The word comes from the Greek περίμετρος perimetros from περί peri 'around' and μέτρον metron 'measure'.The figure shown above is a rectangle with 3 cm length and the measure of one the diagonals is √13 cm. In general, the polygon with n 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. The solution to the quadrilateral isoperimetric problem is the square, and the solution to the triangle problem is the equilateral triangle. 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 isoperimetric problem is sometimes simplified by restricting the type of figures to be used. This problem may seem simple, but its mathematical proof requires some sophisticated theorems. In particular, this can be used to explain why drops of fat on a broth surface are circular.

The solution is intuitive it is the circle. The isoperimetric problem is to determine a figure with the largest area, amongst those having a given perimeter. In the animated picture on the left, all the figures have the same convex hull the big, first hexagon.įurther information: Isoperimetric inequality The convex hull of a figure may be visualized as the shape formed by a rubber band stretched around it. In the case of very irregular shapes, confusion between the perimeter and the convex hull may arise. If one removes a piece from a figure, its area decreases but its perimeter may not. 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). Proclus (5th century) reported that Greek peasants 'fairly' parted fields relying on their perimeters. 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. Nevertheless, there is no relation between the area and the perimeter of an ordinary shape. The real area is 10,000 2 times the area of the shape on the map. For example, if a field is drawn on a 1/10,000 scale map, the actual field perimeter can be calculated multiplying the drawing perimeter by 10,000.

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. Confusing them is a common error, as well as believing that the greater one of them is, the greater the other must be. The perimeter and the area are two main measures of geometric figures.

Main articles: Area (geometry) and convex hull

The three cleavers of a triangle all intersect each other at the triangle's Spieker center. The three splitters of a triangle all intersect each other at the Nagel point of the triangle.Ī 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. Where r Ī 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.

Perimeter of a rectangle