# 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.