Geometry/Perimeter and Arclength

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Perimeter of Circle[edit]

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[edit]

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[edit]

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[edit]

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.