Geometry

Symbol	Definition
$A$	area
$a$	side length
$b$	base length
$C$	circumference
$D$	diameter
$h$	height
$n$	number of sides
$R$	radius
$V$	volume
$x, y, z$	distances along orthogonal coordinate system
$\beta$	interior vertex angle

\begin{align} A &= \frac{bh}{2}\\ \text{sum of interior angles} &= 180° \end{align}

\begin{align} A &= bh\\ \text{sum of interior angles} &= 360° \end{align}

$A = ah = ab \sin \beta$

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

\begin{align} A &= \frac{1}{4} n a^2 \cot \left( \frac{180°}{n} \right) \\ R &= \text{radius of circumscribed circle} = \frac{1}{2} a^2 \csc \left( \frac{180°}{n} \right) \\ r &= \text{radius of inscribed circle} = \frac{1}{2} a \cot \left( \frac{180°}{n} \right) \\ \beta &= 180° - \frac{360°}{n} \\ \text{sum of interior angles} &= n 180° - 360° \\ \end{align}

\begin{align} A &= \pi R^2 \\ C &= 2 \pi R = \pi D \\ \text{perimeter of n-sided polygon inscribed within a circle} &= 2 n R \sin \left(\frac{\pi}{n} \right) \\ \text{area of circumscribed polygon} &= n R^2 \tan \left( \frac{\pi}{n} \right) \\ \text{area of inscribed polygon} &= \frac{1}{2} n R^2 \sin \left( \frac{2\pi}{n} \right) \\ \text{equation for a circle with center at (h,k):} \\ R^2 &= \left(x-h \right)^2 + \left(y-k \right)^2 \\ \end{align}

\begin{align} f &= \text{semimajor axis} \\ g &= \text{semiminor axis} \\ e &= \text{eccentricity} = \frac{ \sqrt{f^2 -g^2} }{f} \\ A &= \pi e f \\ \text{equation for ellipse with center at (h,k):} \\ \frac{(x-h)^2 }{f^2} + \frac{(y-k)^2}{g^2} &= 1 \text{ if major axis along x-axis} \\ \text{or } \frac{(y-k)^2 }{f^2} + \frac{(x-h)^2}{g^2} &= 1 \text{ if major axis along y-axis} \\ \text{distance from center to either focus} &= \sqrt{f^2 -g^2} \\ \text{latus rectum} &= \frac{2g^2}{a} \\ \end{align}

\begin{align} p &= \text{distance from vertex to focus} \\ e &= \text{eccentricity} = 1 \\ \text{equation for parabola with vertex at (h,k), focus at (h+p,k):} \\ (y-k)^2 = 4j(x-h) \text{ if } j > 0 \\ \text{equation for parabola with vertex at (h,k), focus at (h,k+p):} \\ (x-h)^2 = 4j(y-k) \text{ if } j < 0 \\ \end{align}

\begin{align} p &= \text{distance between center and vertex} \\ q &= \text{distance between center and conjugate axis} \\ e &= \text{eccentricity} = \frac{ \sqrt{p^2 +q^2} }{p} \\ \text{equation for hyperbola centered at (h, k):} \\ \frac{(x-h)^2}{p^2} - \frac{(y-k)^2}{q^2} &= 1 \text{ if asymptotes slopes} = \pm \frac{q}{p} \\ \text{or } \frac{(y-k)^2}{p^2} - \frac{(x-h)^2}{q^2} &= 1 \text{ if asymptotes slopes} = \pm \frac{p}{q} \\ \end{align}

\begin{align} A &= 4 \pi R^2 \\ V &= \frac{4}{3} \pi R^3 \\ \text{equation for sphere centered at origin:} \\ x^2 + y^2 + z^2 &= R^2 \\ \end{align}

\begin{align} \rho &= \text{smaller radius} \\ A &= 4 \pi ^2 R \rho \\ V &= 2 \pi ^2 R \rho^2 \\ \end{align}