The boundary‐value problem for plane‐wave scattering by a spherical cap is formulated in terms of either the radial velocity in the aperture or the equivalent‐vortex strength of the shell. The resulting complementary integral equations are used to construct complementary variational expressions for the scattering cross section and also are solved approximately by Galerkin's method. Numerical results are given for the scattering cross section, say σ, with special reference to the Helmholtz resonator (small aperture) and the hemispherical shell. The function σ(k), wherekis the dimensionless wavenumber based on the radius of the sphere, rises initially ask4(Rayleigh‐scattering regime) to a first peak, σ(k0), and then executes a decaying oscillation about the asymptotic value, σ∞(twice the transverse area intercepted by the incident wave);σ(k0) = O(1/β)as the polar angle of the aperture, β, tends to zero, whilstσ(k0) = 2.13σ∞for the hemispherical shell.