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A triangle in a Steiner triple system \(S\) is a triple of blocks from \(S\) which meet pairwise and whose intersection is empty. If \(S\) contains \(b\) blocks, and \(b = 3q + 8\), where \(0 \leq 8 \leq 2\), then a triangulation of \(S\) is a collection of \(q\) triangles \(\{T_1, T_2, \ldots, T_q\}\) in \(S\) such that no two distinct triangles share a common block. It is shown that, for \(v \equiv 1\) or \(3 \pmod{6}\), there exists a Steiner triple system which admits a triangulation. Moreover, if \(8 = 2\), there is a triangulated triple system in which the pair of blocks not occurring in a triangle are disjoint, and a triangulated triple system in which they intersect.