This article is concerned with proving that a planar Borel set which contains a circle of every positive radius has Hausdorff dimension exactly 2. Previous results in this direction by the author can only prove \(11/6\), so the present paper closes the problem. Moreover, the result is a continuous analog of the following result from combinatorial geometry by K. Clarkson, H. Edelsbrunner, L. J. Guibas, M. Sharir and E. Welzl [Discrete Comput. Geom. 5, No. 2, 99-160 (1990; Zbl 0704.51003)]: Given \(n\) circles, the number of internally tangent pairs is \(O(n^{3/2}\alpha(n))\) where \(\alpha\) is the very slow-growing functional inverse of Ackermann’s function. The continuous result follows from an extremal inequality with ideas of J. Bourgain [Geom. Funct. Anal. 1, No. 2, 147-187 (1990; Zbl 0756.42014)]. This inequality is proved by using the sampling techniques of Clarkson et al. [log. cit.].