Geometry/Perimeter and Arclength

Perimeter of Circle

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

$\textstyle O=2 \pi r$

where $\textstyle \pi = 3.1415926535 \dots$ and $\textstyle r$ the radius of the circle.

Perimeter of Polygons

The perimeter of a polygon $\textstyle S$ with $\textstyle n$ number of sides abbreviated $s_1,s_2,\dots,s_n$ can be caculated using the following formula

$S=\sum_{k=1}^n s_k$.

Arclength of Circles

The arclength $\textstyle b$ of a given circle with radius $\textstyle r$ 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 $\textstyle \mathbb{R}^3$ have a parameter form $\textstyle \mathbf{r}\big(t\big)=\big(x\big(t\big),y\big(t\big),z\big(t\big)\big)$ for $\textstyle t \in \big[a,b\big]$, then the arclength can be calculated using the following fomula

$S=\int_{a}^{b} \sqrt{ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2 } \, dt=\int_{\textstyle \gamma} \sqrt{ \left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2 } \, dt$.

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