The Inradius of a Hyperbolic Truncated $$n$$n-Simplex

Hyperbolic truncated simplices are polyhedra bounded by at most $$2n+2$$ 2 n + 2 hyperplanes in hyperbolic $$n$$ n -space. They provide important models in the context of hyperbolic space forms of small volume. In this work, we derive an explicit formula for their inradius by algebraic means and by using the concept of reduced Gram matrix. As an illustration, we discuss implications for some polyhedra related to small volume arithmetic orientable hyperbolic orbifolds.