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We present a general technique for approximating various descriptors of the extent of a set P of n points in R^d when the dimension d is an arbitrary fixed constant. For a given extent measure μ and a parameter ϵ > 0, it computes in time O(n + 1/ϵ^O(1)) a subset Q ⊆ P of size 1/ϵ^O(1), with the property that (1 − ϵ)μ(P) ≤ μ(Q) ≤ μ(P). The specific applications of our technique include ϵ-approximation algorithms for (i) computing diameter, width, and smallest bounding box, ball, and cylinder of P, (ii) maintaining all the previous measures for a set of moving points, and (iii) fitting spheres and cylinders through a point set P. Our algorithms are considerably simpler, and faster in many cases, than previously known algorithms.