Slant submanifolds in almost Hermitian manifolds were introduced in [\textit{B.-Y. Chen}, Bull. Aust. Math. Soc. 41, 135-147 (1990; Zbl 0677.53060)] as submanifolds with constant slant angle. In the present article, the author defines the notion of slant submanifolds in an almost contact metric manifold with structure \((\phi,\xi,\eta, g)\) in a similar way. The author obtains several basic properties of slant submanifolds in a \(K\)-contact manifold when the characteristic vector field \(\xi\) is tangent to the slant submanifolds. He also investigates slant submanifolds in an almost contact metric manifold \(M\times\mathbb{R}\) whose contact structure is obtained from the almost Hermitian structure on an almost Hermitian manifold \(M\).