Following　the　minimum　zone　criterion　set　forth　in　the　current　ANSI　and　ISO　standards,　evaluation　of　roundness　error　is　formulated　as　a　non-differentiable　unconstrained　optimization　problem　and　hard　to　handle.　The　maximum　inscribed　circle　and　minimum　circumscribed　circle　are　all　easily　solved　by　iterative　comparisons,　so　the　relationship　between　the　minimum　zone　circle　and　maximum　inscribed　circle,　minimum　circumscribed　circle　is　proposed　to　solve　efficiently　the　minimum　zone　problem.　Based　on　the　known　minimum　zone　circle,　the　maximum　inscribed　circle　and　minimum　circumscribed　circle　can　be　easily　determined.　The　relationship　is　implemented　and　validated　with　the　data　available　in　the　literature.　(C)　2008　Elsevier　Inc.　All　rights　reserved.