Geometry/Perimeter and Arclength

Perimeter of Circle

The circles perimeter $\textstyle O$ can be calculated using the following formula

$O=2\pi r$

where $r$ the radius of the circle.

Perimeter of Polygons

The perimeter of a polygon $\textstyle S$ with $\textstyle n$ number of sides abbreviated $s_{1},\dots ,s_{n}$ can be calculated using the following formula

$S=\sum _{k=1}^{n}s_{k}$ .

Arclength of Circles

The arclength $b$ of a given circle with radius $r$ can be calculated using

$b={\frac {v}{2\pi }}2\pi r=vr$

where $\textstyle v$ is the angle given in radians.

Arclength of Curves

If a curve $\textstyle \gamma$ in $\mathbb {R} ^{3}$ has the parametric form $\mathbf {r} (t)={\big (}x(t),y(t),z(t){\big )}$ for $t\in [a,b]$ , then the arclength can be calculated using the following fomula

$S=\int \limits _{a}^{b}{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}}}\,dt=\int _{\gamma }{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}+\left({\frac {dz}{dt}}\right)^{2}}}\,dt$

Derivation of formula can be found using differential geometry on infinitely small triangles.