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.

{\ 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 }

.

Web links

Wiktionary: scope - explanations of meanings, word origins, synonyms, translations

Eric W. Weisstein : Perimeter . In: MathWorld (English).