Area of a Pentagon Formula:
A = 52sa
Where,
s is the side of the pentagon.
a is the apothem length.

Area of a Trapezoid = A = $\frac{1}{2}$ $\times$ h $\times$ (a + b)
Where:
h = height (Note – This is the perpendicular height, not the length of the legs.)
a = the short base
b = the long base

The Area of a Rectangle Formula is, A = l $\times$ b
Where,
l = Length
b = Breadth

Area of sector = $\frac{\theta}{360}$ $\times$ $\pi$ $r^2$
Length of an arc of a sector = $\frac{\theta}{360}$ $\times$ 2$\pi$r

Area of a Square = $Side^{2}$

Area of a Kite = $\frac{1}{2}$diagonal
Area of a Parallelogram = base $\times$ height

Area of a Rectangle = base $\times$ height

Area of a Trapezoid = \[\frac{base_{1}+base_{2}}{2}height\]

The Area of a Rhombus Formula is,
4 × area of ∆ AOB
= 4 × $\frac{1}{2}$ × AO × OB sq. units
= 4 × $\frac{1}{2}$ × $\frac{1}{2}$ $d_{1}$ × $\frac{1}{2}$ $d_{2}$ sq. units
= 4 × $\frac{1}{8}$ $d_{1}$ × $d_{2}$ square units
= $\frac{1}{2}$ × $d_{1}$ × $d_{2}$

Therefore
Where, Area of a Rhombus = A = $\frac{1}{2}$ × $d_{1}$ × $d_{2}$
$d_{1}$ and $d_{2}$ are the diagonals of the rhombus.

The Area of a Parallelogram is,
Area = b $\times$ h
Where, b is the length of any base and h is the corresponding altitude or height

The Area of a SquareFormula is,
Area = $a^2$
Where, a is the length of the side of a square

Area of Circle = $\pi$ r² = $\frac{\pi d^2}{4}$ = $\frac{C \times r}{2}$
Where,
r is the radius of the circle.
d is the diameter of the circle.
C is the circumference of the circle.

Area of a regular polygon

A = $\frac{l^{2}n}{4tan(\frac{\pi }{n})}$

Where,
l is the side length
n is the number of sides

The Area of a Hexagon
A = 3sr
Where,
s is the side of the hexagon.
r is the radius of the hexagon.

Area of Octagon =
2$a^2 (1 + \sqrt{2})$ Where,
r is the radius of the circle.
d is the diameter of the circle.
C is the circumference of the circle.

Area under the Curve Formula

\[Area = \int_{a}^{b}f(x)dx\]

\[\LARGE a_{n}=a_{1}+(n-1)d\]

\[\large P(x) = \frac{n!}{r!(n-r)!} . p^{r}(1-p)^{n} = C(n, r).p^{r}(1-p)^{n-r}\]

Where,
n = Total number of events
r = Total number of successful events.
p = Probability of success on a single trial.

_nC_r = $\frac{n!}{r!(n − r)!}$

\[\large \frac{d}{dx}r^{n} = nx^{n-1}\]

\[\large \frac{d}{dx}(fg) = fg^{1} + gf^{1}\]

\[\large \frac{d}{dx}\left (\frac{f}{g} \right ) = \frac{gf^{1}-fg^{1}}{g^{2}}\]

\[\large \frac{d}{dx} f(g(x))= f^{1}(g(x)) g^{1}(x)\]

\[\large \frac{d}{dx} (\sin\: x)= \cos\: x\]

\[\large \frac{d}{dx} (\cos\: x)= -\sin\: x\]

\[\large \frac{d}{dx} (\tan\: x)= -\sec^{2}\: x\]

\[\large \frac{d}{dx} (\cot\: x)= \csc^{2}\: x\]

\[\large \frac{d}{dx} (\sec\: x)= \sec\: x \tan\: x\]

\[\large \frac{d}{dx} (\csc\: x)= -\csc\: x \cot\: x\]

\[\large \frac{d}{dx} (e^{x}) = e^{x}\]

\[\large \frac{d}{dx} (a^{x}) = a^{x} \ln \: a\]

\[\large \frac{d}{dx} \ln \: x = \frac{1}{x}\]

\[\large \frac{d}{dx} (\arcsin x) = \frac{1}{\sqrt{1-x^{2}}}\]

\[\large \frac{d}{dx} (\arcsin x) = \frac{1}{1+x^{2}}\]

Integration Formulas

\[\large \int a\: dr = ax + C\]

\[\large \int \frac{1}{x}\: dr = \ln |x| + C\]

\[\large \int e^{x}\: dx = e^{x} + C\]

\[\large \int a^{x}\: dx = \frac{e^{x}}{\ln a} + C\]

\[\large \int \ln x\: dx = x \ln x-x+C\]

\[\large \int \sin\: x\: dx = -\cos \: x +C\]

\[\large \int \cos\: x\: dx = \sin \:x + C\]

\[\large \int \tan \: dr + \ln |\sec\: x| + C\: or\: -\ln |\cos \: x| + C\]

\[\large \int\cot\:x\:dr = \ln |\sin\:x|+C\]

\[\large \int\sec\:x\:dx = \ln |\sec\:x + \tan\:x|+C\]

\[\large \int\csc\:x\:dx = \ln |\csc \:x – \cot \:x|+C\]

\[\large \int\sec^{2}\:x\:dx = \tan\:x+C\]

\[\large \int\sec\:x\:\tan\:x\:dx = \sec\:x+C\]

\[\large \int\csc^{2}\:x\:dr = -\cot\:x+C\]

\[\large \int\tan^{2}\:x\:dr = \tan\:x-x+C\]

\[\large \int \frac{dr}{\sqrt{a^{2}-x^{2}}} = \arcsin \left ( \frac{x}{a} \right )+C\]

\[\large \int \frac{dr}{\sqrt{a^{2}+x^{2}}} = \frac{1}{a}\arcsin \left ( \frac{x}{a} \right )+C\]

\[\large Adding Fractions: \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\]

\[\large Subtracting Fractions: \frac{a}{b} – \frac{c}{d} = \frac{ad – bc}{bd}\]

\[\large Multiplying Fractions: \frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\]

\[\large Dividing Fractions: \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\]

\[\large e^{x} = b\]

Taking log on both the sides, we have

\[\large log _{e}\; e^{x} = log_{e} \; b\]

\[\large x = log_{e} \; b\]

Trivial Identities

\[\large \log _{b} (1) = 0; \; because \; b^{0}=1; \; b> 0\]
\[\large \log _{b} (b) = 1; \; because \; b^{1}=b\]

Basic Logarithm Formulas

\[\large \log _{b} (xy) = \log _{b}(x) + \log _{b}(y)\]
\[\large \log _{b}\left ( \frac{x}{y} \right ) = \log _{b}(x) – \log _{b}(y)\]
\[\large \log_{b}(x^{d})= d \log_{b}(x)\]
\[\large \log_{b}(\sqrt[y]{x})= \frac{\log_{b}(x)}{y}\]
\[\large c\log_{b}(x)+d\log_{b}(y)= \log_{b}(x^{c}y^{d})\]

Changing the Base

\[\large \log_{b}a = \frac{\log_{d}(a)}{\log_{d}(b)}\]

Addition & Subtraction

\[\large \log_{b} (a+c) = \log_{b}a + \log_{b}\left ( 1 + \frac{c}{a} \right )\]
\[\large \log_{b} (a-c) = \log_{b}a + \log_{b}\left ( 1 – \frac{c}{a} \right )\]

Exponents

\[\large x^{\frac{\log(\log(x))}{\log(x)}} \; = \; \log(x)\]

Polygon formula to find area:

\[\large Area\;of\;a\;regular\;polygon=\frac{1}{2}n\; sin\left(\frac{360^{\circ}}{n}\right)s^{2}\]

Polygon formula to find interior angles:

\[\large Interior\;angle\;of\;a\;regular\;polygon=\left(n-2\right)180^{\circ}\]

Polygon formula to find the triangles:

\[\large Interior\;of\;triangles\;in\;a\;polygon=\left(n-2\right)\]

Where,
n is the number of sides and S is the length from center to corner.

The general Polynomial Formula is written as,

$ax^{n} + bx^{n-1} + ….. + rx + s = 0$

If n is a natural number, aⁿ – bⁿ = (a – b)(a^n-1 + a^n-2 +…+ b^n-2a + b^n-1)

If n is even (n = 2k), aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b +…+ b^n-2a – b^n-1)

If n is odd (n = 2k + 1), aⁿ + bⁿ = (a + b)(a^n-1 – a^n-2b +…- b^n-2a + b^n-1)

(a + b + c + …)² = a² + b² + c² + … + 2(ab + ac + bc + ….

\[\large y=x\:tan\,\theta-\frac{gx^{2}}{2v^{2}\,cos^{2}\,\theta}\]

Where,
y is the horizontal component,
x is the vertical component,
g= gravity value,
v= initial velocity,
$\theta$ = angle of inclination of the initial velocity from horizontal axis,

Trajectory related equations are:

\[\large Time\;of\;Flight: t=\frac{2v_{0}\,sin\,\theta}{g}\]

\[\large Maximum\;height\;reached: H=\frac{_{0}^{2}\,sin^{2}\,\theta}{2g}\]

\[\large Horizontal\;Range: R=\frac{V_{0}^{2}\,sin\,2\,\theta}{g}\]

Where,
V_o is the initial Velocity,
sin $\theta$ is the y-axis vertical component,
cos $\theta$ is the x-axis horizontal component.