displaystyle {(a+b)}^3}$	${\displaystyle {(a+b)}^3=a^3+b^3+3ab(a+b)}$
${\displaystyle {(a–b)}^3}$	${\displaystyle {(a–b)}^3=a^3–b^3–3ab(a–b)}$

Square Roots Property:

A square number always ends with 0, 1, 4, 5, 6, and 9 at its units place. And it is the inverse operation of the square number.

Geometry Formula

Geometry Formula
Cuboid	H=Height, L=Length, B=Breadth, S=Side
	Total Surface area of Cuboid	${\displaystyle A=2((L\times B)+(B\times H)+(L\times H))}$
	Lateral Surface area of Cuboid	${\displaystyle A=2\times H(L+B)}$
	Volume of Cuboid	${\displaystyle V=L\times B\times H}$
	Perimeter of Cuboid	${\displaystyle P=4\times (L+B+H)}$
	Space Diagonals of Cuboid	${\displaystyle D=\sqrt{L^2+B^2+H^2}}$
	Face Diagonals of Cuboid	${\displaystyle D=\sqrt{L^2+B^2}}$
Cube	S= Side
	Surface area of Cube	${\displaystyle A=6{\left(S\right)}^2}$
	Lateral Surface area of Cube	${\displaystyle A=4{\left(S\right)}^2}$
	Volume of Cube	${\displaystyle V={\left(S\right)}^3}$
	Perimeter of Cube	${\displaystyle P=12\times \left(S\right)}$
	Space Diagonals of Cube	${\displaystyle D=\sqrt{3}\left(S\right)}$
Sphere	r= Radius
	Surface area of Sphere	${\displaystyle S=4\pi r^2}$
	Diameter of a Sphere	${\displaystyle D=2r}$
	Volume of Sphere	${\displaystyle V=\left(\frac{4}{3}\right)\pi r^3}$
	Circumference of Sphere	${\displaystyle 2\pi r}$
Hemisphere	r= Radius
	Total Surface area of Hemisphere	${\displaystyle C=3\pi r^2}$
	Curved Surface area of Hemisphere	${\displaystyle C=2\pi r^2}$
	Area of Hollow Hemisphere	${\displaystyle A=2\pi (r_2^2+r_1^2)+\pi (r_2^2-r_1^2)}$
	Volume of Hemisphere	${\displaystyle V=\left(\frac{2}{3}\right)\pi r^3}$
Hexagon	a= Length of One Side
	Area of Hexagon	${\displaystyle A=\frac{3\sqrt{3}}{2}\times a^2}$
	Permiter of a Hexagon	${\displaystyle P=6\times a}$
	Short Diagonal of Hexagon	${\displaystyle D=\sqrt{3}\times a}$
	Long Diagonal of Hexagon	${\displaystyle D=2\times a}$
Prism	L= Base Length, W= Base Width, H= Height, A= Apothem Length
	The surface area of a prism = (2×BaseArea) +Lateral Surface Area. The volume of a prism = Base Area× Height
	Rectangular Prism (A Rectangular Prism has 2 parallel rectangular bases and 4 rectangular faces.)	Surface area of Prism	${\displaystyle A=2\times (LW\times WH\times HL)}$
		Volume of Prism	${\displaystyle V=W\times L\times H}$
		Base Area of Prism	${\displaystyle A=L\times W}$
	Triangular Prism A triangular prism has 3 rectangular faces and 2 parallel triangular bases.	Surface area of Prism	${\displaystyle A=AL+3LH}$
		Volume of Prism	${\displaystyle V=\left(\frac{1}{2}\right)ALH}$
		Base Area of Prism	${\displaystyle A=\left(\frac{1}{2}\right)AL}$
	Pentagonal Prism A pentagonal prism has 5 rectangular faces and 2 parallel pentagonal bases.	Surface area of Prism	${\displaystyle A=5AL+5LH}$
		Volume of Prism	${\displaystyle V=\left(\frac{5}{2}\right)ALH}$
		Base Area of Prism	${\displaystyle A=\left(\frac{5}{2}\right)AL}$
	Hexagonal Prism A hexagonal prism has six rectangular faces and two parallel hexagonal bases.	Surface area of Prism	${\displaystyle A=6AL+6LH}$
		Volume of Prism	${\displaystyle V=3ALH}$
		Base Area of Prism	${\displaystyle A=3AL}$

Algebra:

Euler's Formula:

${\displaystyle \cos \left(\theta \right)+i\sin \left(\theta \right)=e^{i\theta }}$

Polar Form to Quadratic Form:

${\displaystyle \text{Convert}5e^{i\frac{\pi }{3}}\text{to the imaginary from}\text{a+bi.}}$
${\displaystyle \text{Sol:}}$
${\displaystyle e^{i\theta }=\cos \left(\theta \right)+i\sin \left(\theta \right)}$
${\displaystyle \theta =\frac{\pi }{3}}$
${\displaystyle \Rightarrow 5(\cos \left(\frac{\pi }{3}\right)+i\sin \left(\frac{\pi }{3}\right))}$
${\displaystyle \Rightarrow 5(\frac{1}{2}+i\frac{\sqrt{3}}{2})}$
${\displaystyle \Rightarrow \frac{5}{2}+5\frac{\sqrt{3}}{2}i}$

Quadratic Form to Polar Form:

${\displaystyle \text{Convert}\sqrt{2}+\sqrt{2}i\text{to the Ploar from}r{e}^{i\theta }.}$
${\displaystyle Solution:}$
${\displaystyle a=\sqrt{2},b=\sqrt{2}}$
${\displaystyle r=\sqrt{a^2+b^2}}$ ${\displaystyle \Rightarrow \sqrt{{\left(\sqrt{2}\right)}^2+{\left(\sqrt{2}\right)}^2}=2}$
${\displaystyle \tan \theta =\frac{\sin \theta }{\cos \theta }}$
${\displaystyle \tan \theta =\frac{\sqrt{2}}{\sqrt{2}}\Rightarrow 1\Rightarrow \theta =\frac{\pi }{4}}$
${\displaystyle \Rightarrow 2e^i\frac{\pi }{4}}$

Trigonometry Formula

All Trigonometry Formulas
Number	Mathematical Representation
${\displaystyle {\cos }^2\theta =1-{\sin }^2\theta }$	${\displaystyle \tan \theta =\frac{\sin \theta }{\cos \theta }}$
${\displaystyle \sec \theta =\frac{1}{\cos \theta }}$	${\displaystyle {\tan }^2\theta ={\sec }^2\theta –1}$
${\displaystyle \text{Ploar Form}=r{e}^{i\theta }}$ ${\displaystyle r=\sqrt{a^2+b^2}}$	${\displaystyle }$

Trigonometric table(sin-cos-tan table) for 0 to 360.

Trigonometric table 0 to 360 Degree with (sin-cos-tan table)
${\displaystyle Degrees}$	${\displaystyle 0^{\circ }}$	${\displaystyle {30}^{\circ }}$	${\displaystyle {45}^{\circ }}$	${\displaystyle {60}^{\circ }}$	${\displaystyle {90}^{\circ }}$	${\displaystyle {120}^{\circ }}$	${\displaystyle {135}^{\circ }}$	${\displaystyle {150}^{\circ }}$	${\displaystyle {180}^{\circ }}$	${\displaystyle {210}^{\circ }}$	${\displaystyle {225}^{\circ }}$	${\displaystyle {240}^{\circ }}$	${\displaystyle {270}^{\circ }}$	${\displaystyle {300}^{\circ }}$	${\displaystyle {315}^{\circ }}$	${\displaystyle {330}^{\circ }}$	${\displaystyle {360}^{\circ }}$
${\displaystyle Radians}$	${\displaystyle 0}$	${\displaystyle \frac{\pi }{6}}$	${\displaystyle \frac{\pi }{4}}$	${\displaystyle \frac{\pi }{3}}$	${\displaystyle \frac{\pi }{2}}$	${\displaystyle \frac{2\pi }{3}}$	${\displaystyle \frac{3\pi }{4}}$	${\displaystyle \frac{5\pi }{6}}$	${\displaystyle \pi }$	${\displaystyle \frac{7\pi }{6}}$	${\displaystyle \frac{5\pi }{4}}$	${\displaystyle \frac{4\pi }{3}}$	${\displaystyle \frac{3\pi }{2}}$	${\displaystyle \frac{5\pi }{3}}$	${\displaystyle \frac{7\pi }{4}}$	${\displaystyle \frac{11\pi }{6}}$	${\displaystyle 2\pi }$
${\displaystyle \sin }$	${\displaystyle 0}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{\sqrt{2}}}$	${\displaystyle \frac{\sqrt{3}}{2}}$	${\displaystyle 1}$	${\displaystyle \frac{\sqrt{3}}{2}}$	${\displaystyle \frac{1}{\sqrt{2}}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle 0}$	${\displaystyle -\frac{1}{2}}$	${\displaystyle -\frac{1}{\sqrt{2}}}$	${\displaystyle -\frac{\sqrt{3}}{2}}$	${\displaystyle -1}$	${\displaystyle -\frac{\sqrt{3}}{2}}$	${\displaystyle -\frac{1}{\sqrt{2}}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle 0}$
${\displaystyle \cos }$	${\displaystyle 1}$	${\displaystyle \frac{\sqrt{3}}{2}}$	${\displaystyle \frac{1}{\sqrt{2}}}$	${\displaystyle \frac{1}{2}}$	${\displaystyle 0}$	${\displaystyle -\frac{1}{2}}$	${\displaystyle -\frac{1}{\sqrt{2}}}$	${\displaystyle -\frac{\sqrt{3}}{2}}$	${\displaystyle -1}$	${\displaystyle -\frac{\sqrt{3}}{2}}$	${\displaystyle -\frac{1}{\sqrt{2}}}$	${\displaystyle -\frac{1}{2}}$	${\displaystyle 0}$	${\displaystyle \frac{1}{2}}$	${\displaystyle \frac{1}{\sqrt{2}}}$	${\displaystyle \frac{\sqrt{3}}{2}}$	${\displaystyle 1}$
${\displaystyle \tan }$	${\displaystyle 0}$	${\displaystyle \frac{1}{\sqrt{3}}}$	${\displaystyle 1}$	${\displaystyle \sqrt{3}}$	${\displaystyle UD}$	${\displaystyle -\sqrt{3}}$	${\displaystyle -1}$	${\displaystyle -\frac{1}{\sqrt{3}}}$	${\displaystyle 0}$	${\displaystyle \frac{1}{\sqrt{3}}}$	${\displaystyle 1}$	${\displaystyle \sqrt{3}}$	${\displaystyle UD}$	${\displaystyle -\sqrt{3}}$	${\displaystyle -1}$	${\displaystyle -\frac{1}{\sqrt{3}}}$	${\displaystyle 0}$
${\displaystyle \cos ec}$	${\displaystyle UD}$	${\displaystyle 2}$	${\displaystyle \sqrt{2}}$	${\displaystyle \frac{2}{\sqrt{3}}}$	${\displaystyle 1}$	${\displaystyle \frac{2}{\sqrt{3}}}$	${\displaystyle \sqrt{2}}$	${\displaystyle 2}$	${\displaystyle UD}$	${\displaystyle -2}$	${\displaystyle -\sqrt{2}}$	${\displaystyle \frac{2}{\sqrt{3}}}$	${\displaystyle -1}$	${\displaystyle -\frac{2}{\sqrt{3}}}$	${\displaystyle -\sqrt{2}}$	${\displaystyle -2}$	${\displaystyle UD}$
${\displaystyle \sec }$	${\displaystyle 1}$	${\displaystyle \frac{2}{\sqrt{3}}}$	${\displaystyle \sqrt{2}}$	${\displaystyle 2}$	${\displaystyle UD}$	${\displaystyle -2}$	${\displaystyle -\sqrt{2}}$	${\displaystyle \frac{2}{\sqrt{3}}}$	${\displaystyle -1}$	${\displaystyle -\frac{2}{\sqrt{3}}}$	${\displaystyle -\sqrt{2}}$	${\displaystyle 2}$	${\displaystyle UD}$	${\displaystyle 2}$	${\displaystyle \sqrt{2}}$	${\displaystyle \frac{2}{\sqrt{3}}}$	${\displaystyle 1}$
${\displaystyle \cot }$	${\displaystyle UD}$	${\displaystyle \sqrt{3}}$	${\displaystyle 1}$	${\displaystyle \frac{1}{\sqrt{3}}}$	${\displaystyle 0}$	${\displaystyle -\frac{1}{\sqrt{3}}}$	${\displaystyle -1}$	${\displaystyle -\sqrt{3}}$	${\displaystyle UD}$	${\displaystyle \sqrt{3}}$	${\displaystyle 1}$	${\displaystyle \frac{1}{\sqrt{3}}}$	${\displaystyle 0}$	${\displaystyle -\frac{1}{\sqrt{3}}}$	${\displaystyle -1}$	${\displaystyle -\sqrt{3}}$	${\displaystyle UD}$

Rotations in 2D

Matrix for rotation of a point about the origin by an angle 'Theta'.
${\displaystyle \left[\begin{array}{c}{x}_2\\ y_2\end{array}\right]=\left[\begin{array}{cc}\cos \left(\theta \right)& -\sin \left(\theta \right)\\ +\sin \left(\theta \right)& \cos \left(\theta \right)\end{array}\right]\left[\begin{array}{c}{x}_1\\ y_1\end{array}\right]}$