Geometry/Perimeter and Arclength

Perimeter of [[../Circles/Radii, Chords and Diameters|Circle]]

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

$O=2\pi r$

where $r$ the radius of the circle.

Perimeter of [[../Polygon|Polygons]]

The perimeter of a polygon $S$ with $n$ 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 $v$ is the angle given in radians.

Arclength of [[../Curves|Curves]]

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

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

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