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

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.

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.

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