<p>A cone singularity is a normal affine variety <i>X</i> with an effective one-dimensional torus action with a unique fixed point <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(x\in X\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> which lies in the closure of any orbit of the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k^*\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>k</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-action. In this article, we prove a boundedness theorem for cone singularities in terms of their dimension, singularities, and isotropies. Given <i>d</i> and <i>N</i> two positive integers and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation> a positive real number, we prove that the class of <i>d</i>-dimensional <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-log canonical cone singularities with isotropies bounded by <i>N</i> forms a bounded family.</p>