We show that the scalar curvature of a K-contact Ricci soliton is constant and satisfies sharp bounds. Next we show that the scalar curvature of a (2n+1)-dimensional K-contact Ricci almost soliton is equal to 2n(2n+1) plus the divergence of a global vector field. Finally, we show that, if a complete connected Sasakian or η-Einstein K-contact manifold of dimension >3 is a proper Ricci almost soliton, then it is isometric to a unit sphere.