Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.

I think it will be a Triangle group, but I'm not sure.

(In spherical geometry, the answer is that there is no smallest tile, because you can make bipyramids with arbitrarily small faces. I don't think this will be the case with the hyperbolic plane though.)