A new similarity invariant metric $$v_G$$ v G is introduced. The visual angle metric $$v_G$$ v G is defined on a domain $$G\subsetneq {{\mathbb {R}}}^n$$ G ⊊ R n whose boundary is not a proper subset of a line. We find sharp bounds for $$v_G$$ v G in terms of the hyperbolic metric in the particular case when the domain is either the unit ball $${\mathbb {B}}^n$$ B n or the upper half space $${\mathbb {H}}^n$$ H n . We also obtain the sharp Lipschitz constant for a Möbius transformation $$f: G\rightarrow G'$$ f : G → G ′ between domains $$G$$ G and $$G'$$ G ′ in $${{\mathbb {R}}}^n$$ R n with respect to the metrics $$v_G$$ v G and $$v_{G'}$$ v G ′ . For instance, in the case $$G=G'={\mathbb {B}}^n$$ G = G ′ = B n the result is sharp.