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 $$\R^3$$ 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.