# Circumference (geometry)

Circumference of the circle:
U  =  d · π (here, d  = 1)
Perimeter of the rectangle :
U  = 2 a  + 2 b  = 2 ( a  +  b )

The circumference of a flat figure, which is bounded by a line , denotes the length of its boundary line.

The formula for the circumference is:

${\ displaystyle U = \ pi \, d = 2 \ pi \, r}$
• ${\ displaystyle U}$ stands for the scope
• ${\ displaystyle r}$for the radius of the circle ,
• ${\ displaystyle \ pi}$for the circle number with the value 3.14159265… and
• ${\ displaystyle d}$for the circle diameter .

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

Heart curve (drawing with )${\ displaystyle \ gamma \ colon [0.2 \ pi] \ rightarrow \ mathbb {R} ^ {2}}$
${\ displaystyle a = 1}$
${\ displaystyle x (t) = 2a \ cos (t) (1+ \ cos (t))}$
${\ displaystyle y (t) = 2a \ sin (t) (1+ \ cos (t))}$
${\ displaystyle U = \ int \ limits _ {0} ^ {2 \ pi} {\ sqrt {x '(t) ^ {2} + y' (t) ^ {2}}} \, \ mathrm {d } t = 16a}$

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

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

so the circumference can be calculated using the following integral : ${\ displaystyle U}$

${\ displaystyle U = \ int \ limits _ {a} ^ {b} {\ sqrt {x '(t) ^ {2} + y' (t) ^ {2}}} \, \ mathrm {d} t }$.