In this paper we consider \beta [0, s], Brownian motion of time length s > 0, in m-dimensional Euclidean space R^m and on the m-dimensional torus T^m. We compute the expectation of (i) the heat content at time t of R^m \ \beta [0, s] for fixed s and m = 2,3 in the limit t \downarrow 0, when \beta [0, s] is kept at temperature 1 for all t > 0 and R^m \ \beta [0, s] has initial temperature 0, and (ii) the inradius of T^m \ \beta [0, s] for m = 2,3,… in the limit s \rightarrow \infty. Key words and phrases. Laplacian, Brownian motion, Wiener sausage, heat content, inradius, spectrum
