$P$: perimeter distance
$A$: area
$r$: radius
$D$: diameter
$n$: number of sides

$h=b\sin \theta =a{\displaystyle \frac{\tan \theta \tan \theta {}_{\text{1}}}{\tan \theta +\tan \theta {}_{\text{1}}}}$

$A=\frac{1}{2}ah=\frac{1}{2}ab\sin \theta ={\displaystyle \frac{a^2}{2}}{\displaystyle \frac{\sin \theta \sin \theta {}_{\text{1}}}{\sin \theta +\sin \theta {}_{\text{1}}}}$

$s={\displaystyle \frac{a+b+c}{2}}$, then $A=s\sqrt{(s-a)(s-b)(s-c)}$
(Heron’s formula)

$P=a+b+\sqrt{a^2+b^2-2ab\cos \theta }=a(1+{\displaystyle \frac{\sin \alpha }{\sin (\theta +\theta {}_{\text{1}})}}+{\displaystyle \frac{\sin \theta }{\sin (\theta +\theta {}_{\text{1}})}})$

$h=\frac{\sqrt{3}}{2}a$ $P=3\text{a}$ $A=\frac{\sqrt{3}}{4}{a}^2={\displaystyle \frac{1}{\sqrt{3}{h}^2}}$

$h=\frac{a}{2}\tan \theta =b\sin \theta$ $\theta ={\tan }^{\text{-}1}\left({\displaystyle \frac{2h}{a}}\right)$

$a=2b\cos \theta$ $b=\sqrt{h^2+{\displaystyle \frac{a^2}{4}}}$

$A=\frac{1}{2}{b}^2\sin \theta$

$P=4\text{a}$ $A=a^2$

$P=2(a+b)$ $A=ab$

$A={\displaystyle \frac{a+b}{2}}h$

$P=na$ $\theta =180°(1-\frac{2}{n})$, peak angle $\varphi ={\displaystyle \frac{180°}{n}}$, central angle
$A={\displaystyle \frac{a{n}^2}{4\tan \left(\frac{\pi }{n}\right)}}$ $a=\sqrt{{\displaystyle \frac{4A\tan \left(\frac{\pi }{n}\right)}{n}}}$

if $n$ is even: $L{}_{\text{sharps}}={\displaystyle \frac{L{}_{\text{flats}}}{\cos \varphi }}$ $a=L{}_{\text{flats}}\tan \varphi$

Area

$P=\pi D=2\pi r$ $A=\pi r^2={\displaystyle \frac{\pi D^2}{4}}$

$A=\pi ab$

Circumference. No closed formula exists, so we have approximations and series expansions.

$L=r\theta$ $c=2r\sin \left(\frac{\theta }{2}\right)$ $A=\frac{1}{2}{r}^2\theta$

$L=r\theta$

$c=2r\sin \left(\frac{\theta }{2}\right)=2r{}_{\text{p}}\tan \left(\frac{1}{2}\theta \right)=2\sqrt{r^2-r{{}_{\text{p}}}^2}=2\sqrt{h(2r-h)}$

$r{}_{\text{p}}=r-h=r\cos \left(\frac{1}{2}\theta \right)=\frac{1}{2}c\cot \left(\frac{1}{2}\theta \right)=\frac{1}{2}\sqrt{4r^2-c^2}$

$\theta ={\displaystyle \frac{L}{r}}=2{\cos }^{\text{-}1}\left({\displaystyle \frac{r{}_{\text{p}}}{r}}\right)=2{\tan }^{\text{-}1}\left({\displaystyle \frac{c}{2r{}_{\text{p}}}}\right)=2{\sin }^{\text{-}1}\left({\displaystyle \frac{c}{2r}}\right)$

$A=\frac{1}{2}{r}^2(\theta -\sin \theta )=\frac{1}{2}(rL-r{}_{\text{p}}c)=r^2{\cos }^{\text{-}1}\left({\displaystyle \frac{r{}_{\text{p}}}{r}}\right)-r{}_{\text{p}}\sqrt{r^2-r{{}_{\text{p}}}^2}=r^2{\cos }^{\text{-}1}\left({\displaystyle \frac{r-h}{r}}\right)-(r-h)\cdot \sqrt{2rh-h^2}$

$A=\mathrm{F}(\theta {}_{\text{1}})-\mathrm{F}(\theta {}_{\text{0}})$, where:

$\mathrm{F}(\theta )={\displaystyle \frac{ab}{2}}(\theta -{\tan }^{\text{-}1}\left({\displaystyle \frac{(b-a)\sin (2\theta )}{b^2{\cos }^2\theta +a^2{\sin }^2\theta }}\right))$