For a given volume, the object with the smallest surface area (and therefore with the smallest SA:V) is a ball, a consequence of the isoperimetric inequality in 3 dimensions. By contrast, objects with acute-angled spikes will have very large surface area for a given volume.

The sphere is perfectly symmetrical, and has the smallest ratio of surface area to volume of any three-dimensional shape. The internal and external forces at work within and around these structures force them to assume the shape that has the smallest possible surface area for the volume contained, which is a sphere.

(13) πr2 = 1. From this, it can be deduced that a circle has the smallest ratio between area and perimeter of any regular shape and therefore a circle has the minimal perimeter for a given area.

In our discussion, we have found out that, not only that a cube and an octagon are less efficient at holding liquids, but that in the limit, the infinitely-sided polyhedron, the sphere, is the most efficient shape for holding a liquid.

dome house

So then the most efficient shape is a rectangle of width twice its height, not a square (in the corner, you have two sides, so a square is better).

hexagon

A hexagon is the shape that best fills a plane with equal size units and leaves no wasted space. Hexagonal packing also minimizes the perimeter for a given area because of its 120-degree angles.

An irregular shape is a shape which has sides and angles of any length and size.

2D Shapes