It is possible to construct a gure in 3 dimensions which is combinatorially equivalent to a regular icosahedron, and whose faces are all congruent but not equilateral. Such icosamonohedra can be convex or nonconvex, and can be deformed continuously. A scalene triangle can construct precisely zero, one, or two convex icosamonohedra, and each occurs. Demonstrated here are two explicit convex examples, the rst of which is the unique such object constructed from scalene right triangles, proving a conjecture of Bancho and Strauss. 1 Introduction: Monohedra A monohedron is essentially a 3-polyhedron whose faces are all congruent. The following conventions allow for more precision in certain cases where adjacent faces are coplanar. A polyhedron is bounded and 3-dimensional, but not necessarily convex (e.g. for Corollary 5.2). Each polyhedron P in this paper is endowed with a polytopal subdivision [Z, Example 5.2] of each of its facets, although these are often trivial. The terms facet and..