The isoperimetric problem asks one to find a closed plane curve of a given perimeter which encloses the greatest area . The solution is a circle .
If the class of curves to be considered is limited to smooth curves, the isoperimetric
problem can be stated symbolically as follows: find an arc with parametric
equations
such that the positively oriented signed area
is a maximum .
Zenodorus proved that the area of the circle is larger than that of any polygon having the same perimeter , but the problem was not rigorously solved until
Steiner published several proofs in 1841 (Wells 1991).
For parallelohedra , the analogous problem minimizes surface area at fixed volume .
In three dimensions, the unique minimizer up to similarity
is the Archimedean truncated octahedron ,
as asserted by the truncated octahedral
conjecture and proved by Hales and Song (2026) and independently Cesaroni and
Novaga (2026). In four dimensions, the unique minimizer up to similarity
is the regular 24-cell (Novaga and Song 2026). These
results concern convex polytopes tiling
by translations and do not assert optimality among
arbitrary space-filling regions.
See also Circle ,
Dido's Problem ,
Double Bubble ,
Isoperimetric
Profile ,
Isoperimetric Quotient ,
Isoperimetric Theorem ,
Isovolume
Problem ,
Perimeter ,
Relative
Perimeter ,
Signed Area ,
Truncated
Octahedral Conjecture
Explore with Wolfram|Alpha
References Bogomolny, A. "Isoperimetric Theorem and Inequality." https://www.cut-the-knot.org/do_you_know/isoperimetric.shtml . Borwein,
J. and Bailey, D. Mathematics
by Experiment: Plausible Reasoning in the 21st Century. Cesaroni, A. and Novaga, M. "The Truncated
Octahedron Minimizes Surface Area Among Parallelohedra of Equal Volume." 2 Sep
2026. https://arxiv.org/abs/2609.02384 . Hales,
T. and Song, L. "The Truncated Octahedral Conjecture." 1 Sep 2026. https://arxiv.org/abs/2609.00997 . Isenberg,
C. "The Maximum Area Contained by a Given Circumference." Appendix V in
The
Science of Soap Films and Soap Bubbles. Littlewood, J. E. Littlewood's
Miscellany. Novaga,
M. and Song, L. "The Isoperimetric Problem for Four-Dimensional Parallelohedra."
1 Oct 2026. https://arxiv.org/abs/2610.01653 . Steinhaus,
H. Mathematical
Snapshots, 3rd ed. Wells,
D. The
Penguin Dictionary of Curious and Interesting Geometry. Referenced on Wolfram|Alpha Isoperimetric Problem
Cite this as:
Weisstein, Eric W. "Isoperimetric Problem."
From MathWorld https://mathworld.wolfram.com/IsoperimetricProblem.html
Subject classifications
,
for ![t in [t_1,t_2]](/images/equations/IsoperimetricProblem/Inline3.svg)
such that 
,
(where no further intersections occur) constrained by
