Perimeter

The perimeter of a plane figure bounded by a line denotes the length of its boundary line.

The formula for the circumference is:

$U = \pi \, d = 2 \pi \, r$

· stands for the perimeter,

· for the radius of the circle,

· $\pi$ for the circle number with the value 3.14159265... and

· for the circle diameter.

The perimeter of a polygon is the sum of its side lengths.

If the boundary line of the figure is described by a closed piecewise smooth parameter curve γ $\gamma \colon [a,b]\rightarrow \mathbb {R} ^{2}$ described by

$\gamma (t)={\begin{pmatrix}x(t)\\y(t)\end{pmatrix}}$ ,

then its perimeter calculated via the following integral:

Circumference of the circle: U = d-π (here d = 1)

Perimeter of the rectangle: U = 2-a + 2-b = 2-(a + b)

Heart curve γ $\gamma \colon [0,2\pi ]\rightarrow \mathbb {R} ^{2}$
(drawing with a=1 )
$x(t)=2a\cos(t)(1+\cos(t))$
$y(t)=2a\sin(t)(1+\cos(t))$
$U=\int \limits _{0}^{2\pi }{\sqrt {x'(t)^{2}+y'(t)^{2}}}\,\mathrm {d} t=16a$

Perimeter

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