Perimeter
A perimeter is a closed
.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
Formulas
shape | formula | variables |
---|---|---|
circle | where is the radius of the circle and is the diameter. | |
semicircle | where is the radius of the semicircle. | |
triangle | where , and are the lengths of the sides of the triangle. | |
square/rhombus |
where is the side length. | |
rectangle | where is the length and is the width. | |
equilateral polygon | where is the number of sides and is the length of one of the sides. | |
regular polygon | where is the number of sides and is the distance between center of the polygon and one of the vertices of the polygon. | |
general polygon | where is the length of the -th (1st, 2nd, 3rd ... nth) side of an n-sided polygon. |
The perimeter is the distance around a shape. Perimeters for more general shapes can be calculated, as any path, with , where is the length of the path and 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 with
then its length can be computed as follows:
A generalized notion of perimeter, which includes hypersurfaces bounding volumes in -
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 and length equals
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
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
In terms of the radius r of the circle, this formula becomes,
To calculate a circle's perimeter, knowledge of its radius or diameter and the number π suffices. The problem is that π is not
Perception of perimeter
The perimeter and the
Proclus (5th century) reported that Greek peasants "fairly" parted fields relying on their perimeters.[1] 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. 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 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.
Etymology
The word comes from the Greek περίμετρος perimetros, from περί peri "around" and μέτρον metron "measure".
See also
References
- ISBN 0-486-24074-6.