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A perimeter is either a path that encompasses/surrounds/outlines a shape (in two dimensions) or its length ( one-dimensional). The perimeter of a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a pl ...
or an
ellipse In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. As such, it generalizes a circle, which is the sp ...
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 \int_0^L \mathrms, where L is the length of the path and 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 \gamma: ,brightarrow \mathbb^2 with : \gamma(t)=\beginx(t)\\y(t)\end then its length L can be computed as follows: : L = \int\limits_a^b\sqrt\,\mathrm dt A generalized notion of perimeter, which includes hypersurfaces bounding volumes in n- dimensional
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimensio ...
s, is described by the theory of
Caccioppoli set In mathematics, a Caccioppoli set is a Set (mathematics), set whose boundary (topology), boundary is measurable set, measurable and has (at least Local property, locally) a ''finite measure (mathematics), measure''. A synonym is set of (locally) fin ...
s.


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 Approximation#Mathematics, approximating them with limit of a sequence, 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 summation, sum of the lengths of its Edge (geometry), sides (edges). In particular, the perimeter of a rectangle of width w and length \ell equals 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 (geometry), centre and each of its Vertex (geometry), vertices. The length of its sides can be calculated using trigonometry. If is a regular polygon's radius and is the number of its sides, then its perimeter is :2nR \sin\left(\frac\right). A splitter (geometry), 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 concurrent lines, all intersect each other at the Nagel point of the triangle. A cleaver (geometry), 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 A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a pl ...
, often called the circumference, is proportional to its diameter and its radius. That is to say, there exists a constant number pi, (the ancient greek, Greek ''p'' for perimeter), such that if is the circle's perimeter and its diameter then, :P = \pi\cdot.\! In terms of the radius of the circle, this formula becomes, :P=2\pi\cdot r. To calculate a circle's perimeter, knowledge of its radius or diameter and the number suffices. The problem is that is not rational number, rational (it cannot be expressed as the quotient of two integers), nor is it algebraic number, algebraic (it is not a root of a polynomial equation with rational coefficients). So, obtaining an accurate approximation of is important in the calculation. The computation of the digits of is relevant to many fields, such as mathematical analysis, algorithmics and computer science.


Perception of perimeter

The perimeter and the area (geometry), area are two main measures of geometric figures. Confusing them is a common error, as well as believing that the greater one of them is, the greater the other must be. Indeed, a commonplace observation is that an enlargement (or a reduction) of a shape make its area grow (or decrease) as well as its perimeter. For example, if a field is drawn on a 1/ scale map, the actual field perimeter can be calculated multiplying the drawing perimeter by . The real area is times the area of the shape on the map. Nevertheless, there is no relation between the area and the perimeter of an ordinary shape. For example, the perimeter of a rectangle of width 0.001 and length 1000 is slightly above 2000, while the perimeter of a rectangle of width 0.5 and length 2 is 5. Both areas equal to 1. Proclus (5th century) reported that Greek peasants "fairly" parted fields relying on their perimeters. However, a field's production is proportional to its area, not to its perimeter, so many naive peasants may have gotten fields with long perimeters but small areas (thus, few crops). If one removes a piece from a figure, its area decreases but its perimeter may not. In the case of very irregular shapes, confusion between the perimeter and the convex hull may arise. The convex hull of a figure may be visualized as the shape formed by a rubber band stretched around it. In the animated picture on the left, all the figures have the same convex hull; the big, first hexagon.


Isoperimetry

The isoperimetric problem is to determine a figure with the largest area, amongst those having a given perimeter. The solution is intuitive; it is the
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre; equivalently it is the curve traced out by a point that moves in a pl ...
. In particular, this can be used to explain why drops of fat on a broth surface are circular. This problem may seem simple, but its mathematical proof requires some sophisticated theorems. The isoperimetric problem is sometimes simplified by restricting the type of figures to be used. In particular, to find the quadrilateral, or the triangle, or another particular figure, with the largest area amongst those with the same shape having a given perimeter. The solution to the quadrilateral isoperimetric problem is the square, and the solution to the triangle problem is the equilateral triangle. In general, the polygon with sides having the largest area and a given perimeter is the regular polygon, which is closer to being a circle than is any irregular polygon with the same number of sides.


Etymology

The word comes from the Ancient Greek, Greek περίμετρος ''perimetros'' from περί ''peri'' "around" and μέτρον ''metron'' "measure".


See also

* Arclength * Area * Coastline paradox * Girth (geometry) * Pythagorean theorem * Surface area * Volume * Wetted perimeter


References


External links

* * {{MathWorld , urlname=Semiperimeter , title=Semiperimeter Elementary geometry Length