Hermite-Hadamard type inequalities are given for the higher order convex functions and the general three, four and five point quadrature rules, as well as for their corrected counterparts. Specifically, Hermite-Hadamard type estimates are obtained for various classical quadrature rules, such as the Simpson, dual Simpson, Simpson 3/8, Maclaurin, the Gauss 2 and 3-point, the Lobatto 4 and 5-point rule, and their corrected variants.