  We provide an explicit expression for the Pleijel constant for the planar
disk and some of its sectors, as well as for $N$-dimensional rectangles. In
particular, the Pleijel constant for the disk is equal to 0.4613019... Also, we
characterize the Pleijel constant for some rings and annular sectors in terms
of asymptotic behavior of zeros of certain cross-products of Bessel functions.
