## General Equation for Closed Plane Shapes

1. General Equation for Any Closed Plane Shape

[Eq.1]

A = area, p = perimeter, and k = perimeter efficiency. Multiply k by 100 to express efficiency as a per cent.
Perimeter efficiency, k, is a mathematical characteristic of shape. Once found for a given shape, it need never be calculated again. The value of k gives the length of perimeter of a shape relative to all other closed shapes in plane geometry. Shapes with high k values are very compact and have little perimeter with respect to the area enclosed. Shapes with low k values have a great deal of perimeter with respect to the area enclosed. Table PE - 1 gives values of k for a variety of shapes found in plane geometry.

Values of k can also be used to mathematically relate area and perimeter. For many shapes, area and perimeter are easily calculated. But for others, even simple ones such as ellipses and parabolas, calculation of the length of curve is cumbersome. The circumference of an ellipse, for example, is usually found by resorting to tables of elliptical integrals. Values of k can just as well be substituted for the elliptical integrals with the added feature that the value of k gives additional information about the relative length of perimeter of the shape.

 If k and A are known: [Eq. 1a] If k and p are known: [Eq. 1b]