Geometry/Perimeter and Arclength

Perimeter of Circle

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

${\displaystyle O=2\pi r}$

where ${\displaystyle r}$  the radius of the circle.

Perimeter of Polygons

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

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

Arclength of Circles

The arclength ${\displaystyle b}$  of a given circle with radius ${\displaystyle r}$  can be calculated using

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

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

Arclength of Curves

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

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

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