The isoperimetric inequality for a plane figure of area 
and perimeter 
states that
 |
where 
is known as the isoperimetric quotient.
The equation becomes an equality
only for a circle.
At fixed area, the isoperimetric inequality gives a lower bound for the perimeter. Reverse isoperimetric inequalities instead give upper bounds under additional geometric restrictions. Convexity alone is insufficient, since rectangles of fixed area can have arbitrarily large perimeter.
For a nondegenerate triangle 
with side lengths 
, let 
be a convex body whose
area is a fraction 
of the area of 
. Ait El Manssour et al. (2026) proved
 |
where 
is the perimeter. Equality is attained by retaining
the longest side of 
and placing the third vertex at
fraction 
along the shortest side from their common endpoint, or by a congruent triangle
contained in 
.
