The isoperimetric quotient of a closed curve is defined as the ratio of its enclosed area to the area of a circle with the same perimeter,
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
where 
and 
are the enclosed area and perimeter
of the curve, and 
and 
are the radii of circles
with the same area and perimeter,
respectively, so that 
and 
. The isoperimetric
inequality gives 
, with equality only in the case of the circle.
For a regular polygon with 
edges and inradius 
,
the area is given by
 |
(4)
|
 |
(5)
|
and the perimeter is given by
 |
(6)
|
Thus,
 |
(7)
|
which converges to 1 for 
.
The isoperimetric quotient of a polyhedron is the dimensionless quantity
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
where 
and 
are the volume and surface
area of the polyhedron, and 
and 
are the radii of spheres
with the same volume and surface
area, respectively, so that 
and 
.
