Why Do All Triangles Form a Triangle?

We provide a geometric explanation, based on symmetry, for why the moduli space of all triangles up to similarity is itself a triangle. Symmetries occur because the lengths of the sides define triples in R-3 so are acted on by the symmetric group S-3, which is isomorphic to the symmetry group D-3 of an equilateral triangle. The moduli space for triangles is a fundamental domain for the action of D-3 on an equilateral triangle in R-3 determined by all triangles with unit perimeter and is chosen from a subdivision into six congruent triangles. Isosceles and equilateral triangles occupy special locations determined by their symmetries. The sides of a right triangle lie on one of three double cones in R-3, and those of unit perimeter lie on a segment of a hyperbola in the moduli space.