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We explore the correspondence between one-loop Feynman integrals and(hyperbolic) simplicial geometry to describe the "all-mass" case: integralswith generic external and internal masses. Specifically, we focus on$n$-particle integrals in exactly $n$ space-time dimensions, as these integralshave particularly nice geometric properties and respect a dual conformalsymmetry. In four dimensions, we leverage this geometric connection to give aconcise dilogarithmic expression for the all-mass box in terms of theMurakami-Yano formula. In five dimensions, we use a generalized Gauss-Bonnettheorem to derive a similar dilogarithmic expression for the all-mass pentagon.We also use the Schl\"afli formula to write down the symbol of these integralsfor all $n$. Finally, we discuss how the geometry behind these formulas dependson space-time signature, and we gather together many results related to theseintegrals from the mathematics and physics literature.

All-Mass $n$-gon Integrals in $n$ Dimensions