Perimetern

Perimetern is a geometric quantity that measures the boundary length of a two-dimensional figure. In this article, Perimetern refers to the length of the enclosing boundary and is commonly denoted by P. It is defined for any shape that has a boundary, whether that boundary is composed of straight segments or a smooth curve.

For polygons, Perimetern is obtained by summing the lengths of all edges: P = s1 + s2 + ... + sn. For smooth shapes, it is computed as the integral along the boundary: P = ∮ ds. Classic special cases include the circle, where P = 2πr, and rectangular shapes, where P = 2(l + w). Perimetern has the same dimensional units as the underlying length measurement and depends on the chosen metric.

In practice, Perimetern can be approximated on grids or digital representations. On a pixel grid, measurements may vary with connectivity choices (such as 4-connected or 8-connected boundaries) and with the resolution of the representation. Algorithms for computing Perimetern range from edge-summing to more sophisticated contour tracing and chain-code methods.

Applications of Perimetern include land surveying, architecture, computer graphics, and geographic information systems, where boundary length informs design, cost, and spatial analysis. In theory, Perimetern is related to area through inequalities such as the isoperimetric inequality, which states that among figures with a given area, the circle has the smallest Perimetern. Related concepts include circumference, boundary, and surface measures in higher dimensions.

The notion of boundary length has a long history in geometry, evolving from practical measurement to formal mathematical treatment in calculus and geometry.