A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden. The perimeter of a wheel/circle (its circumference) describes how far it will roll in one revolution. Similarly, the amount of string wound around a spool is related to the spool's perimeter; if the length of the string was exact, it would equal the perimeter.
Formulas
The perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path, with ∫ 0 L d s {\textstyle \int _{0}^{L}\mathrm {d} s} , where L {\displaystyle L} is the length of the path and d s {\displaystyle ds} is an infinitesimal line element. Both of these must be replaced by algebraic forms in order to be practically calculated. If the perimeter is given as a closed piecewise smooth plane curve γ : [ a , b ] → R 2 {\displaystyle \gamma :[a,b]\to \mathbb {R} ^{2}} with
γ ( t ) = ( x ( t ) y ( t ) ) {\displaystyle \gamma (t)={\begin{pmatrix}x(t)\\y(t)\end{pmatrix}}}
then its length L {\displaystyle L} can be computed as follows:
L = ∫ a b x ′ ( t ) 2 + y ′ ( t ) 2 d t {\displaystyle L=\int _{a}^{b}{\sqrt {x'(t)^{2}+y'(t)^{2}}}\,\mathrm {d} t}
A generalized notion of perimeter, which includes hypersurfaces bounding volumes in n {\displaystyle n} -dimensional Euclidean spaces, is described by the theory of Caccioppoli sets.
Polygons
Polygons are fundamental to determining perimeters, not only because they are the simplest shapes but also because the perimeters of many shapes are calculated by approximating them with sequences of polygons tending to these shapes. The first mathematician known to have used this kind of reasoning is Archimedes, who approximated the perimeter of a circle by surrounding it with regular polygons. The perimeter of a polygon equals the sum of the lengths of its sides (edges). In particular, the perimeter of a rectangle of width w {\displaystyle w} and length ℓ {\displaystyle \ell } equals 2 w + 2 ℓ . {\displaystyle 2w+2\ell .}
An equilateral polygon is a polygon which has all sides of the same length (for example, a rhombus is a 4-sided equilateral polygon). To calculate the perimeter of an equilateral polygon, one must multiply the common length of the sides by the number of sides. A regular polygon may be characterized by the number of its sides and by its circumradius, that is to say, the constant distance between its centre and each of its vertices. The length of its sides can be calculated using trigonometry. If R is a regular polygon's radius and n is the number of its sides, then its perimeter is
2 n R sin ( 180 ∘ n ) . {\displaystyle 2nR\sin \left({\frac {180^{\circ }}{n}}\right).}
A splitter of a triangle is a cevian (a segment from a vertex to the opposite side) that divides the perimeter into two equal lengths, this common length being called the semiperimeter of the triangle. The three splitters of a triangle all intersect each other at the Nagel point of the triangle. A cleaver of a triangle is a segment from the midpoint of a side of a triangle to the opposite side such that the perimeter is divided into two equal lengths. The three cleavers of a triangle all intersect each other at the triangle's Spieker center.
Circumference of a circle
The perimeter of a circle, often called the circumference, is proportional to its diameter and its radius. That is to say, there exists a constant number pi, π (the Greek p for perimeter), such that if P is the circle's perimeter and D its diameter then,
P = π ⋅ D . {\displaystyle P=\pi \cdot {D}.\!}
In terms of the radius r of the circle, this formula becomes,
P = 2 π ⋅ r . {\displaystyle P=2\pi \cdot r.}
… excerpt ends here. Continue reading the full article.
