We consider the Friedrichs inequality for functions defined on a disk of unit radius Ω and equal to zero on almost all boundary except for an arc λe of length ɛ, where ɛ is a small parameter. Using the method of matched asymptotic expansions, we construct a two-term asymptotics for the Friedrichs constant \(C(\Omega ,\partial \Omega \backslash \bar \gamma _\varepsilon )\) for such functions and present a strict proof of its validity. We show that \(C(\Omega ,\partial \Omega \backslash \bar \gamma _\varepsilon ) = C(\Omega ,\partial \Omega ) + \varepsilon ^2 C(\Omega ,\partial \Omega )(1 + O(\varepsilon ^{2/7} ))\). The calculation of the asymptotics for the Friedrichs constant is reduced to constructing an asymptotics for the minimum value of the operator −Δ in the disk with Neumann boundary condition on λe and Dirichlet boundary condition on the remaining part of the boundary.