Summary: In this paper, we extend Fibonacci’s method for computing Pythagorean triples of the form \((x,y,y+1)\) to compute Pythagorean triples of the form \((x,y,y+\ell)\) in three ways. Each of these three constructions exploits a certain relationship between \(x\) and \(\ell\) which must be true of any Pythagorean triple. Constructions 1 and 2 provide new proofs of already known formulae, whereas Construction 3 provides a novel method of computing Pythagorean triples.