The approximate Carath\\'eodory theorem states that given a compact convex set\n$\\mathcal{C}\\subset\\mathbb{R}^n$ and $p\\in\\left[2,+\\infty\\right[$, each point\n$x^*\\in\\mathcal{C}$ can be approximated to $\\epsilon$-accuracy in the\n$\\ell_p$-norm as the convex combination of $\\mathcal{O}(pD_p^2/\\epsilon^2)$\nvertices of $\\mathcal{C}$, where $D_p$ is the diameter of $\\mathcal{C}$ in the\n$\\ell_p$-norm. A solution satisfying these properties can be built using\nprobabilistic arguments or by applying mirror descent to the dual problem. We\nrevisit the approximate Carath\\'eodory problem by solving the primal problem\nvia the Frank-Wolfe algorithm, providing a simplified analysis and leading to\nan efficient practical method. Furthermore, improved cardinality bounds are\nderived naturally using existing convergence rates of the Frank-Wolfe algorithm\nin different scenarios, when $x^*$ is in the interior of $\\mathcal{C}$, when\n$x^*$ is the convex combination of a subset of vertices with small diameter, or\nwhen $\\mathcal{C}$ is uniformly convex. We also propose cardinality bounds when\n$p\\in\\left[1,2\\right[\\cup\\{+\\infty\\}$ via a nonsmooth variant of the algorithm.\nLastly, we address the problem of finding sparse approximate projections onto\n$\\mathcal{C}$ in the $\\ell_p$-norm, $p\\in\\left[1,+\\infty\\right]$.\n